Proof. Cyclicity of trace proves stability under composition. Stability under tensor product follows by taking a partial trace: for a map \(a:M\to N\) and a test map \(b:N\widehat\otimes P\to M\widehat\otimes P\), the trace of \(b(a\widehat\otimes\mathop{\mathrm{id}}_P)\) is the trace of \(a\) composed with the partial trace of \(b\). These formulas hold for bundles in codimension one and therefore over \(R\). Duality is compatible with the same ideal.

Write \(A=A_M=\mathop{\mathrm{End}}_R(M)\) and \(J=\mathop{\mathrm{rad}}A\). The algebra \(A\) is finite over the henselian local ring \(R\), so \(\mathfrak m A\subset J\), \(A/J\) is a finite-dimensional semisimple \(\mathbb C\)-algebra, and \(J^q\subset\mathfrak m A\) for some \(q\). We recall why idempotents lift in this setting. First lift an idempotent through the nilpotent radical of \(A/\mathfrak m A\). For an ensuing lift \(a\in A\) of an idempotent modulo \(\mathfrak m A\), the commutative finite algebra \(R[a]\) splits into the factors supported at \(0\) and \(1\) in its closed fiber by henselianity. The resulting idempotent lifts the given one. Induction in complementary corners lifts an entire orthogonal system. Thus primitive idempotents of \(A/J\) lift to actual direct summands of \(M\).

For \(x\in J\), all sufficiently large powers \(x^j\) belong to \(\mathfrak m A\), so their traces vanish modulo \(\mathfrak m\). The generic characteristic polynomial of \(x\) has coefficients in \(R\): this is true at every height-one localization, where \(M\) is free, and \(R\) is the intersection of those valuation rings. If \(\lambda_1,\ldots,\lambda_r\) are the roots of its reduction over \(\mathbb C\), Newton identities give \(\sum_i\lambda_i^j=0\) for every sufficiently large \(j\). Grouping the distinct nonzero roots, a Vandermonde matrix applied to consecutive such powers forces all their multiplicities to be zero. There are consequently no nonzero roots, and \(\mathop{\mathrm{tr}}(x)=0\) modulo \(\mathfrak m\). The same holds for \(yx\) for any \(y\in A\), because \(J\) is an ideal. Hence \(J\subset\mathcal R(M,M)\).

The residual trace now descends to \(A/J\). On each simple block \(\operatorname{Mat}_t(\mathbb C)\) it is a scalar multiple of the ordinary matrix trace, by cyclicity. Evaluate it on a primitive diagonal idempotent. Its lift projects onto a nonzero reflexive direct summand \(M_0\), so the scalar is \(\mathop{\mathrm{rk}}(M_0)>0\). The trace pairing is therefore nondegenerate on every block. This proves the reverse inclusion and the equality with \(J\).

Apply the argument to finite direct sums of objects. Their endomorphism algebras in the quotient are semisimple matrix algebras, and their primitive projectors lift. In particular every object is a finite sum of simple objects, distinct simple objects have no morphisms, and the endomorphisms of each simple object are \(\mathbb C\). This also proves that the quotient is abelian and that all its exact sequences split. The ideal contains \(\mathfrak m\mathop{\mathrm{Hom}}_R(M,N)\), so its Hom spaces are finite-dimensional. Finally the trace of the identity is \(\mathop{\mathrm{rk}}M\), as asserted. ◻

Proof. Let \(a:M\to N\) and lift a left inverse in the quotient to \(b:N\to M\). Then \(ba=\mathop{\mathrm{id}}_M+r\) with \(r\in\mathop{\mathrm{rad}}\mathop{\mathrm{End}}_R(M)\), so \(ba\) is invertible. Replacing \(b\) by \((ba)^{-1}b\) gives an actual left inverse. If the quotient map is an isomorphism, apply the same argument to both composites of lifts of it and its inverse; the lift is an isomorphism. Apply this to an isomorphism with \(R^r\) for the final statement. ◻

Proof. Lemma 9 gives an abelian rigid symmetric \(\mathbb C\)-linear category with \(\mathop{\mathrm{End}}({\bf1})=\mathbb C\). Tensor product is exact because every exact sequence splits. For an object of rank \(r\), its \((r+1)\)st exterior power is zero: it is zero on the common bundle locus and hence its reflexive extension is zero. Deligne’s criterion (Deligne 1990, Theorem 7.1) therefore applies. The category is tensor-generated by one object; Deligne’s Corollary 6.20 supplies a fiber functor over a finite extension of \(\mathbb C\), hence over \(\mathbb C\) itself. Tensor reconstruction gives the stated group. A tensor generator makes it an algebraic subgroup of its general linear group, and semisimplicity of the representation category makes it reductive in characteristic zero. The fiber dimension equals the categorical dimension, namely \(\mathop{\mathrm{rk}}V=n\). ◻

Proof. Pull a connection on \(M_U\) back to a finite smooth dominating germ \(\pi:(Y,y)\to(C,c)\). The complement of \(\pi^{-1}(U)\) has codimension at least two. On \(Y\) let \(F=(\pi^*M)^{**}\). In local smooth coordinates each coefficient of the connection extends across that complement by the Hartogs property of \(F\). Its target \(F\otimes\Omega_Y^1\) is reflexive because \(\Omega_Y^1\) is locally free. Thus the connection extends to \(F\) and still satisfies Leibniz’s rule.

Here is the elementary freeness argument, which does not require a flat connection. For a coordinate derivation \(\partial\), choose a presentation of \(F\) and lift the operator \(\nabla_{\partial}\) to the free module of generators. If \(A\) is the corresponding matrix on generators and \(P\) a matrix of relations, stability of the relations gives a matrix \(Q\) with \[\partial P+AP=PQ.\] The derivative of every minor of \(P\) is therefore a linear combination of minors of the same size. Each Fitting ideal of \(F\) is stable under every coordinate derivation. A nonzero ideal in a smooth characteristic-zero local ring with this property is the unit ideal: repeatedly differentiating a nonzero element of minimal order produces an element with nonzero constant term. If \(r\) is the generic rank of \(F\), its \(r\)th Fitting ideal is nonzero and hence is the unit ideal. Thus \(F\) is generated by \(r\) elements. The resulting surjection from a free module of rank \(r\) has a torsion kernel, which is zero because it is a submodule of a free module over a domain. This proves that \(F\) is free.

Take a normal Galois closure \(W\to C\) of \(Y\to C\), with group \(\Gamma\). Its normal domain is local over the henselian germ, so the whole group fixes the unique marked point. The reflexive pullback \(P\) of \(M\) to \(W\) is free: it agrees in codimension one with the pullback of the free module \(F\), and both reflexive extensions agree. It has its natural semilinear \(\Gamma\)-action. At a prime divisor, divisorial inertia acts trivially on its restriction to that divisor, because there \(P\) is the ordinary pullback of a bundle from \(C\). The same inertia therefore acts trivially on the fiber of \(P\) at the marked point. To see the last specialization explicitly, for an inertia element of order \(m\), restrict \(P\) to its fixed locus and apply the idempotents \[\frac1m\sum_{j=0}^{m-1}\zeta^{-j}g^j.\] Their ranks are locally constant. On the divisor all nonidentity eigenvalues have rank zero, hence they have rank zero at its specialization to the marked point as well.

Since \(C\) has no further quasi-étale cover, these inertia groups normally generate \(\Gamma\), by the construction in Proposition 8. The action on the marked fiber of \(P\) is consequently trivial. Lift a basis of that fiber to sections of \(P\) and average over \(\Gamma\). The averaged sections still lift the basis and hence, by Nakayama, form a basis of the free module \(P\). This basis is \(\Gamma\)-invariant. The invariant basis identifies \(P^\Gamma\) with \(R^r\), because the invariants of the normal cover ring are \(R\). Thus \(P^\Gamma\) is reflexive. The natural map \(M\to P^\Gamma\) is an isomorphism in codimension one by ordinary finite descent for the pullback of a bundle. Both modules are reflexive, so this isomorphism extends over the normal germ. Hence \(M=P^\Gamma\) is free, as claimed. ◻

Proof. The natural tensor action defining \(\Phi_M\) is a bundle map on \(U\); its reflexive extension is the displayed morphism in \(\mathcal C\). Choose local connections \(\nabla_\alpha\) on \(M_U\). Their differences represent \(a(M_U)\). On \(T_j(M_U)\) the corresponding differences are the infinitesimal tensor actions of \(\nabla_\alpha-\nabla_\beta\). Evaluating on the global section \(t_j\) gives the coboundary of the local sections \(\nabla_\alpha t_j\). This proves the asserted vanishing after restricting (1) to bundles on \(U\) and taking ordinary \(H^1\).

An injection in the semisimple quotient has a left inverse there. Lemma 10 lifts it to an actual left inverse between reflexive coefficient modules. Restriction to \(U\) gives a split map of bundles, which induces an injection on ordinary \(H^1\). Its kernel contains \(a(M_U)\), so that class is zero. The argument uses only that the individual \(t_j\) are actual sections. No algebraic identities among their images in the quotient need hold for the chosen lifts. ◻

Proof. For a reductive subgroup of a general linear group, the invariant mixed tensors have that subgroup as their simultaneous pointwise stabilizer. Indeed, projectors onto direct summands and all maps between tensor constructions are themselves such tensors. Preserving them is precisely the condition to give a tensor automorphism of the forgetful functor, and tensor reconstruction identifies those automorphisms with the group. Differentiating, its Lie algebra is the common kernel of the infinitesimal tensor-action maps. Since the endomorphism space is finite-dimensional, finitely many tensors suffice for this last kernel.

Suppose a representation \(N\) of \(G\) has finite image. Realize it as a direct summand of tensor constructions in \(V_{\mathrm{fib}}\), and lift its projector to obtain an actual module \(M\). Choose finitely many invariant mixed tensors of \(N\) whose infinitesimal stabilizer is the zero Lie algebra. By the definition of the quotient Hom spaces these tensors lift to actual maps from \(R\), hence to actual sections on the chart. The map \(\omega(\overline{\Phi_M})\) of Lemma 13 has as its kernel this infinitesimal stabilizer inside the full space \(\mathop{\mathrm{End}}_{\mathbb C}(N)\), and is therefore injective. Thus \(\overline{\Phi_M}\) is injective. Tensoring it with \(\overline V\) preserves injectivity, so \(\overline{\Psi_M}\) is injective and that lemma and Lemma 12 make \(M\) free. Therefore \(N\) is trivial. The regular representation of the finite component group \(G/G^\circ\) is one such representation. It is trivial only if that component group is trivial. Hence \(G\) is connected.

A character of \(G\) corresponds to a rank-one reflexive module on the chart. By analytic \(\mathbb Q\)-factoriality some positive reflexive tensor power of that module is free. Thus the character has finite order. A connected algebraic group has no nontrivial finite-image character. The character group of \(G\) is therefore zero. For a connected reductive group its quotient by the derived group is a torus, whose characters detect it; that torus must be trivial. Thus \(G\) is semisimple. ◻

Proof. Local coordinate connections on \(\Omega_U^1\) are torsion-free. Their differences show that \(a(V_U)\) belongs to the direct summand \[H^1\bigl(U,\mathop{\mathrm{Sym}}^2V_U\otimes V_U^\vee\bigr) \subset H^1\bigl(U,V_U\otimes\mathop{\mathrm{End}}(V_U)\bigr).\] On the other hand, choose local connections preserving the actual splitting \(V_U=\bigoplus_i(V_i)_U\). The off-diagonal endomorphism components of the same cohomology class are zero. These two conditions imply \[ a(V_U)=\sum_i a_i,\qquad a_i\in H^1\bigl(U,\mathop{\mathrm{Sym}}^2(V_i)_U\otimes(V_i)_U^\vee\bigr). \tag{2}\] Here is the index check. Block preservation allows only coefficients \(V_a\otimes V_b\otimes V_b^\vee\). Symmetry interchanges the first two covariant slots. If \(a\ne b\), its image is an off-diagonal coefficient \(V_b\otimes V_a\otimes V_b^\vee\), whose class is zero. Because these are actual direct sums of bundles, the original mixed component is also zero. This leaves precisely the terms in (2). In particular \(a_i\) is the class \(a((V_i)_U)\) under the indicated inclusion of its coefficients.

Choose lifts \(t_j\) of finitely many \(G\)-invariant tensors in natural mixed powers of \(V_{\mathrm{fib}}\) cutting out \(\mathfrak g\) infinitesimally. Use them to define \(Q\) and \(\Phi_V\) as in Lemma 13. For the \(i\)th block set \[S_i=\mathop{\mathrm{Sym}}^{[2]}V_i\widehat\otimes V_i^\vee.\] The symmetric inclusion in the first two slots, followed by the diagonal inclusion of the \(i\)th endomorphism block, gives \[\jmath_i:S_i\longrightarrow V_i\widehat\otimes\underline{\mathop{\mathrm{End}}}(V).\] The required one-block coefficient map is \[ \kappa_i=(\mathop{\mathrm{id}}_{V_i}\widehat\otimes\Phi_V)\circ\jmath_i: S_i\longrightarrow V_i\widehat\otimes Q. \tag{3}\] On \(U\) the source of this map is \(\mathop{\mathrm{Sym}}^2(V_i)_U\otimes(V_i)_U^\vee\). Projecting the first slot of \((\Psi_{V,U})_*a(V_U)=0\) onto \((V_i)_U\) and using (2) gives \((\kappa_{i,U})_*a_i=0\) in ordinary \(H^1\).

Write \(L=V_{\mathrm{fib}}\) and \(L_i=V_{i,\mathrm{fib}}\). The kernel of \(\omega(\overline{\Phi_V})\) is \(\mathfrak g\) inside \(\mathop{\mathrm{End}}_{\mathbb C}(L)\). Hence the fiber kernel of (3) is the explicit intersection \[ \ker\omega(\overline{\kappa_i})= \bigl(\mathop{\mathrm{Sym}}^2L_i\otimes L_i^\vee\bigr) \cap\bigl(L_i\otimes\mathfrak g\bigr) \quad\text{inside }L_i\otimes\mathop{\mathrm{End}}_{\mathbb C}(L). \tag{4}\] Partial evaluation of the first covariant slot by \(x\in T_i=L_i^\vee\) is supported on the \(i\)th endomorphism block. It belongs to \(\mathfrak g\) precisely when it belongs to \(\mathfrak h_i\). The symmetry of the first two slots is the defining symmetry in \(\mathfrak h_i^{(1)}\) on \(T_i\). Passing from the action on \(L_i\) to the dual action on \(T_i\) transposes and changes sign, which leaves this linear kernel unchanged. Thus (4) identifies with \(\mathfrak h_i^{(1)}\).

If the kernel is zero, faithfulness of \(\omega\) makes \(\overline{\kappa_i}\) injective, hence split injective in the semisimple quotient. Lemma 10 lifts a left inverse for \(\kappa_i\) between the reflexive coefficient modules. Restricting that left inverse to \(U\) makes the induced ordinary \(H^1\) map injective. Thus \(a_i=0\) in ordinary \(H^1\), and Lemma 12 makes \(V_i\) free. A free irreducible object is the tensor unit. This proves the assertion. ◻

Proof. Write a prolongation tensor as \(A(x,y)=A_x(y)\) with \(A_x\in\mathfrak h\). Fix \(x=u\otimes w_1\) with \(w_1\ne0\) and choose \(w_2\) independent of \(w_1\). For every \(v\in U\), symmetry gives \[(A_xv)\otimes w_2 =A(u\otimes w_1,v\otimes w_2) =(A_{v\otimes w_2}u)\otimes w_1.\] The two indicated tensor subspaces have zero intersection, so \(A_xv=0\) for every \(v\). Thus \(A_x=0\) for every pure tensor \(x\), and hence for every \(x\). No basis of \(W\) invariant under any larger Lie algebra has been assumed. ◻

Proof. Use homogeneous polynomial vector fields on the affine space \(T\). Let \(\mathfrak p_{-1}=T\) be the translations, \(\mathfrak p_0=\mathfrak h\), and for \(k\ge1\) set \[\mathfrak p_k= \{A\in\mathop{\mathrm{Sym}}^{k+1}T^*\otimes T: A(x_1,\ldots,x_k,{-})\in\mathfrak h \text{ for every }x_1,\ldots,x_k\in T\}.\] These spaces form a graded Lie algebra \(\mathfrak p=\bigoplus_{k\ge-1}\mathfrak p_k\) under the vector-field bracket. Indeed, differentiating a bracket gives the sum of brackets with derivatives. Induction on the total degree, starting with closure of \(\mathfrak h\) under brackets, proves \([\mathfrak p_i,\mathfrak p_j]\subset\mathfrak p_{i+j}\). The representation also gives \([\mathfrak h,\mathfrak p_k]\subset\mathfrak p_k\) directly.

Suppose first that \(\mathfrak p\) is finite-dimensional. Its solvable radical is graded, because it is preserved by the scaling automorphisms of this grading. From any nonzero homogeneous polynomial vector field in that radical, successive brackets with translations produce a nonzero constant field. Irreducibility under \(\mathfrak h\) then puts all of \(T\) in the radical. Since \(\mathfrak p_1\ne0\), the bracket \([T,\mathfrak p_1]\) is nonzero. It lies in the intersection of the radical with \(\mathfrak h\), a solvable ideal of a semisimple algebra, which is impossible. Thus the radical is zero.

The grading derivation \(D\), acting as \(k\mathop{\mathrm{id}}\) on \(\mathfrak p_k\), must now be inner. Recall a short proof of this fact. For any derivation \(D\) of a semisimple algebra \(\mathfrak a\), nondegeneracy of the Killing form gives \(z\in\mathfrak a\) such that \(E=D-\operatorname{ad}(z)\) satisfies \(\mathop{\mathrm{tr}}(E\operatorname{ad}(x))=0\) for all \(x\). Then \[B(Ex,y)=\mathop{\mathrm{tr}}([E,\operatorname{ad}(x)]\operatorname{ad}(y)) =\mathop{\mathrm{tr}}(E\operatorname{ad}([x,y]))=0,\] so \(E=0\). Apply this to \(\mathfrak a=\mathfrak p\) and write \(D=\operatorname{ad}(z)\). Comparing homogeneous degrees and using the zero center shows \(z\in\mathfrak p_0=\mathfrak h\). But \(D|_T=-\mathop{\mathrm{id}}_T\). This would give a nonzero scalar in the semisimple subalgebra \(\mathfrak h\subset\mathfrak{sl}(T)\), again impossible. Hence \(\mathfrak p\) is infinite-dimensional.

We make the symbol argument for this last alternative explicit. Let \(S=\mathop{\mathrm{Sym}}T\), and let \(\mathfrak h^\perp\subset T\otimes T^*\) be the annihilator of \(\mathfrak h\subset T^*\otimes T\). Form the finitely generated graded \(S\)-module \[\mathcal M=(S\otimes T^*)/ S\mathfrak h^\perp,\] where the relations have degree one. Perfect pairing with homogeneous polynomial vector fields identifies \(\mathcal M_d^*=\mathfrak p_{d-1}\) for \(d\ge0\): annihilating a relation multiplied by \(d-1\) linear factors is exactly the displayed partial-evaluation condition. Since \(\mathfrak p\) is infinite-dimensional, \(\mathcal M\) is not a finite-length module supported at the origin. Its support thus contains a nonzero point \(\xi\in T^*=\mathop{\mathrm{Spec}}(S)(\mathbb C)\).

The fiber at \(\xi\) is \[T^*/\{(\xi\otimes\mathop{\mathrm{id}})(r):r\in\mathfrak h^\perp\},\] and is nonzero. Choose a nonzero vector \(u\in T\) annihilating its relation space. Then \(\langle r,u\otimes\xi\rangle=0\) for every \(r\in\mathfrak h^\perp\). Hence \(u\otimes\xi\in\mathfrak h\). Since both \(u\) and \(\xi\) are nonzero, this is the required rank-one endomorphism. ◻

Proof. We give the weight argument, including the exceptional short-root possibilities. Consider in \(\mathbb P(\mathfrak h)\) the closure of the orbit of a rank-one operator. It consists of rank-one operators, since the rank-at-most-one locus is closed and the zero operator does not occur in projective space. A Borel subgroup has a fixed point on this closure. In this particular linear setting one can see the fixed point directly: triangularize the solvable Borel action, choose the smallest term of an invariant complete flag whose projectivization meets the closure, and intersect there. This projective intersection avoids the preceding hyperplane and is therefore also affine. It is finite; a connected group fixes each of its points. In the adjoint representation a Borel-fixed line is the highest-root line of one simple ideal. The highest root is long. We have therefore found a highest-long-root operator of rank one in one simple ideal.

An irreducible representation of a direct sum of simple Lie algebras is an external tensor product of their irreducible representations. The rank of this root operator on \(T\) is its rank on the indicated factor multiplied by the dimensions of all other factors. Rank one forces those other factors to have dimension one. They act trivially, and faithfulness eliminates them. Thus \(\mathfrak h\) is simple.

All long roots are Weyl-conjugate. For every long root \(\alpha\), the raising operator \(e_\alpha\) has rank one. Restriction to the root \(\mathfrak{sl}_2\) is a direct sum of irreducibles; a summand of highest weight \(m\) contributes rank \(m\) to \(e_\alpha\). There is consequently exactly one two-dimensional standard summand and otherwise only trivial summands. The Cartan subalgebra commuting with this root \(\mathfrak{sl}_2\) acts on the unique standard summand by a single character. Thus the root reflection \(s_\alpha\) interchanges exactly two full Cartan weights, each of multiplicity one and differing by \(\alpha\), and fixes every other full weight. This assertion does not yet exclude a zero weight of higher multiplicity.

Consider one irreducible component of the subsystem of long roots. Its simple reflections act by distinct transpositions on the set of weights. Adjacent simple roots have a product of reflections of order three, so their transpositions share exactly one vertex. Orthogonal simple roots give commuting transpositions. Two different simple roots cannot give the same transposition, since the difference of the transposed weights would be both roots up to sign. Connectedness of the diagram therefore gives a single connected moving set of vertices. The transpositions generate the full symmetric group on this set.

Map the augmentation representation on these vertices to the weight space by sending a vertex difference to the corresponding weight difference. The map is nonzero and equivariant. The augmentation representation of a symmetric group is irreducible, so the map is injective. Its image is exactly the span of the root component: consecutive vertex differences are its simple roots, and all differences lie in that span. Conjugating a simple-root transposition shows that every pair difference is a root. Conversely every root in this component arises this way, by conjugacy of its root reflections. The component is therefore of type \(A\), with the moving weights the vertices of its standard simplex, up to their common centroid.

Distinct long-root components have disjoint moving sets. Their transpositions commute, so an overlap could only be an identical pair; the difference of that pair would then belong to two orthogonal root spans, which is impossible. The long roots span the entire Cartan dual. Hence the centroid of a moving simplex, being fixed by its component and by every other component, is zero. A weight moved by no long-root component is zero.

The ordinary classification of reduced irreducible root systems (Čap 2016, Theorem 3.10) now leaves very few cases. For completeness, their long-root subsystems are \[\begin{array}{c|c} \text{root system}&\text{long-root subsystem}\\ \hline A_\ell&A_\ell\\ B_\ell\ (\ell\ge3)&D_\ell\\ C_\ell&A_1^{\ell}\\ D_\ell\ (\ell\ge4)&D_\ell\\ E_6,E_7,E_8&E_6,E_7,E_8\\ F_4&D_4\\ G_2&A_2 \end{array}\] Here \(B_2=C_2\) and the rank-one cases are \(A_1\). For the non-simply-laced rows, this table follows directly from the coordinate roots: the long roots of \(B_\ell\) and \(F_4\) are \(\pm\epsilon_i\pm\epsilon_j\), those of \(C_\ell\) are \(\pm2\epsilon_i\), and the long roots of \(G_2\) form a regular hexagon. In the simply-laced cases every root is long. Thus only \(A\), \(C\), \(B_3\) (where \(D_3=A_3\)), and \(G_2\) are possible. The Weyl groups of \(B_3\) and \(G_2\) contain \(-\mathop{\mathrm{id}}\). In those cases the nonzero weights would be respectively a tetrahedron or a triangle with centroid zero. Such a simplex is not preserved by \(-\mathop{\mathrm{id}}\); adjoining zero weights cannot make it centrally symmetric. These two cases are excluded.

For type \(A_\ell\), the remaining simplex weights are the standard weights or their negatives. The highest weight is therefore the first or last fundamental weight. Highest-weight uniqueness (Čap 2016, Theorem 4.3) identifies \(T\) with the standard module or its dual. For type \(C_\ell\), the long roots are \(\pm2\epsilon_j\); each centered two-vertex simplex is \(\{\epsilon_j,-\epsilon_j\}\). The highest weight is \(\epsilon_1\), so highest-weight uniqueness gives the standard symplectic module. In both cases this last uniqueness statement excludes any additional zero-weight multiplicity. The root-system and highest-weight facts used here are the usual finite-dimensional characteristic-zero theorems; the rank-one and simplex deductions have been proved above. ◻

Proof. By Proposition 14, \(\mathfrak g\) is semisimple. The one-block algebra \(\mathfrak h_i\) of Lemma 15 is an ideal of \(\mathfrak g\), hence semisimple. Split off its complementary ideal. On the irreducible dual of the \(i\)th block the representation is \(U\otimes W\), with \(\mathfrak h_i\) acting on \(U\). If \(\dim W>1\), Lemma 16 makes the first prolongation zero, contrary to Lemma 15 for a nontrivial block. Therefore \(\mathfrak h_i\) acts irreducibly there. It acts faithfully, since its elements act by zero on every other block and the total representation is faithful. Lemmas 17 and 18 identify it with the full special linear or symplectic algebra in its standard module.

The ideals so obtained for distinct blocks commute, since each is zero on all the other blocks. A remaining complementary semisimple ideal would commute with these irreducible standard actions. By Schur’s lemma it would act by scalars on each block; a semisimple Lie algebra has no scalar character. It also acts trivially on the trivial blocks, so faithfulness eliminates it. If two nontrivial blocks were isomorphic as \(\mathfrak g\)-modules, an element acting trivially on one would act trivially on the other; their one-block ideals would be zero. This again contradicts Lemma 15.

It remains to check the group, rather than just its Lie algebra. Take the simply connected covering group of the connected semisimple group \(G\). Its factors are the indicated \(\operatorname{SL}\) and \(\operatorname{Sp}\) groups. Each factor acts faithfully by its standard representation on its own block, and trivially on the others. Thus the kernel on the full representation is trivial, including any proposed central or diagonal central subgroup. The covering map is therefore an isomorphism and its image is the blockwise product in (5). Finally a trivial summand in the trace quotient is an actual free summand by Lemma 10, and conversely. This identifies \(s\) as claimed. ◻

Proof. For a standard \(\operatorname{SL}_m\) block with \(m>2\), select its degree-one representation. The exterior powers in degrees \(1,\ldots,m-1\) are irreducible with distinct highest weights, while degrees \(0\) and \(m\) are trivial. The selected representation therefore occurs once and in degree one only.

For a standard \(\operatorname{Sp}_{2m}\) block \(L_i\), select \[P_i^m=\ker\bigl(\Lambda^mL_i\longrightarrow\Lambda^{m-2}L_i\bigr),\] where the arrow is symplectic contraction and negative exterior powers are zero. The primitive decomposition \[\Lambda^bL_i\simeq \bigoplus_{a=0}^{\lfloor\min(b,2m-b)/2\rfloor} P_i^{\min(b,2m-b)-2a}\] has irreducible, mutually distinct primitive terms \(P_i^s\) of highest weight \(\varpi_s\) (Fulton and Harris 1991, Theorem 17.5). Thus \(P_i^m\) occurs once and only for \(b=m\). This includes \(\operatorname{SL}_2=\operatorname{Sp}_2\).

Take the external tensor product of the selected representations. It is simple for the product group and occurs in \(\Lambda^\bullet W\) once, in a single degree \(j_0\). In \(\Lambda^{j_0}V_{\mathrm{fib}}\) it still has multiplicity one: a positive exterior degree from the trivial summand would require its occurrence in a smaller degree of \(W\). The factorwise rule is preserved by permutations. In the symplectic case the kernel defining \(P_i^m\) is also unchanged when the symplectic form is rescaled, so the selection has the asserted intrinsic meaning. ◻

Proof. Suppose first that the characteristic is \(p>0\). Write \(d=\dim L\) and \(\tau=\operatorname{trdeg}_k\kappa(L)\), where \(k\) is the perfect ground field. All residue fields of the \(A_i\) are finite extensions of \(\kappa(L)\), so their \(p\)-degrees are \(p^\tau\). For \(q=p^a\), finite flat Frobenius on a regular local ring, as in Kunz’s theorem (Kunz 1969), gives \[ F_*^aA_i\simeq A_i^{\oplus q^{d+\tau}}. \tag{18}\] Here the module structure on the left is through \(x\mapsto x^q\). The formula follows either from a regular system of parameters and a residue-field \(q\)-basis, or by taking the degree of Frobenius on the fraction field. Completion does not change this formula. In particular, the exponent is the same for both algebras, including at a nonclosed point.

The two modules in (17) have the same generic rank. Clearing denominators in a generic isomorphism and its inverse gives maps \[u:A_1^{\oplus d_2}\longrightarrow A_2^{\oplus d_1},\qquad v:A_2^{\oplus d_1}\longrightarrow A_1^{\oplus d_2}, \qquad vu=r\mathop{\mathrm{id}},\quad uv=r\mathop{\mathrm{id}}\] for one nonzero \(r\in L\). We estimate the size of an invertible block of \(F_*^a(r)\) on \(F_*^aA_i\), viewed as a free \(A_i\)-module. If \(\mathfrak n_i\) is the maximal ideal, its cokernel needs \[\dim_{\kappa(A_i)} \bigl(F_*^a(A_i/rA_i)/\mathfrak n_iF_*^a(A_i/rA_i)\bigr) =q^\tau\operatorname{length}_{A_i} A_i/(r,\mathfrak n_i^{[q]})\] generators. Since \(\mathfrak n_i\) has \(d\) generators, \(\mathfrak n_i^{d(q-1)+1}\subset\mathfrak n_i^{[q]}\). The Hilbert–Samuel polynomial of \(A_i/rA_i\), of dimension at most \(d-1\), therefore bounds this number by \(O(q^{d+\tau-1})\). The case \(d=0\) is immediate, since \(r\) is a unit. Elementary invertible row and column operations over \(A_i\) now exhibit an identity block of size \(q^{d+\tau}-O(q^{d+\tau-1})\) in \(F_*^a(r)\).

Apply \(F_*^a\) to \(u,v\). The identity block just found, repeated \(d_2\) times, factors through \(F_*^a(A_2^{\oplus d_1})\). Consequently \[A_1^{\oplus d_2(q^{d+\tau}-O(q^{d+\tau-1}))} \quad\text{is an $L$-module summand of}\quad A_2^{\oplus d_1q^{d+\tau}}.\] Finite modules over the complete local ring \(L\) have Krull–Schmidt decompositions. For each indecomposable isomorphism class \(T\), let \(m_i(T)\) be its multiplicity in \(A_i\). The displayed summand relation, divided by \(q^{d+\tau}\) and followed by \(a\to\infty\), gives \(d_2m_1(T)\le d_1m_2(T)\). Interchanging the indices gives equality. This proves (17).

Here are details of the characteristic-zero passage. Henselize the algebraic base at the chosen point. The idempotents separating the finitely many points of a finite algebra descend over an étale neighborhood. Thus individual factors can be treated in a finite algebraic diagram; after shrinking that diagram they remain regular and dominant over a normal integral base. Let \(M=A_1^{\oplus d_2}\) and \(N=A_2^{\oplus d_1}\) denote the corresponding finite modules before completion. Choose finite generators \(\phi_1,\ldots,\phi_s\) of the ordinary module \(\mathop{\mathrm{Hom}}(M,N)\). Its completion is \(\mathop{\mathrm{Hom}}(\widehat M,\widehat N)\), because \(M\) is finitely presented and completion is flat.

An isomorphism \(\widehat M\simeq\widehat N\) exists precisely when some linear combination of the \(\phi_i\) is surjective modulo the maximal ideal. For the forward implication, reduce the coefficients of a completed isomorphism. For the converse, Nakayama gives surjectivity, and a surjective map of the same generic rank between torsion-free modules has zero kernel. Existence of the required linear combination is a finite matrix condition: at least one maximal minor of \(\sum z_i\overline\phi_i\) is a nonzero polynomial in the \(z_i\).

Spread the finite presentations, the marked point, and this Hom calculation over a finitely generated characteristic-zero domain. Generic freeness applied to the kernels and cokernels of the finite presentation matrices makes the Hom calculation and its reduction at the marked point commute with specialization. Retain normality, regularity, ranks, and the dimensions of the two residual modules. If all the maximal-minor polynomials vanished in characteristic zero, their finitely many coefficients would vanish in every remaining specialization. At any good positive-characteristic specialization, however, the first part supplies an isomorphism of the completed modules. This is a contradiction. A factor that splits further under specialization causes no difficulty: the first part applies to all its regular factors and their sum. Finally, comparisons with one fixed factor and addition prove the assertion for finite products. ◻

Proof. Set \(\mathcal K_0=\mathcal P\). Relative generation on projective space constructs \(n+2\) exact sequences \[ 0\longrightarrow\mathcal K_{r+1}\longrightarrow\mathcal F_r \longrightarrow\mathcal K_r\longrightarrow0, \qquad 0\le r\le n+1, \tag{23}\] where each \(\mathcal F_r\) is a finite sum of \(\mathsf H_Y(-a_{rj})\). To obtain such a surjection, choose ordinary coherent generators after twisting and extend their maps \(\mathsf H_Y\)-linearly. Since \(\mathcal K_r\) is locally projective over \(\mathsf H_Y\), the sequence locally splits over that algebra. Induction shows that every kernel \(\mathcal K_{r+1}\) is again locally finite projective over \(\mathsf H_Y\).

The sheaf \(\mathsf H_Y\) is flat as a right \(\mathsf H\)-module: its tensor functor is \(\mathcal O_Y\otimes_R-\), and \(Y\) is \(R\)-flat. Therefore all the kernels just constructed are \(\mathsf H\)-flat, and every sequence in (23) stays exact after tensoring with an arbitrary \(Q\). Choose \(m_0\ge a_{rj}\) for all the finitely many twists. For \(m\ge m_0\) each \(\mathcal F_r(m)\otimes_{\mathsf H}Q\) is a sum of \(\mathcal O_Y(m-a_{rj})\otimes_RQ\). The coordinate Čech calculation on projective space gives zero higher cohomology for these sheaves for every \(R\)-module \(Q\); their sections are the degree \(m-a_{rj}\) homogeneous polynomials with coefficients in \(Q\).

For \(j>0\), the long exact sequences consequently give \[ H^j(Y,\mathcal K_r(m)\otimes_{\mathsf H}Q) \simeq H^{j+n+2-r}(Y,\mathcal K_{n+2}(m)\otimes_{\mathsf H}Q). \tag{24}\] The \(n+1\) standard affine opens of \(Y\) have affine intersections, so quasi-coherent cohomology vanishes in degrees greater than \(n\). Thus the left side is zero for \(r=0,1,2\) and \(j>0\). In particular this proves (21). We have made no assertion about the projective dimension or the low-degree cohomology of the final kernel \(\mathcal K_{n+2}\).

The first two sequences, together with the vanishings for \(\mathcal K_1\) and \(\mathcal K_2\), give \[ H^0(Y,\mathcal P(m)\otimes_{\mathsf H}Q) =\operatorname{coker}\bigl( H^0(Y,\mathcal F_1(m))\otimes_{\mathsf H}Q \longrightarrow H^0(Y,\mathcal F_0(m))\otimes_{\mathsf H}Q\bigr). \tag{25}\] For \(Q=\mathsf H\) this is a finite presentation of \(G_m=H^0(Y,\mathcal P(m))\) by finite free right \(\mathsf H\)-modules. Right exactness of tensoring this same presentation proves (22), with its canonical map. Finally, tensoring any short exact sequence of coefficient modules with \(\mathcal P(m)\) remains exact by \(\mathsf H\)-flatness, and taking sections remains exact by (21). Equation (22) says that \(G_m\otimes_{\mathsf H}-\) is exact. Hence \(G_m\) is flat; being finitely presented by (25), it is projective. ◻

Proof. Use the constant polarizable Hodge module on \(Z\) in right \(\mathcal D_Z\) notation. Its underlying module is \(\omega_Z\), with lowest filtration index \(-2n\). The filtered direct-image construction and projective strictness give \[ F_{-2n}\mathcal H^0(a_+\omega_Z)=a_*\omega_Z. \tag{33}\] To track the index, use the transfer Spencer resolution. Its term in degree \(-i\) is filtered by pieces with source filtration lowered by \(i\) and by the nonnegative order of target differential operators. At total index \(-2n\), all \(i>0\) terms vanish; in degree zero only order zero remains. The lowest filtered complex is consequently \(\mathbf Ra_*\omega_Z\). Strictness identifies its degree-zero cohomology with the left side of (33). These conventions and strictness are the ones in (Saito 2017, secs. B.2–B.3 and Theorem 1.4), the latter being (Saito 1988, Theorem 5.3.1).

The strict-support decomposition is compatible with the filtration. The sheaf \(a_*\omega_Z\) is torsion-free on \(P\): multiplication by a nonzero base function is injective on the dominant integral smooth source. Thus its summands on proper supports are zero. After changing from right to left modules by \(\omega_P^{-1}\), the sheaf \(R\eta\) lies in the full-support part of \(\mathcal H^0(a_+\omega_Z)\), which is generically the \(n\)-th relative de Rham cohomology. This part is an intersection extension of semisimple algebraic local systems, hence a semisimple regular holonomic \(\mathcal D_P\)-module with no simple constituent on a proper support. Here we use strict support (Saito 2017, sec. 1.2) and algebraic semisimplicity (Deligne 1971, Proposition 4.2.5 and Theorem 4.2.6); restricting to an arbitrary small analytic open would not justify this semisimplicity assertion.

Inside that semisimple module let \(U\) and \(U_g\) be the \(\mathcal D_P\)-submodules generated by \(R\eta\) and \(gR\eta\). At the generic point of \(P\), the finite domain \(R\) becomes the field \(\operatorname{Frac}(R)\), so \(g\) is invertible and these restrictions agree. The quotient \(U/U_g\) is supported properly. Semisimplicity makes it a sum of constituents of the full-support module, so it is zero. Consequently \(\eta\) belongs locally to \(\mathcal D_P(gR\eta)\). Writing this membership using finitely many generators gives (31). Quasi-compactness gives a finite principal-open cover and a finite maximum \(o_g\) for the orders. On an open \(D(f)\), a local section of \(\mu_*\mathcal O_{\mathop{\mathrm{Spec}}R}\) has the form \(r/f^m\) with \(r\in R\). Absorb multiplication by \(f^{-m}\) into the operator acting on \(gr\eta\); its coefficients remain regular on \(D(f)\) and its order does not increase. Thus the \(t_j\) may be taken in \(R\). Algebraic differential operators on these opens of affine space have the expressions (32); their finitely many coefficient denominators are units on the corresponding opens. The restriction of the identity to the generic smooth locus involves only algebraic de Rham classes, connections, and finite-order operations, and thus is finite algebraic data that can be reduced after localization. ◻

Proof. Put \(b=b_2(X)\) and choose a prime \(\ell>2b\). After a finite extension, both the residual representation on a stable lattice of \(H^2_{\mathrm{et}}(X_{\overline K},\mathbb Q_\ell)\) modulo \(\ell\) and \(\mu_\ell\) are constant. At a degree-one good place of norm \(p\), geometric Frobenius has an integral trace \(a_p\) satisfying \[ |a_p|\le bp,\qquad a_p\equiv b\pmod\ell, \qquad p\equiv1\pmod\ell. \tag{36}\] We use integrality and the weight-two Weil bound, and the equality of the étale and crystalline characteristic polynomials (Katz and Messing 1974, Theorem 1), using its formulation and the crystalline applicability stated in (Milne 2025, VI, Theorem 3.9 and Section 4).

Discard the finitely many ramified places of the number field and the places where relative de Rham cohomology or its base change has torsion. At a degree-one remaining place the local base is \(\mathbb Z_p\). Crystalline–de Rham comparison for the smooth proper lift identifies the reduction of the crystalline trace with the trace of Frobenius on \(H^2_{\mathrm{dR}}(X_{\mathbb F_p})\) (Bhatt and Jong 2011, Corollaries 3.8 and 3.10). Modulo \(p\), Frobenius on the de Rham complex factors as its projection onto \(\mathcal O_X\) followed by the map taking functions to their \(p\)-th powers in degree zero. On cohomology write this factorization as \(\beta\alpha\) through \(H^2(\mathcal O_X)\). The reverse composite \(\alpha\beta\) is coherent Frobenius. The identity \(\mathop{\mathrm{tr}}(\beta\alpha)=\mathop{\mathrm{tr}}(\alpha\beta)\) shows that zero coherent Frobenius forces \(p\mid a_p\).

In that case \(a_p/p\) is an integer in \([-b,b]\) congruent to \(b\) modulo \(\ell\). Since \(\ell>2b\), it equals \(b\), so \(a_p=bp\). If coherent Frobenius vanished at all but finitely many degree-one good places, the semisimplification of \(H^2_{\mathrm{et}}(X_{\overline K},\mathbb Q_\ell)\) would therefore have the same character as \(\mathbb Q_\ell(-1)^{\oplus b}\). Indeed degree-one places have density one among primes of a number field; the others have norm at least the square of the underlying rational prime and density zero. Chebotarev for number fields and the characteristic-zero character criterion (Serre 2014, Theorem 3.2 and Section 5.1.1.3) give the assertion: finite-quotient Frobenius density and continuity identify the two trace functions in the projective limit.

This contradicts Hodge–Tate comparison at a place above \(\ell\): \(h^2(\mathcal O_X)=1\) contributes a weight-zero summand, whereas all Tate summands just found have weight one, in the convention where \(\mathbb Q_\ell(-1)\) has weight one. Comparison applies to the smooth proper variety over that local field (Scholze 2013, Theorem 1.6 and Corollary 1.8); de Rham representations are stable under subquotients and their filtered dimensions are additive in exact sequences (Brinon and Conrad 2009, Theorem 5.2.1(2) and Propositions 6.3.2–6.3.3), so passing to semisimplification has not lost this obstruction. Thus coherent Frobenius is nonzero at infinitely many, and hence arbitrarily large, good residue characteristics.

By (34), its \(n\)-fold product is nonzero on top coherent cohomology. Serre duality identifies the transpose of this map with Cartier trace on global top forms. The space of global top forms is the frame line and \(X\) is geometrically integral and proper, so (35) sends \(1\) to a nonzero constant. ◻

Proof. Fix \(b\) and one of the principal opens in Lemma 32 containing \(\mu(b)\). In this proof \(t_j,D_j\) denote its data, so the \(t_j\) are regular at \(b\) and the coefficients of the \(D_j\) are regular at \(\mu(b)\).

First work over \(K_0=\operatorname{Frac}(R)\), where the family \(Z_0=Z\times_RK_0\) is smooth and proper. The retained étale locus of \(\mu\) identifies \(du\) with a basis of \(\omega_{K_0/k}\) and gives the relative frame \(\eta=\sigma_Z/du\). Serre duality gives dual lines \[V_0=H^n(Z_0,\mathcal O_{Z_0})=K_0v,\qquad W_0=H^0(Z_0,\omega_{Z_0/K_0})=K_0\eta,\qquad \langle v,\eta\rangle=1.\] Put \[Z_0^{(p)}=Z_0\times_{\mathop{\mathrm{Spec}}K_0,F_{K_0}}\mathop{\mathrm{Spec}}K_0,\qquad V_0^{(p)}=V_0\otimes_{K_0,F_{K_0}}K_0,\qquad W_0^{(p)}=W_0\otimes_{K_0,F_{K_0}}K_0.\] Flat field base change identifies these twisted vector spaces with \(H^n(Z_0^{(p)},\mathcal O_{Z_0^{(p)}})\) and \(H^0(Z_0^{(p)},\omega_{Z_0^{(p)}/K_0})\), respectively. Write \(v^{(p)}=v\otimes1\) and \(\eta^{(p)}=\eta\otimes1\). They remain dual under the base-changed Serre pairing. Relative Frobenius \(F_{\mathrm{rel}}:Z_0\to Z_0^{(p)}\) and its finite dual trace have the following \(K_0\)-linear maps on these lines: \[ \begin{aligned} F_{\mathrm{rel}}^*:V_0^{(p)}&\longrightarrow V_0, &v^{(p)}&\longmapsto s\,v,\\ \operatorname{Tr}_{F_{\mathrm{rel}}}:W_0&\longrightarrow W_0^{(p)}, &\eta&\longmapsto s\,\eta^{(p)}. \end{aligned} \tag{38}\] Here the second map is induced by \(F_{\mathrm{rel}*}\omega_{Z_0/K_0}\to\omega_{Z_0^{(p)}/K_0}\). It is the Serre-dual transpose of the first, which proves that their scalars are the same \(s\in K_0\).

Taking \(p\)-th powers of a Čech cocycle for \(v\) gives a class \(\lambda\) in middle relative de Rham cohomology. Its projection to \(H^n(Z_0,\mathcal O_{Z_0})\) is \(F_{\mathrm{rel}}^*v^{(p)}\). Pairing \(\lambda\) with the top form \(\eta\), with the order that gives Serre evaluation, therefore gives \(\langle\lambda,\eta\rangle=s\). This distinguishes the coherent Frobenius map in (38) from its de Rham lift. The lift is horizontal: its \(p\)-power representatives are closed even for absolute differentials over \(k\), so the connecting operation defining the Gauss–Manin derivative annihilates them. The de Rham pairing is compatible with the connection. Pairing the reduced identity (31) with this horizontal class gives \[ s=\sum_jD_j(gt_js). \tag{39}\] The pairing, connection, and (31) on the smooth generic family are reductions of their finite characteristic-zero descriptions. The scalar identity (39) is obtained in this fiber from the newly constructed horizontal class.

Factor absolute Frobenius as \(Z_0\xrightarrow{F_{\mathrm{rel}}}Z_0^{(p)} \xrightarrow{w}Z_0\), where \(w\) is the base change of \(F_{K_0}\). This is the relative Frobenius factorization of (The Stacks Project Authors 2026, Tag 0CC6). The relative trace in (38) sends \(m\eta\) to \(sm\eta^{(p)}\). The trace for \(w\), by flat base change and the projection formula on top forms, applies the Cartier map \(\operatorname{Car}_{K_0}:F_*\omega_{K_0/k}\to\omega_{K_0/k}\) to its remaining base form. Transitivity of finite trace therefore gives the rational identity below; compatibility of trace with this flat base change is (The Stacks Project Authors 2026, Tag 0E5L). \[ \psi_h(m)\,du =\operatorname{Car}_{K_0}\bigl(F_*(s m\,du)\bigr) \qquad(m\in K_0). \tag{40}\] We suppress \(F_*\) in the input notation for \(\psi_h\). The scalar \(s\) stays inside the base Cartier operator. At this point it is only a rational element of \(K_0\).

Suppose (37) fails, so \(\psi_h(gR_b)\subset\mathfrak m_{R_b}\). Complete the regular base at \(\mu(b)\): \[A=k[[v_1,\ldots,v_n]],\qquad v_i=u_i-u_i(\mu(b)),\qquad K=\operatorname{Frac}(A),\qquad \mathfrak m=(v_1,\ldots,v_n).\] The finite completed algebra \(R\otimes_{k[u]}A\) is a product of the completed local rings at the points above \(\mu(b)\). They are normal complete local domains; keep the factor \(L\) at \(b\) and put \(K'=\operatorname{Frac}(L)\). After tensoring with \(K\), the whole product is finite étale of total dimension \(e_\mu\): the nonzero \(\Delta_\mu\) is invertible in \(K\), and its free étale rank was retained. Thus its factors are finite separable fields and \[ [K':K]\le e_\mu<p. \tag{41}\] Write \(T=\operatorname{Tr}_{K'/K}\).

We next transport (40) to these fields; this step does not require \(s\) to be regular in \(L\). Finite Frobenius and the cofinality of the ordinary and Frobenius adic filtrations give \(F_*R_b\otimes_{R_b}L\simeq F_*L\), by \((F_*r)\otimes x\mapsto F_*(rx^p)\). Hence \(\psi_h\) extends to \(L\) and then to a \(K'\)-linear map \(F_*K'\to K'\). The monomials \(v^\alpha\), \(0\le\alpha_i<p\), are a \(p\)-basis of \(K\): decompose power series according to their exponents modulo \(p\) and use perfection of \(k\). They remain a \(p\)-basis of the finite separable extension \(K'\). They are also a \(p\)-basis of the embedded \(K_0\), since \(K_0\) is finite separable over \(k(u_1,\ldots,u_n)\) and translation by constants of \(k\) changes \(u_i\) to \(v_i\). The Cartier maps computed in these bases agree on \(K_0\). The extended \(\psi_h\) and \(m\mapsto\operatorname{Car}_{K'}(F_*(sm\,du))/du\) agree on the \(p\)-basis monomials by (40); they are both \(K'\)-linear maps on \(F_*K'\), so they agree everywhere. The same uniqueness of separable extension shows that the extended derivations in (39) agree on \(K_0\), so that identity also holds in \(K'\) with these operators.

The failed containment extends to \(\psi_h(gL)\subset\mathfrak m_L\). For every base monomial \(a=v^\beta\) and each \(t_j\) it follows that \[ T(\psi_h(gat_j))\in\mathfrak m. \tag{42}\] Indeed \(T(L)\subset A\) by integrality and normality. An element of \(\mathfrak m_L\) has all its conjugates in the maximal ideal above \(\mathfrak m\): the integral closure in a finite normal closure is finite over the complete regular ring \(A\) and is local by henselianity. Their sum consequently lies in \(\mathfrak m\).

The common \(p\)-basis also proves that Cartier commutes with this rational trace. If \(x=\sum v^\alpha c_\alpha^p\) in \(K'\), then \(T(x)=\sum v^\alpha T(c_\alpha)^p\), since \(T(c^p)=T(c)^p\). Cartier for \(du=dv_1\wedge\cdots\wedge dv_n\) extracts the coefficient with \(\alpha=(p-1)\mathbf1\) in either field. Hence \[T\bigl(\operatorname{Car}_{K'}(F_*(x\,du))/du\bigr) =\operatorname{Car}_{K}(F_*(T(x)\,du))/du.\] Derivations likewise commute with \(T\), by uniqueness of their separable extensions. Combining this identity with (40) on \(K'\) and (42), and setting \(z_j=T(gt_js)\in K\), gives \[\operatorname{Car}_{K}(F_*(a z_j\,du))/du\in\mathfrak m \quad\text{for every base monomial }a.\] Write uniquely \[z_j=\sum_{0\le\alpha_i<p}v^\alpha c_{j,\alpha}^{p}, \qquad c_{j,\alpha}\in K.\] Multiplication by \(v^{(p-1)\mathbf1-\alpha}\) followed by Cartier extracts exactly \(c_{j,\alpha}\). For another exponent \(\beta\) the exponent vector lies componentwise between \(0\) and \(2p-2\) and has Cartier residue \((p-1)\mathbf1\) only when \(\beta=\alpha\). Consequently every \(c_{j,\alpha}\) belongs to \(\mathfrak m\). This proves the required regularity of the traced products, namely \[ z_j\in\mathfrak m^{[p]} =(v_1^p,\ldots,v_n^p)A. \tag{43}\] No regular extension of \(s\) or \(\eta\) across \(b\) has entered this argument.

The reduced expressions (32) have coefficients in \(A\) at \(\mu(b)\) and ordinary differential order at most \(o_g<p\). They preserve \(\mathfrak m^{[p]}\), since the ordinary partial derivatives annihilate every \(v_i^p\). Their coefficients lie on the regular base \(P\), so they pass through the \(K\)-linear trace \(T\). Taking \(T\) in (39) and using (43) gives \(T(s)\in\mathfrak m^{[p]}\). Cartier consequently sends \(T(s)du\) into \(\mathfrak m\,du\). But (40) with \(m=1\) gives \[\operatorname{Car}_{K}(F_*(T(s)du))/du =T(\psi_h(1))=[K':K]\delta\ne0\pmod{\mathfrak m},\] where (41) and \(\delta\in k^*\) give the final inequality. This contradiction proves the lemma. ◻

Proof. The preceding argument makes \(C_\nu(\mathcal V)\) homotopy equivalent to a zero-differential complex near every closed point. Its coefficient identifications satisfy (15) and (16). Lemma 28 therefore gives the canonical map (48) for every module on each such neighborhood. Since the base is affine, the cohomology modules correspond to the quasi-coherent direct-image sheaves; the map is multiplication by the pulled-back coefficient.

The local homotopy splitting extends to a neighborhood of each closed point: its maps involve finite modules, the local function \(t\), and the inverse of the unit \(u\). These neighborhoods are chosen before any coefficient module \(M\). They cover the finite-type chart, since any nonempty closed complement would have a closed point. Consequently the local statement holds as a sheaf statement for every module. The maps are compatible with the étale overlaps and deck actions because they are canonical multiplication maps. Proposition 25 therefore descends them to every supported thickening. The ordinary supported pullback on \(X\) is exactly the one constructed there. Taking the Euler characteristic of Leray and using the spread chartwise direct-image isomorphisms of Proposition 27 gives (49). Here \(N_i\otimes M\) has the ordinary bounded-support meaning fixed in Section 4; all supports are proper.

The only choices required before reduction were the finite diagrams, resolutions, one cleared homotopy per chart, and bounded differential identities. The proof of Lemma 31 works for all coefficient modules in each retained fiber, with the old bound for \(C_{\nu_0}\) retained by the split sequences. The exponent \(\nu\), the comparison at \(p\nu\), and the local functions \(t\) were constructed anew in that fiber. No spreading of infinitely many Frobenius identities or of the individual modules \(M\) occurs. A finite chart cover and Lemma 33 give the asserted simultaneous choice of reductions. ◻

Proof. Ambient vector fields are tangent to the orbit and restrict onto its tangent bundle. Their restriction kernel is the trace kernel: a vector field vanishing at an orbit point pairs to zero there with every reflexive one-form, whereas differentials of ordinary functions detect every nonzero tangent vector. The perfect barred pairing between \(V\) and \(V^\vee\) therefore identifies the two multiplicity bundles with \(\Omega^1_{\mathcal S}\) and \(T_{\mathcal S}\). Evaluation and contraction give mutually inverse inclusion and projection maps for the trivial-isotypic part. They are intrinsic, so agree on overlaps.

Exterior multiplication applied to \(V=\mathcal U\otimes\mathbf1\oplus W\) in this quotient category gives \(\Lambda^iV=\bigoplus_{a+b=i}\Lambda^a\mathcal U\otimes\Lambda^bW\). Writing this identity in local dual frames gives matrices of barred Hom maps and their inverses. Applying these matrices to a left module proves (51); an unbarred algebra splitting is not needed.

At each point, Corollary 20 supplies a simple constituent of \(\Lambda^\bullet W\) which occurs once and in exactly one degree \(j_0\). The fixed transverse descriptions transport this fiberwise selection, and its degree is locally constant, hence constant on the connected \(\mathcal S\). In \(\Lambda^{j_0}V\) its multiplicity is still one: a positive exterior degree of \(\mathcal U\) would require its occurrence in a smaller degree of \(W\). Its isotypic projector is \(u\), and Schur’s lemma gives (52) fiberwise.

The selection in the corollary is intrinsic to the independent standard factors and is unchanged by their permutations. The centers of the semisimple block algebras are finite étale over \(\mathcal S\); in a product presentation their primitive components are constant. The invariant selection therefore defines an algebraic idempotent on the whole component. Finally every automorphism acts on an endomorphism algebra by conjugation and fixes the identity of the selected simple object. This proves the assertion about the character of its identity line, including for ineffective stabilizers. ◻

Proof. First let \((R,\mathfrak m)\) be a complete transverse local ring and \(M\) the corresponding finite direct sum of transverse modules. Put \[E=\mathop{\mathrm{End}}_R(M),\qquad E_j=\mathop{\mathrm{End}}_R(M/\mathfrak m^jM).\] For every fixed \(j\), the images of \(E_l\to E_j\), \(l\ge j\), form a descending sequence of subspaces of a finite-dimensional vector space. Write \(D_j\) for its stable value. The maps \(D_{j+1}\to D_j\) are surjective. Indeed, for \(a\in D_j\), the images in \(E_{j+1}\) of its lifts to all sufficiently large \(E_l\) are nonempty descending affine subspaces; they stabilize and give a lift in \(D_{j+1}\). Recursively choose compatible lifts. Completeness of \(M\) identifies \(\varprojlim_jE_j\) with \(E\), so \(D_1\) is exactly the image of \(E\to E_1\). This is an all-level argument; stabilization at level one alone would not justify the lifting assertion.

Choose \(k\) for which \(\mathop{\mathrm{im}}(E_k\to E_1)=D_1\). A map in the kernel lowers \(M/\mathfrak m^kM\) into its \(\mathfrak m\)-multiple, so the \(k\)-th power of this kernel is zero. Also \(\ker(E\to D_1)\subset\mathop{\mathrm{rad}}(E)\): if \(a\) lowers \(M\) into \(\mathfrak mM\), then \(1-ba\) is invertible for every \(b\in E\), by the convergent \(\mathfrak m\)-adic geometric series. The trace-radical calculation consequently identifies \[D_1/\mathop{\mathrm{rad}}(D_1)=E/\mathop{\mathrm{rad}}(E).\] The composite \(E_k\to D_1\to D_1/\mathop{\mathrm{rad}}(D_1)\) is onto and has nilpotent kernel. Indeed its kernel modulo the lowering kernel is the radical of a finite-dimensional algebra and is nilpotent. Notice that no claim has been made that every element of \(E_k\) lifts to \(E\); only its residual action must lift.

On a product presentation, finite quotients and their Hom modules are these fixed transverse objects extended by the smooth parameter ring. Flat extension commutes with their finite presentations and kernels. Thus the chosen exponent works throughout that presentation. Take the maximum of the exponents for a finite cover of \(\mathcal S\). Assertions about the quotient, its kernel, and a fixed nilpotence exponent descend from faithfully flat completions.

To check canonicity, let \(\mathcal A_{\mathrm{res}}\) be the common image of the full and ambient endomorphism algebras in \(\mathop{\mathrm{End}}(T/IT)\). The original trace quotient \(\mathcal E\to\overline{\mathcal E}\) kills the kernel of \(\mathcal E\to\mathcal A_{\mathrm{res}}\), by the transverse calculation, and hence factors uniquely through \(\mathcal A_{\mathrm{res}}\). Composing with \(\mathop{\mathrm{End}}(T)\to\mathcal A_{\mathrm{res}}\) defines (53), including on maps without ambient lifts. Its kernel is the transverse nilpotent ideal extended along the smooth parameters; it need not be the ordinary Jacobson radical of the full algebra over the parameter ring. The quotient is locally a product of matrix algebras over a reduced smooth base and has no nonzero nilpotent two-sided ideal. Thus this kernel is intrinsically the largest nilpotent two-sided ideal. Transition isomorphisms preserve it and the factorization through the residual-action image, so the map glues on the stack independently of the product coordinates.

After setting the smooth parameters to zero, let \(C\) be the image of ambient endomorphisms in the full finite algebra \(\mathop{\mathrm{End}}(T_o)\). Product structure identifies \(C\) with the transverse ambient image. It maps onto the barred algebra because the original \(E\) does. Its kernel is its intersection with the nilpotent kernel of (53). Changing the smooth parameter lifts while keeping the transverse coordinates is a formal automorphism preserving \(I\); differential tensors transform functorially under it. This conjugates the calculation and does not change any exponent or create an additional lifting obstruction. ◻

Proof. We first prove, in complex cohomology with supports, that for one positive rational number \(\rho\) \[ [F_0]_c=\rho\,f^*[S_0]. \tag{55}\] The supported class \([S_0]\) exists because \(B\) is an oriented rational homology manifold by Proposition 7. Equidimensionality implies that every component of \(f^{-1}(S_0)\) of dimension \(2n-c=n+d\) dominates \(S_0\): one whose image has smaller dimension has dimension at most \(n+d-1\).

Choose one maximal chart and one lifted component \(S'\) of its stratum. For each top-dimensional component of \(f^{-1}(S_0)\), restrict it to the chosen base neighborhood and choose a component of its inverse image under the source torsor of Proposition 25. Its base image is a stratum lift. The deck group is transitive on such lifts over a general coarse point, so translating the chosen source component makes its base image \(S'\). This translation leaves its image on \(X\) unchanged. Thus one chosen lift \(S'\) sees every top component, without requiring transitivity on components of the fibers of \(f\). Use the finite projection of Section 5 on this source chart, \[\pi:Z\longrightarrow Y=\mathbb P^n_V.\] Each chosen lifted component has dimension \(n+d\) and is finite over \(S'\times\mathbb P^n\). It therefore dominates that entire irreducible support. We can now choose one common general closed point \(y\in S'\times\mathbb P^n\) at which all its reduced preimage support branches are smooth and étale over the support. In analytic germs each of these branches maps isomorphically to the support germ.

Let \(R=\widehat{\mathcal O}_{Y,y}\) and let \(A_\alpha=\widehat{\mathcal O}_{Z,z_\alpha}\) run through the regular local factors above this same point. Write \(r_\alpha\) for their generic ranks over \(R\), and \(J\subset R\) for the support ideal. The quotient \(R/J\) is regular. If \(m_\alpha\) is the scheme multiplicity of the ordinary reference pullback on this branch, the preceding degree-one assertion gives \[m_\alpha=\mathop{\mathrm{rk}}_{R/J}(A_\alpha/JA_\alpha).\] Proposition 29, in characteristic zero and over this single target ring, gives \[A_\alpha^{\oplus r_\beta}\simeq A_\beta^{\oplus r_\alpha}.\] Quotienting by \(J\) and taking ranks yields \[ r_\beta m_\alpha=r_\alpha m_\beta. \tag{56}\] Thus \(m_\alpha/r_\alpha\) is one common scalar.

We compare the cohomological multiplicities by the finite local-degree rule. For a finite local branch, the projection formula with supports, capped with the complex orientation, multiplies the target support class by the generic degree of the branch after pushforward. Dividing by the degree of its reduced support map gives the coefficient of the pulled class. Applied first to the chosen base-chart branch, this gives \(\lambda[S']\) as the pullback of \([S_0]\), with one positive factor \(\lambda\) on \(S'\). The intervening étale base refinement does not change that factor. Applied next to a branch of \(\pi\), whose reduced support map has degree one at the common point, the same rule gives coefficient \(r_\alpha\). Hence the coefficient of \(f^*[S_0]\) on that branch is \(\lambda r_\alpha\). Equation (56) makes the scheme-to-cohomology ratio \(m_\alpha/(\lambda r_\alpha)\) independent of the branch. The source charts are étale, so both kinds of multiplicities descend unchanged. Top support classes are determined at these generic points; this proves (55) globally.

By Propositions 7 and 2, ordinary \(H^{2c}(B,\mathbb Q)\) is one-dimensional, and \([S_0]\) is a multiple of \(c_1(H)^c\). Its pullback is therefore a multiple of \(e^c\). For every integer \(t\), the IHS Riemann–Roch polynomial gives \[\chi(X,f^*H^{\otimes t})=\operatorname{RR}_X(q(te)) =\operatorname{RR}_X(0),\] since \(q(e)=0\); see (Huybrechts 1999, sec. 1.11). The coefficient of \(t^c\) in ordinary Hirzebruch–Riemann–Roch is \(\frac1{c!}\int_Xe^c\mathop{\mathrm{td}}_{2n-c}(X)\). It vanishes for \(c>0\). Together with (55) this proves (54).

For completeness, specialization of this number does not use rational smoothness of a characteristic-\(p\) base. The ordinary coherent sheaf \(F_0\) on the smooth projective \(X\) has a finite locally free resolution. Include its finitely many terms, maps, kernels, the descended quotient ideal defining \(F_0\), and the identification with the ordinary pullback in a finite coherent model. Generic flatness and coherent base change on a nonempty principal open preserve the resolution of that quotient on fibers and support codimension at least \(c\). Thus its restriction represents the actual reference pullback on every retained fiber. Its Chern character below codimension \(c\) vanishes and its component in codimension \(c\) is its actual support cycle. The degree \[\deg\bigl(\mathop{\mathrm{ch}}_c(F_{0,s})\mathop{\mathrm{td}}_{2n-c}(T_{X_s})\bigr)\] is constant: it is the degree of a fixed relative Chern polynomial in the vector bundles of that resolution and the relative tangent bundle on a smooth proper family. This gives the last assertion. The argument neither keeps track of individual cohomology classes on the reduced base nor requires a constant number of support components. ◻

Proof. There are finitely many connected components. The functional and its representation dependence are componentwise; if the estimate holds on them with constants \(C_j\), summing gives the stated bound with \(C_{\mathcal D}=\sum_j C_j\) under our rank convention. We may therefore assume \(\mathcal D\) connected.

Let \(A\) be the pullback of an ample line bundle from the coarse space and set \[\Theta_m(G)=\sum_{a=0}^d(-1)^a \chi\bigl(\mathcal D,\Lambda^a\Omega^1_{\mathcal D} \otimes A^{-ma}\otimes G\bigr).\] The coarse pushforward is exact by tameness (Abramovich et al. 2008, Definition 3.1 and Theorem 3.2). The usual Euler polynomial on its projective coarse space shows that \(\Theta_m(G)\) is a polynomial in \(m\) of degree at most \(d\).

For \(m\) sufficiently large, global sections of \(T_{\mathcal D}\otimes A^m\) evaluate onto the invariant tangent vectors at every residual gerbe. To see the uniform assertion, push this bundle to the coarse space, use an ample twist to generate that coherent sheaf, and use exactness of invariants to restrict to a residual gerbe. At a point \(x\) with stabilizer \(J_x\), the possible values are exactly \((T_x\mathcal D)^{J_x}\). The locus where this dimension is at most \(r\) has dimension at most \(r\). On a finite quotient chart, its stabilizer strata lie in fixed loci of subgroups; these are smooth for tame actions and their tangent spaces are the invariant tangent spaces. There are finitely many relevant subgroup strata on a finite atlas.

For a stratum of dimension \(s\) with invariant tangent dimension \(r\), the incidence of sections vanishing at a point has codimension \(r\) in the space of sections over that point. Since \(s\le r\), a general section has a zero scheme \(Z_m\) of dimension at most zero. At every zero its \(d\) components generate an ideal of height \(d\) in a regular local ring. They are a regular sequence, and their Koszul complex gives \[\Theta_m(G)=\chi(Z_m,G|_{Z_m}).\] This remains valid when \(Z_m\) is empty.

The right side is a finite sum of invariant dimensions on zero-dimensional quotient charts. Filtering their Artin rings by powers of the maximal ideal expresses each summand as an integral linear functional of the fiber representation \(G_x\). Its absolute value is bounded by a fixed constant times \(\dim G_x\). In particular, it depends only on these fiber representations, not on the transition functions of \(G\). Choose \(d+1\) distinct sufficiently large integers \(m\). Polynomial interpolation expresses \(\Theta_0\) as a fixed rational linear combination of the corresponding \(\Theta_m\). This proves both assertions for bundles, with one constant independent of \(G\). Additivity extends the representation dependence to virtual classes, and the triangle inequality gives the displayed presentation bound. The construction uses the full finite groups at zeroes and applies without an effectiveness assumption. ◻

Proof. If \(I^bM=0\), choose generators \(x_1,\ldots,x_v\) for the maximal ideal of the transverse ring. Its Frobenius pullback is killed by \((\mathfrak m^b)^{[q]}\), which contains \((x_1^{bq},\ldots,x_v^{bq})\) and hence \(\mathfrak m^{vbq}\). A fixed number of generators for \(M\) also generates its ordinary Frobenius pullback. Hilbert–Samuel growth in transverse dimension \(c\) therefore bounds its length by \(O(q^c)\). Tensoring with each of the fixed \(N_i\) changes this by only a fixed generator bound. The total rank of the graded pieces of \(L(F^{a*}M)\), and hence the sum of the ranks in its expression (57), is \(O(q^c)\) on a finite atlas.

The annihilating thickness just obtained grows with \(q\). The all-modules, all-orders conclusion of Proposition 35 therefore matters here: it applies to each new coefficient \(F^{a*}M\) in this same reduction. The Frobenius square commutes with ordinary pullback on the ambient charts and with their étale descent, so \(f^*F^{a*}M=F_X^{a*}f^*M\). Applying (58) and Lemma 40 proves the bound with the original test and prime fixed. ◻

Proof. If \(M\) has support of dimension less than \(d\), its ordinary pullback has codimension greater than \(c\) by equidimensionality, and its codimension-\(c\) character is zero. Otherwise repeat the common-chart and lifted-component choice from Lemma 39 in this fixed reduction, using the retained labelled source torsor. Translating a lift of each top component to the same \(S'\) again makes every component visible there. Now take the generic point \(\eta\) of \(S'\times\mathbb P^n\). Complete its target local ring and call it \(R\); its residue field is \(K=\kappa(\eta)\). Let \(A_\alpha\) be all the regular complete local factors over it. Write \[r_\alpha=\mathop{\mathrm{rk}}_RA_\alpha,\qquad h_\alpha=[\kappa(A_\alpha):K],\qquad m_\alpha(Q)=\operatorname{length}_{A_\alpha} (A_\alpha\otimes_RQ)\] for a finite-length \(R\)-module \(Q\). Here \(Q\) is the module induced by \(M\); the reference module is \(K\). The finite-regular-module identity and length over \(R\) give, for every pair of factors, \[ r_\beta h_\alpha m_\alpha(Q) =r_\alpha h_\beta m_\beta(Q),\qquad r_\beta h_\alpha m_\alpha(K) =r_\alpha h_\beta m_\beta(K). \tag{61}\] The second lengths are positive. Division cancels both generic ranks and residue degrees and shows \[\frac{m_\alpha(Q)}{m_\alpha(K)} =\frac{m_\beta(Q)}{m_\beta(K)}.\] Proposition 29 applies at these nonclosed points with their residue-field Frobenius degrees; no perfection of \(K\) is being assumed. All top components are visible over this one generic point, because every chosen lift dominates the support under the finite projection. If factors split in reduction, include every new factor in (61). Descent through the étale source charts shows that \[[f^*M]_c=\rho_M[F_0]_c\] as support cycles, for a scalar \(\rho_M\) depending possibly on \(M\) and on the good prime. Lemma 39 proves (59).

It remains to explain why this one coefficient suffices. Frobenius is flat on the smooth \(X\), and Chern characters of ordinary Frobenius pullbacks satisfy \(\mathop{\mathrm{ch}}_i(F_X^{a*}F)=q^i\mathop{\mathrm{ch}}_i(F)\); one can check this on a finite vector-bundle resolution by the splitting principle. Hirzebruch–Riemann–Roch, applied additively to that finite resolution (Baum et al. 1975, sec. 0.1), therefore gives the polynomial \[ \chi(X,F_X^{a*}f^*M)= \sum_{i=c}^{2n}q^i \int_X\mathop{\mathrm{ch}}_i(f^*M)\mathop{\mathrm{td}}_{2n-i}(X),\qquad q=p^a. \tag{62}\] The growth bound of Lemma 41 kills its coefficients for \(i>c\). Equation (59), proved independently, kills the coefficient for \(i=c\). Thus the polynomial is identically zero; evaluating at \(q=1\) and using (58) proves (60). The scalar \(\rho_M\) is fixed before the Frobenius exponent varies. Uniformity of that scalar in \(M\), in the prime, or in Frobenius pullbacks is unnecessary. ◻

Proof. In a product chart the restricted ordinary cotangent is \(\Omega^1_{S_U}\oplus (\mathfrak m_{t_0}/\mathfrak m_{t_0}^2)\otimes\mathcal O_{S_U}\), hence is locally free. This is the ordinary cotangent, which records all embedding directions, not its reflexive hull. A finite-order automorphism of a complete local ring that is trivial on its cotangent space and has order invertible in the residue field is trivial: it is the identity on the associated graded, and at the first nonzero order of its difference from the identity, iteration multiplies that difference by the order, a contradiction. Equivalently, average lifts of a cotangent basis and apply complete Nakayama. A deck stabilizer acts faithfully on the ambient germ, so its cotangent representation is faithful.

The surjection from the ambient maximal ideal to the cotangent space of the smooth stratum admits an \(H\)-equivariant linear section by averaging. Its finite-dimensional image gives the desired lifted functions. Their restrictions are coordinates by the formal inverse function theorem. They can be chosen as algebraic local functions before completion, since a finite group average of such functions is again algebraic locally. Lemma 26 permits this change of parameters. The arguments use only finite group averaging and invertibility of the parameter Jacobian, and hence apply in the indicated reductions as well. ◻

Proof. Choose a minimal \(A\)-generating list \(q_1,\ldots,q_m\) of \(Q_i\). Lift these constant elements through the given surjection. Their residue classes in \(C^{r_i}/\mathfrak m_CC^{r_i}\) are independent, since their images form a basis of \(Q_i/\mathfrak m_AQ_i\). Complete them to a basis of the free module. The first \(m\) columns are then exactly the \(q_j\); subtract suitable combinations of these columns from the others to make the latter zero. These invertible column operations on \(C^{r_i}\) induce invertible operations on \((E_i\otimes_A C)\otimes_C C^{r_i}\) even if \(E_i\) is not \(A\)-flat. Evaluation respects the same column operations, so the resulting map has constant coefficients in the asserted sense. Flatness of \(A\to A[[t]]\) preserves its kernel, image and cokernel. A finite \(A\)-module extended in this way is finite free over \(K[[t]]\) as a \(K[[t]]\)-module, which proves exact specialization. ◻

Proof. By Lemma 43, the ordinary ambient cotangent restricted to \(\mathcal S\) is a vector bundle faithful on every full stabilizer. Since \(q_0\ge k\), ambient Frobenius factors through a map \(\Phi_{q_0}:\mathcal S_k\to\mathcal S\). Pulling back that bundle gives a bundle on \(\mathcal S_k\) whose stabilizer representations remain faithful: raising roots of unity of order prime to \(p\) to the \(q_0\)-th power is a permutation. A finite sum of bounded tensor powers contains every irreducible representation of each stabilizer. One elementary justification is that matrix coefficients of a faithful representation separate elements of a finite group, hence generate its function algebra; their products are matrix coefficients of tensor powers. There is one bound for a finite quotient atlas and its finitely many subgroup types. Dual powers may equally be included.

Let \(G\) be such a generating bundle. If \(\mathcal A\) is any coherent sheaf on \(\mathcal S_k\), exact coarse pushforward and sufficiently ample coarse twists generate the invariant Hom sheaf from \(G\) to \(\mathcal A\). The evaluation maps then surject on every residual gerbe, since \(G\) contains all its simple types. Nakayama makes them surjective on stalks. Taking suitable finite sums of twists of \(G\) gives the \(P_i\). This is the generating-sheaf criterion of (Olsson and Starr 2003, Definition 5.1 and Theorem 5.2), here realized by the faithful ambient cotangent bundle.

For the product assertion, complete at a point and forget the action temporarily. Put \[A=R_{\mathrm{tr}}/\mathfrak m_{\mathrm{tr}}^k, \qquad D=K[[t_1,\ldots,t_d]],\qquad C=A\otimes_KD.\] Every \(T_i\) is \(M_i\otimes_KD\) for a finite \(A\)-module \(M_i\). In these coordinates \(u\) is a constant primitive central idempotent in the \(N_{j_0}\) block: its center is the fixed transverse center tensored with the local ring \(D\), and its idempotents are constant. Thus the finite-dimensional transverse Hom calculation and (53) give \(\mathcal A_i=A_i\otimes_KD\), where \(A_i\subset\mathop{\mathrm{Hom}}_A(M_i,M_{j_0})\) is the kernel of the transverse map \(a\mapsto u\bar a\) in (64). Trivialize \(P_i\), so its surjection is \(C^{r_i}\to A_i\otimes_KD\). Apply Lemma 44 with \(E_i=M_i\), \(F=M_{j_0}\) and \(Q_i=A_i\). It makes the evaluation a fixed transverse map extended by \(D\) and proves that its kernel, image and cokernel are product modules, flat over \(D\), whose sequences specialize exactly modulo \((t_1,\ldots,t_d)\).

These basis changes establish formal product form only; they need not be equivariant. The actual bundles, maps and kernel in (66) keep their original equivariant structures. For character calculations, choose equivariant lifts of the smooth stratum parameters by tame averaging, as in Lemma 43. Their ideal is stable under the stabilizer. The formal automorphism fixing the transverse ring in Lemma 26 transports the tensor, full Hom, trace and ideal data to these new product coordinates. Reapplying Lemma 44 there proves flat specialization for this stable ideal. Completion detects the coherent local properties claimed here; it does not assert an algebraic equivariant product splitting. ◻

Proof. Let the stabilizer be \(J\), choose the equivariant smooth parameters from Lemma 45, and denote specialization modulo them by a subscript \(o\). Let \(A\) be the resulting transverse Artin ring over the algebraically closed residue field \(K\), and let \(\delta=\det(\Omega^1_{\mathcal S,o})\) be the one-dimensional cotangent determinant representation of \(J\).

The completed finite map to the smooth parameter disk computes the canonical module \(\omega_C\) as finite Hom to the disk’s canonical module (The Stacks Project Authors 2026, Tag 0AX0). The ring is finite free over the parameter ring, so this dualizing complex is concentrated in its canonical degree. The chosen parameter span is stabilizer-stable, with determinant \(\delta\), and the parameters form a regular sequence on \(C=A[[t_1,\ldots,t_d]]\). Equivariant Koszul adjunction gives \[\bigl(\omega_C/(t_1,\ldots,t_d)\omega_C\bigr) \otimes\delta^{-1} \simeq\omega_A\simeq A^*, \qquad A^*=\mathop{\mathrm{Hom}}_K(A,K).\] Thus the specialized dualizing module is \(A^*\otimes_K\delta\) with its actual stabilizer action. Product form justifies specialization of Hom, and finite duality gives \(\mathop{\mathrm{Hom}}_A(T'_o,A^*)\simeq(T'_o)^*\) equivariantly, where \((-)^*=\mathop{\mathrm{Hom}}_K(-,K)\). The other line in (67) has character \(\delta^{-q_0}\). Consequently \[ (M_{T'})_o\simeq(T'_o)^*\otimes\delta^{1-q_0} \simeq(T'_o)^*. \tag{69}\] The last isomorphism is an assertion about the stabilizer representation: (65) makes \(\delta^{1-q_0}\) trivial. It does not assert that the corresponding global line bundle is trivial.

For each \(j\), consider the image \(C_j\) of the ambient \(N_j\) endomorphisms in \(\mathop{\mathrm{End}}_A(T_{j,o})\). By Lemma 38, it surjects onto the barred corner with nilpotent kernel. This is a \(J\)-equivariant surjection. Exactness of tame invariants makes \(C_j^J\) surject onto the invariant barred corner, whose idempotent \(p_j\) we want to lift. The invariant kernel is still nilpotent, so idempotent lifting gives \[p'_j\in C_j^J,\qquad \overline{p'_j}=p_j.\] An explicit lifting proof starts with a lift \(a\) and corrects \(a^2-a\) successively modulo powers of the nilpotent ideal; \(2a-1\) is invertible modulo that ideal because its square is \(1\) there. The corrections take place inside \(C_j^J\). A global unbarred projector is not asserted.

Being in the ambient image has an additional consequence. Choose an ambient endomorphism inducing \(p'_j\) after specialization. It preserves every \(\mathcal J^s(N\otimes M_{T'})\) because \(\mathcal J\) is a two-sided ideal. On the specialized associated graded its action is \(p_{j,o}\), since the residual specialization of this representative is \(p_{j,o}\); no assertion about its residual action before specialization is needed. Product form makes the filtration specialize exactly. The idempotent \(p'_j\) therefore splits the specialized filtered module with graded image \(p_{j,o}L(M_{T'})_o\). Thus in the representation ring, \[ \bigl[(p_jL(M_{T'}))_o\bigr] =\bigl[(p'_jT_{j,o})\otimes_A(T'_o)^*\bigr] =\bigl[\mathop{\mathrm{Hom}}_A(p'_jT_{j,o},T'_o)^*\bigr]. \tag{70}\] The second equality is ordinary finite-dimensional tensor–Hom duality: dualizing its left-hand tensor gives \(\mathop{\mathrm{Hom}}_A(p'_jT_{j,o},T'_o)\).

Put \(Q_j=p'_jT_{j,o}\). Lemma 45 identifies \(Z_{0,o}\) with the actual kernel of the specialized evaluation \(\Phi_o:Y_{0,o}\to T_{j_0,o}\). Apply (63) with \(Q=Q_j\) and this specialized evaluation. We compute the image of its Hom map explicitly. Each evaluation column belongs to the right ideal (64), and precomposing it by any map from \(Q_j\) remains in that ideal. Conversely, a right-ideal map \(Q_j\to T_{j_0,o}\) extends by zero on \((1-p'_j)T_{j,o}\) to an element of \(\mathcal A_{j,o}\). Express that element in the generating columns of \(P_{j,o}\) and restrict to \(Q_j\). These expressions lift it through \(Y_{0,o}\). This verifies equality of the image with the indicated ideal, using all \(A\)-linear maps and with their genuine \(J\)-actions.

The full quotient (53) now identifies that cokernel with \(u\overline{\mathcal E}_op_j\). Surjectivity on this corner follows by lifting any barred map to a full truncated map and precomposing with \(p'_j\); its kernel is exactly the inverse-image ideal just described. Dualizing the signed identity and using (70) leaves \([(u\overline{\mathcal E}_op_j)^*]\). By (52) this is zero unless \(j=j_0\), when it is the invariant identity line. Summing with the signs \((-1)^j\) in (57) proves (68). ◻

Proof of Proposition 36. Equation (60) vanishes for the three modules (67). Take their signed sum. By Lemma 46 its coefficient has at every point the representation \((-1)^{j_0}\mathbf1\). Lemma 40 therefore gives \[ \chi\bigl(\mathcal S, \lambda_{-1}(\Omega^1_{\mathcal S})\bigr)=0. \tag{71}\] This coherent Euler characteristic is retained from characteristic zero in the chosen model. Indeed the finitely many exterior bundles, proper coarse pushforward and coherent cohomology can be spread with base change; tame coarse pushforward is exact. Thus (71) holds for the original complex stack.

Finally on smooth finite quotient charts the holomorphic de Rham complex resolves the constant sheaf, and taking invariants is exact in characteristic zero. Hypercohomology Euler characteristics, and proper GAGA (Porta and Yu 2016, Theorem 7.1), identify \[\sum_{a,b}(-1)^{a+b}\dim H^b(\mathcal S,\Omega^a_{\mathcal S}) =\sum_i(-1)^i\dim H^i(S_0,\mathbb C).\] The rational or complex cohomology of a finite stabilizer has no positive degrees; hence passage to the coarse space in this identity does not insert stabilizer weights. This remains true for stabilizers acting trivially on a chart. The left side is (71), proving the claimed ordinary Euler vanishing. ◻

Proof. Given a realized pair \((P,\kappa)\), nearby planes \(T'\) contain oriented period planes \(P'\) close to \(P\) and orthogonal positive vectors \(\kappa'\) close to \(\kappa\). Local Torelli realizes \(P'\) by a small deformation of that manifold. Openness of the relative Kähler cone makes \(\kappa'\) Kähler, proving openness.

Recall the two parts of the MBM theorem used here: the negative rational wall classes are invariant under deformation that keeps them of type \((1,1)\), and the Kähler cone is a component of the positive cone after removing their orthogonal hyperplanes (Amerik and Verbitsky 2015, Theorems 1.17 and 1.19). No bound on their squares is used. A realized plane cannot be orthogonal to such a class: that class would be of type \((1,1)\) and orthogonal to a Kähler class. Conversely, suppose \(T\) has no such orthogonality. Choose a generic oriented plane \(P\subset T\), so that every rational class orthogonal to \(P\) is already orthogonal to \(T\). Indeed, each rational vector not orthogonal to \(T\) excludes a proper closed subset of the sphere of planes in \(T\), and there are only countably many such vectors. Period surjectivity gives a marked manifold with period \(P\). Its rational \((1,1)\) classes include no MBM class. The cone theorem therefore identifies its Kähler cone with the whole positive component. The orthogonal ray to \(P\) in \(T\), with sign fixed by the orientation, is in that component. This realizes \(T\).

For rank four one can avoid even the MBM formulation in this step. If the negative line \(T^\perp\) is irrational, the same generic choice of \(P\) has zero Picard lattice. Its Kähler cone is the whole positive component, so \(T\) is realized. Thus a nonrealized plane has rational negative normal line.

Finally, an intersection between the Kähler cones of two marked models with the same period gives a Hodge parallel-transport isometry carrying a Kähler class to a Kähler class. Torelli makes it the isometry of an isomorphism. The cones then agree. ◻

Proof. Choose \(b\in U_{\mathbb Q}\) with \(q(a,b)=1\) and \(q(b)=0\); subtracting \(q(b_0)a/2\) from any \(b_0\) paired to \(1\) with \(a\) gives such a vector. Let \(V=\langle a,b\rangle^\perp\cap U\), a positive rational plane. For \(x\in V_{\mathbb Q}\) the null rotation is \[ \begin{split} T_{a,x}a&=a,\\ T_{a,x}b&=b+x-\tfrac12 q(x)a,\\ T_{a,x}v&=v-q(v,x)a\quad(v\in V), \end{split} \qquad T_{a,x}|_{U^\perp}=\mathop{\mathrm{id}}. \tag{97}\] Direct pairing verifies that it preserves \(q\), and substitution gives \(T_{a,x}T_{a,y}=T_{a,x+y}\). Thus \(T_{a,x}^m=T_{a,mx}\). Its matrix entries in an integral basis are polynomials in \(m\) of degree at most two with rational coefficients and constant term the identity. For sufficiently divisible \(m\), both \(T_{a,mx}\) and its inverse are integral. This accounts also for the finite gluing between the sublattices in \(U\) and \(U^\perp\).

For \(x\ne0\), choose a rational positive vector \(\ell\in V\cap x^\perp\). The rotation fixes \(\ell\). Replace it by a primitive integral vector on the same line. A generic period in \(\ell^\perp\) has Picard lattice \(\mathbb Z\ell\). Period surjectivity and the cone theorem give an ample marked deformation with polarization \(\ell\) or \(-\ell\); choose the ample sign. This sign does not alter the stabilizer of \(\ell\).

We now use the prescribed-polarization conclusion of (OpenAI 2026, Lemma 4.2). It applies to this primitive positive class \(\ell\), ample on a manifold in the chosen marked component. Write \(D_\ell\) for the component of its type-IV period domain containing that ample period, and put \[G_\ell(\mathbb Z)=\{g\in O(\Lambda,q):g\ell=\ell, \ gD_\ell=D_\ell\}.\] The cited lemma supplies a smooth projective family of IHS manifolds over a smooth connected quasi-projective variety \(S\), with a relative polarization of class \(k\ell\) for some \(k>0\) and a reference marking obtained by parallel transport from \(X_0\). It also supplies a torsion-free finite-index subgroup \(\Gamma\subset G_\ell(\mathbb Z)\) containing the family monodromy and an algebraic period map \[\Phi:S\longrightarrow Q_\ell:=\Gamma\backslash D_\ell.\] Here \(Q_\ell\) is smooth and quasi-projective, the map \(D_\ell\to Q_\ell\) is a covering with free \(\Gamma\)-action, and a local marked lift of \(\Phi\) is submersive. The type-IV domain \(D_\ell\) is contractible, including in rank four, so this covering identifies \(\pi_1(Q_\ell)\) with \(\Gamma\).

The remaining finite-index assertion concerns the image of this actual family monodromy, and we prove it here. The submersive local lift makes the algebraic map \(\Phi\) dominant.

Choose an irreducible algebraic slice \(S'\subset S\) of dimension \(\dim Q_\ell\) dominating \(Q_\ell\). For example, intersect with general hyperplanes through a smooth submersive point, transverse to its period fiber. After normalization and restriction, this slice is smooth and generically finite over \(Q_\ell\). There is a nonempty Zariski-open \(Q^\circ\subset Q_\ell\) over which its restriction is a nonempty finite étale cover: generic finiteness gives a finite open after removing the boundary in a proper compactification, and characteristic zero allows removal of the branch locus. Therefore its fundamental-group image has finite index in \(\pi_1(Q^\circ)\). The map \(\pi_1(Q^\circ)\to\pi_1(Q_\ell)\) is surjective, since a loop in a complex manifold can be moved off a proper algebraic subset. It follows that \[ [\Gamma:\Phi_*\pi_1(S)]<\infty. \tag{98}\] The period map on the universal cover of \(S\), based at this reference marking, is equivariant for the actual second-cohomology monodromy. Because \(D_\ell\to Q_\ell\) is a covering with free action, the deck transformation induced by each loop is exactly that cohomology operator. Hence the subgroup in (98) consists of actual parallel transports, with the original marking retained.

Our rotation fixes \(\ell\); after taking an integral power, a further power preserves \(D_\ell\) and therefore lies in \(G_\ell(\mathbb Z)\). Since \(\Gamma\) has finite index in \(G_\ell(\mathbb Z)\), a further power lies in \(\Gamma\); by (98), one more positive power lies in the family monodromy. This uses only that every element has a positive power in any finite-index subgroup; normality of the subgroup is unnecessary. Transport to the original reference fiber finishes the proof. The zero parameter gives the identity. ◻

Proof. Choose a rational null partner \(b\) of \(e\) on the line of \(w_*\) with \(q(e,b)=1\), and a rational orthogonal basis \(v_1,v_2\) of their positive perpendicular plane. Write \(q(v_1)=A>0\), \(q(v_2)=B>0\). For a rational number \(t\ne0\), set \(x=v_1\), \(y=v_1+t v_2\). Choose \(t\) so that \[ \frac{2(A-Bt^2)}{A+Bt^2}\notin\mathbb Z. \tag{99}\] This is possible: the expression lies in \((-2,2)\) and assumes each of the three possible integral values at only finitely many real \(t\). By Lemma 65, for some positive integers \(m_1,m_2\) the opposite rotations \(T_{e,m_1x}\) and \(T_{b,m_2y}\) are integral monodromies.

For completeness, the spectral calculation can be performed in a real orthonormal transverse basis without any rationality assumption on \(\sqrt A\) or \(\sqrt B\). Identify \(a e+b' b+c v_1+d v_2\) with the Hermitian matrix \[\begin{pmatrix} \sqrt2 a&\sqrt A c+i\sqrt B d\\ \sqrt A c-i\sqrt B d&-\sqrt2 b' \end{pmatrix}.\] Its determinant is the negative of \(q\). The action \(H\mapsto MHM^*\) of \(SL_2(\mathbb C)\) induces \(SO^+(3,1)\). Formula (97) corresponds, for the first rotation, to an upper triangular unipotent with parameter \(-m_1\sqrt A/\sqrt2\); the opposite rotation corresponds to a lower triangular unipotent with parameter \(m_2(\sqrt A-i t\sqrt B)/\sqrt2\). Put \[\gamma_k=T_{e,km_1x}T_{b,km_2y}.\] Its spin trace is \[ \tau_k=2+k^2m_1m_2\zeta, \qquad \zeta=-\tfrac12(A-i t\sqrt{AB}). \tag{100}\] Let \(\lambda_k,\lambda_k^{-1}\) be its spin eigenvalues, choosing \(|\lambda_k|>1\). For large \(k\), \(\lambda_k/\tau_k\to1\). The four eigenvalues on \(U_{\mathbb R}\) are \[|\lambda_k|^2,\quad |\lambda_k|^{-2},\quad \lambda_k/\overline{\lambda_k},\quad \overline{\lambda_k}/\lambda_k.\] The sum of the last pair converges to \(2\cos(2\arg\zeta)=2(A-Bt^2)/(A+Bt^2)\). By (99), it is nonintegral for all sufficiently large \(k\). In particular, this conjugate pair is nonreal and neither eigenvalue is \(1\) or \(-1\).

The characteristic polynomial on \(U\) is monic with rational coefficients, and its roots are algebraic integers because \(\gamma_k\) preserves \(\Lambda\). It therefore belongs to \(\mathbb Z[z]\). It is reciprocal, has constant term \(1\), and has no rational root: a rational root would be \(1\) or \(-1\). If it factored into two quadratics over \(\mathbb Q\), real coefficients would force the nonreal pair to belong to one quadratic. That monic factor has integral coefficients, so the sum of the pair would be an integer, a contradiction. Thus it is irreducible. Fix a sufficiently large \(k\).

Finally, a nontrivial null rotation fixing \([w_*]\) has powers converging to \([w_*]\) on every other point of the null boundary. In boundary coordinates with \([w_*]\) at infinity these powers are translations \(z\mapsto z+jc\), with \(c\ne0\). Such a rotation has an integral monodromy power by Lemma 65. Neither eigenline of \(\gamma_k\) is rational, by irreducibility, so neither is \([w_*]\). Conjugating \(\gamma_k\) by a sufficiently high power of this rotation brings both eigenlines as close to \([w_*]\) as desired and preserves all the asserted properties. Call the conjugate \(\gamma\). ◻

Proof. Choose future eigenvectors for \(a_+,a_-\) with eigenvalues \(\rho,\rho^{-1}\), respectively. The remaining real invariant plane is positive definite and \(\gamma\) acts on it by a rotation. Since \(e\) is a future null vector different from both eigenlines, its coefficients along both future eigenvectors are positive. Indeed, pairing its eigenspace decomposition with either eigenvector shows the coefficient of the other to be positive. It follows that \[ \rho^{-N}\gamma^Ne\longrightarrow v_+, \qquad \rho^{-N}\gamma^{-N}e\longrightarrow v_-, \tag{104}\] where \(v_+,v_-\) are positive representatives of \(a_+,a_-\).

If the assertion failed, choose \(N\to\infty\) and \(z_N=[s_Ne+\gamma^Ne]\in J\), with \(s_N>0\). We first claim \[ s_N\rho^{-N}\longrightarrow0, \qquad s_N\rho^N\longrightarrow\infty. \tag{105}\] To prove the first limit, normalize \(\gamma^Ne\) by \(c_N=-q(e,\gamma^Ne)\); by (104), \(c_N/\rho^N\) tends to a positive constant, and \([\gamma^Ne]\to a_+\in\mathcal V\). The tail exclusion (103) bounds \(s_N/c_N\) above. If a subsequence of \(s_N/\rho^N\) had a positive limit, \(z_N\) would converge to an interior point of \((e,a_+)\). Closedness of \(J\) would put that point in \(J\), contradicting (101). This proves the first limit. Apply the same argument to \[\gamma^{-N}z_N =[s_N\gamma^{-N}e+e] =[s_N^{-1}e+\gamma^{-N}e]\in J.\] The uniform tail at \(e\) now uses \(a_-\in\mathcal V\) and gives \(s_N^{-1}\rho^{-N}\to0\), which is the second limit.

Choose integers \(k_N\) such that \[1\le s_N\rho^{2k_N-N}<\rho^2.\] The two limits in (105) imply \(k_N\to\infty\) and \(N-k_N\to\infty\). Passing to a subsequence, suppose \(s_N\rho^{2k_N-N}\to r\in[1,\rho^2]\). Apply \(\gamma^{-k_N}\) to \(z_N\) and divide its representative by \(\rho^{N-k_N}\). Equation (104) gives \[\frac{s_N\gamma^{-k_N}e+\gamma^{N-k_N}e}{\rho^{N-k_N}} =s_N\rho^{2k_N-N} \frac{\gamma^{-k_N}e}{\rho^{k_N}} +\frac{\gamma^{N-k_N}e}{\rho^{N-k_N}} \longrightarrow r v_-+v_+.\] The limit has negative square, so its line is an interior point of the axis \((a_-,a_+)\). Invariance and closedness of \(J\) put that line in \(J\), contradicting (101). ◻

Proof. Nonzero variation means that, at some regular fiber, the derivative in a base direction of its weight-one period map is nonzero. This condition persists under small deformations of the fibration: on a small smooth product neighborhood in the family supplied by Proposition 48, the relative Hodge bundle and that derivative vary holomorphically, so a chosen nonzero matrix entry stays nonzero.

The fiber polarization can also be transported. Its class is the restriction of an integral ample class on \(X_0\). By Lemma 47 and topological transport, the restriction image on a nearby smooth fiber is a rational one-dimensional space with kernel \(e^\perp\). The restriction of a Kähler class spans its real form, since it pairs positively with \(e\), so the transported integral class remains of type \((1,1)\). Positivity on the smooth torus is open. This gives a polarized family near the chosen fiber, and the same argument applies at the conjugate fibration.

In rank four the preliminary point was \(X_*=X_0\). Complex conjugation of its hyperkähler complex structure reverses the period-plane orientation and the Kähler sign. The resulting marked manifold \(\overline X_*\) has period \(\overline P_*\), ample class \(-h\), and conjugate fibration \(\overline f_*:\overline X_*\to\overline B\) with fiber class \(-e\). It lies in the same marked deformation component: a half-circle of the twistor sphere joins the complex structure to its negative. Nonzero variation of \(\overline f_*\) is equivalent to that of \(f_*\).

We choose the endpoint neighborhood in Lemma 66 small enough for the following comparison with this conjugate fibration. Under the actual parallel transport \(\gamma^{-N}\), the underlying plane of \(P_N\) becomes \[\langle \gamma^{-N}e,e\rangle^\perp\cap U_{\mathbb R}.\] It approaches an unoriented plane near \(P_*\), since \([\gamma^{-N}e]\to a_-\) and \(a_-\) can be chosen arbitrarily close to \([w_*]\). The transported positive cone is \[ \gamma^{-N}\mathcal C_N =\{a\gamma^{-N}e-b e:a,b>0\}. \tag{110}\] By (96), this cone contains a vector close to \(-h\). Preservation of the orientation of maximal positive three-spaces therefore makes the transported oriented period close to \(\overline P_*\).

Choose the endpoint neighborhood sufficiently small, and then \(N\) sufficiently large, that this period lies in a neighborhood where the \((-e)\)-deformation of \(\overline f_*\) exists and has nonzero variation. Let \(Y_N\) be that actual small deformation of \(\overline X_*\) with period \(\gamma^{-N}P_N\). It has a Kähler class near \(-h\), in the component (110). In rank four, \(H^{1,1}(X_N,\mathbb R)=\langle e,u_N\rangle_{\mathbb R}\), so (107) says that the entire positive component of \(X_N\) is Kähler. The Hodge parallel-transport isometry between \(Y_N\) and \(X_N\) induced by \(\gamma^N\) consequently carries a Kähler class to a Kähler class. Torelli identifies these manifolds by an isomorphism inducing that isometry. The class \(-e\) on \(Y_N\) becomes \(u_N\) on \(X_N\).

Line bundles on an IHS manifold are determined by their Chern classes. Thus the transported fibration and the fibration obtained from \(u_N\) have the same connected fibers: positive generated powers define the same Stein factorization. It follows that \(g\) has nonzero polarized variation. The isometry in this comparison is the actual parallel transport constructed in Lemma 65. ◻

Proof. Take a local Galois chart \(U\to U/G\) as above. Its non-étale locus on the coarse space has codimension at least two. The inverse image of that locus has codimension at least two in \(X\), by equidimensionality, and in \(A\), by finiteness. Consequently, the full normalizations of the components of the pullbacks of \(U\to U/G\) that dominate these smooth sources are finite, by excellence, and étale over the sources, by purity of the branch locus (The Stacks Project Authors 2026, Tag 0BMB). Their deck actions are torsor actions: the torsor identities hold on the dense étale locus and extend uniquely to the finite étale covers. The torsors and their equivariant maps to \(U\) are precisely the data defining the two lifts. Uniqueness on normalized overlaps makes these data compatible and gives the stated representable lifts. The normalized-source package of Proposition 8 gives the asserted properties of \(\widetilde f\) on each chart: its source is smooth and symplectic, its map to \(U\) is projective with Lagrangian equidimensional fibers, and Stein factorization gives connected fibers and direct image \(\mathcal O_U\).

The pullback of \(\widetilde p\) to \(U\) is finite, since it is finite over the ordinary finite base change \(A\times_B U\). It is a morphism between smooth \(n\)-dimensional varieties and is flat by miracle flatness (The Stacks Project Authors 2026, Tag 00R4). Thus \(\widetilde p\) is finite flat, and it is surjective because its coarse morphism is surjective. ◻

Proof. Use the global isomorphisms \(\phi_j:R^j f_*\mathcal O_X\xrightarrow{\sim}\Omega_B^{[j]}\) chosen once on the original base in Section 2, with \(\phi_0\) the unit isomorphism. We apply the chart comparison of Proposition 27. Choose a connected affine quotient chart \(U\to U/G\) with \(U/G\to B\) an integral separated étale neighborhood, and view \(U\to\mathfrak B\) as its atlas map. Write \(f_U:X_U\to U\) for the normalized lift, put \(U^0=U\times_B B_{\rm reg}\), and let \(\pi:U^0\to B_{\rm reg}\) be the induced étale morphism. The complement of \(U^0\) has codimension at least two. Over this open the lift is the ordinary base change of the original fibration. Ordinary flat base change for \(\pi\), followed by the étale cotangent identification, gives \[(R^j(f_U)_*\mathcal O_{X_U})|_{U^0} \simeq\pi^*(R^j f_*\mathcal O_X|_{B_{\rm reg}}) \xrightarrow{\ \pi^*\phi_j\ }\Omega^j_{U^0}.\] This open includes the discriminant over the regular part of \(B\); the flat map used here is \(\pi\), so no smoothness of the fibers is required for the base-change step (The Stacks Project Authors 2026, Tag 02KH).

For \(j>0\), Ou’s local reflexivity theorem applies to \(f_U\): Proposition 8 supplies its projective, equidimensional map with connected fibers and smooth canonical-trivial source, and \(U\) is smooth (Ou 2019, Theorem 1.3). Both \(R^j(f_U)_*\mathcal O_{X_U}\) and \(\Omega_U^j\) are therefore their reflexive extensions from \(U^0\). The displayed isomorphism extends uniquely across the complement. For \(j=0\) the same conclusion is the unit isomorphism from \((f_U)_*\mathcal O_{X_U}=\mathcal O_U\). On a chart overlap the extended maps agree over the inverse image of \(B_{\rm reg}\), by transitivity of flat base change from the same \(\phi_j\). Every component of the overlap meets this open, since it maps étale to a chart and cannot be supported over its codimension-at-least-two complement. Uniqueness of reflexive extension makes the maps agree everywhere. Thus they glue to degreewise isomorphisms \[R^j\widetilde f_*\mathcal O_X\simeq\Omega^j_{\mathfrak B} \qquad(0\le j\le n).\] For \(j>n\), the restriction to \(U^0\) is zero by Proposition 2, and reflexivity gives zero on \(U\). Thus no higher direct-image terms occur.

These isomorphisms first give (130) in coherent \(K\)-theory. In this setting coherent and vector-bundle \(K\)-theory agree. Indeed, \(T_{\mathfrak B}\) is faithful on every stabilizer by linearization of finite group actions. A finite direct sum \(\mathcal G\) of bounded tensor powers of this bundle and its dual contains every irreducible stabilizer representation occurring on a finite system of quotient charts. Exactness of finite-group invariants makes the evaluation \[c^*c_*(\mathcal G^\vee\otimes\mathcal F)\otimes\mathcal G \longrightarrow\mathcal F\] surjective for every coherent \(\mathcal F\). For sufficiently large \(k\), the coherent sheaf \(c_*(\mathcal G^\vee\otimes\mathcal F)\otimes H^k\) is globally generated on \(B\). Its generators yield a surjection from a finite sum of \(\mathcal G\otimes c^*H^{-k}\) onto \(\mathcal F\). Iterating and using regularity of the smooth stack gives a finite locally free resolution. This proves the claimed comparison and allows the identity to be pulled back in vector-bundle \(K^0\), even along an inertia morphism which is not flat.

We spell out the nilpotence needed for (131). If \(V\) is a vector bundle of rank \(r\) on a smooth projective variety \(Y\), then \([V]-r\) is nilpotent in \(K^0(Y)\). Indeed, pull back to the full flag bundle of \(V\). This pullback is injective on \(K^0\), since pushforward of its structure sheaf is \(\mathcal O_Y\), with no higher direct images. The pulled-back class is a sum of classes \([L]-1\) for line bundles. For a globally generated line bundle \(L\), a finite set of generating sections and its exact Koszul complex show that \((1-[L^{-1}])^N=0\) for some \(N\). An arbitrary line bundle is a quotient of two sufficiently positive globally generated line bundles. Therefore its class minus one is also nilpotent: it is an invertible multiple of a difference of two commuting nilpotent elements. A finite sum of commuting nilpotents is nilpotent. This proves the assertion.

Apply it on the original projective \(X\) to \(\widetilde f^*T_{\mathfrak B}\). The projection formula and (130) give (131), with one exponent \(m\) before any inertia component is chosen. ◻

Proof. An inertia component is a smooth proper stack: on a quotient chart its presentation is a fixed locus \(U^h\) modulo the appropriate centralizer. The fixed eigensubbundle of \(j^*T_{\mathfrak B}\) is \(T_S\). Also, \(W\) is a scheme, since it is finite over \(A\). The morphism \(W\to S\) is finite flat, being a base change of \(\widetilde p\). Thus \(W\) is projective, pure-dimensional of dimension \(d\), and Cohen–Macaulay. These assertions hold even if \(W\) is nonreduced.

Restrict (131) to \(S\) and take the character of the tautological automorphism. This is the inertia eigenbundle character used in Toën’s Riemann–Roch framework (Toën 1999, sec. 3.3 and Lemma 4.6); only its explicitly described ring homomorphism and moving-factor cancellation are needed here. More explicitly, for a bundle \(V\) on \(\mathfrak B\), decompose \(j^*V=\bigoplus_{\zeta} V_\zeta\) into the \(h\)-eigenbundles and use the ring homomorphism \[[V]\longmapsto\sum_{\zeta}\zeta[V_\zeta] \quad\hbox{in }K^0(S)\otimes_{\mathbb Z}\mathbb C.\] Pull this identity to \(W\) and take its Chern character in \(H^{\rm even}(W,\mathbb C)\). The degree-zero term coming from \([T_{\mathfrak B}]-n\) is \(\operatorname{tr}(h|T_{\mathfrak B})-n\), which is nonzero. Indeed, the tangent action is faithful and all eigenvalues have absolute value one; their sum can be \(n\) only if every eigenvalue is one. This would make \(h\) the identity by finite-order linearization. The character of each moving cotangent factor in the alternating exterior sum has nonzero degree-zero term \((1-\zeta)^{\operatorname{rk}V_\zeta}\), for \(\zeta\ne1\). The eigenbundle ranks are constant on connected \(S\), so each of these degree-zero terms is the same nonzero scalar on every connected component of \(W\). Each full class is therefore a unit: every positive-degree element of the cohomology of the finite-dimensional projective scheme \(W\) is nilpotent. Cancelling them gives \[\operatorname{ch} \bigl(\lambda_{-1}(q^*\Omega^1_S)\bigr)=0.\] The term of degree \(2d\) on the left is \(c_d^{\rm top}(E)\), since for Chern roots \(x_i\) of \(E\) the expression is \(\prod_i(1-e^{-x_i})\). Hence \[ c_d^{\rm top}(E)=0 \quad\hbox{in }H^{2d}(W,\mathbb C). \tag{133}\] In particular, a nonempty nonidentity inertia component of dimension zero is impossible, as in that case this says \(1=0\).

Consider the bundle map \[\alpha:\operatorname{Lie}(A)\otimes\mathcal O_W =a^*T_A\longrightarrow E\] used in (132). For every reduced irreducible subvariety \(Y\subset W\) of dimension \(s\), this map has rank at least \(s\) at its generic point. In fact, \(Y\to S\) is finite onto its image and is generically separable. On a smooth dense open of \(Y\), and after an étale chart on \(S\), its differential consequently has rank \(s\). The differential identity for the fiber square says \[\alpha\circ d(a|_Y)=d(q|_Y),\] because the latter differential has values in the fixed tangent subbundle. This proves the rank assertion.

It follows that the locus on \(W_{\rm red}\) where \(\operatorname{rk}\alpha\le r\) has dimension at most \(r\): apply the preceding assertion to each of its irreducible components. Over the stratum of rank \(r\), the incidence variety \[\{(w,u)\in W_{\rm red}\times\operatorname{Lie}(A): \alpha_w(u)=0\}\] has dimension at most \(r+(n-r)=n\). There are finitely many rank strata. Thus a general \(u\) gives a section whose zero scheme has at most zero-dimensional support. It is a finite scheme because \(W\) is proper.

At every such zero, the \(d\) local entries of the section are a system of parameters in a \(d\)-dimensional Cohen–Macaulay local ring, hence a regular sequence (The Stacks Project Authors 2026, Tag 02JN). Let \([W]_d=\sum_i\operatorname{length}(\mathcal O_{W,\eta_i})[W_i]\) be the scheme-theoretic fundamental cycle, where the \(W_i\) are the irreducible components with generic points \(\eta_i\). The algebraic zero-section formula gives \[c_d^{\rm CH}(E)\cap[W]_d=[Z(s_{u,S})]_0 \quad\hbox{in }\mathrm{CH}_0(W)\] (The Stacks Project Authors 2026, Tag 0FA9). The complex Borel–Moore cycle class commutes with vector-bundle Chern classes and proper pushforward (Fulton 1998, chap. 19). With the same generic multiplicities in \(\operatorname{cl}_{\rm BM}([W]_d)\), this gives \[\operatorname{length} Z(s_{u,S}) =\deg\bigl(c_d^{\rm CH}(E)\cap[W]_d\bigr) =\bigl\langle c_d^{\rm top}(E), \operatorname{cl}_{\rm BM}([W]_d)\bigr\rangle=0,\] where the last equality is (133). Each nonempty isolated zero has strictly positive length. Thus there are no zeros. The inertia has only finitely many connected components, so the intersection of the corresponding nonempty open sets of choices of \(u\) supplies the single field \(v\) in the statement. ◻

Proof. Let \(Z\) be the normalization of such a component, let \(z_0\in Z\) map to the given arrow, and suppose that \(\delta_Z:Z\to A\) is dominant. Take its Stein factorization \[Z\xrightarrow{\beta}T\xrightarrow{\gamma}A.\] Here \(T\) is normal, \(\gamma\) is finite and dominant, and \(\beta\) has connected fibers. The connected fiber through \(z_0\) lies entirely over nonidentity arrows: its image lies in \(\delta^{-1}(0)\), where the identity and nonidentity parts are open and closed.

Put \(t_0=\beta(z_0)\), and take the field \(v\) of Lemma 77. A finite dominant map to the smooth \(n\)-fold \(A\) is locally open. For completeness, a sufficiently small local branch of \(T\) at \(t_0\) is finite over a neighborhood of \(0\); its analytic image is closed and has dimension \(n\), so contains a neighborhood of \(0\). We can therefore lift a sequence \(\exp(\varepsilon_\nu v)\), with \(0\ne\varepsilon_\nu\to0\), to points of \(T\) tending to \(t_0\). Lift these points farther to \(Z\). Properness gives a subsequence tending to a point \(z_\infty\in\beta^{-1}(t_0)\). Its arrow is nonidentity, and the arrows in the sequence have coordinates \[(x_\nu,y_\nu),\qquad y_\nu=x_\nu+\exp(\varepsilon_\nu v), \qquad x_\nu,y_\nu\longrightarrow x_\infty .\]

Choose a local quotient chart at \(\widetilde p(x_\infty)\), a lift \(\psi\) of \(\widetilde p\) on a small analytic neighborhood of \(x_\infty\), and finite-group linearizing coordinates at \(\psi(x_\infty)\). After passing to a subsequence the labels are one fixed group element \(h\ne1\), fixing \(\psi(x_\infty)\), and \[\psi(y_\nu)=h\psi(x_\nu).\] Projecting to the \(h\)-fixed coordinate space yields \[\operatorname{pr}_{h=1} \bigl(\psi(x_\nu+\exp(\varepsilon_\nu v))-\psi(x_\nu)\bigr) =0.\] Division by \(\varepsilon_\nu\) and passage to the limit give \(\operatorname{pr}_{h=1}d\psi_{x_\infty}(v)=0\). This is a zero of the section (132) on the inertia component containing \(h\), contrary to Lemma 77. ◻

Proof. Let \(U_Z\subset D_Z\) be a nonempty open over which every irreducible fiber component of \(\delta_Z\) has dimension \(n-r\), let \(a\in U_Z\), and let \(F\) be any one of those components, with its reduced structure. On \(F\) the second coordinate map is \(t_a\) composed with the first. The equality of the two pulled-back polarizations therefore says \[ (s|_F)^*\lambda_P(a)\simeq\mathcal O_F . \tag{134}\] Let \(H_F\subset A\) be the connected abelian subvariety generated by differences of points of \(s(F)\). Since \(s\) is finite, \[\dim H_F\ge n-r.\] Resolve \(F\) and choose a base point on its resolution. Its map to \(A\) factors through a surjective homomorphism from its Albanese variety onto \(H_F\), followed by a translation. The dual homomorphism \(\widehat H_F\to\operatorname{Pic}^0(\widetilde F)\) has finite kernel. Equation (134) thus implies that the restriction of \(\lambda_P(a)\) to \(H_F\) is torsion.

For a fixed abelian subvariety \(H\subset A\), the set of points with this property is a countable union of torsion translates of \(H^\perp\). To check the assertion including disconnected kernels, the homomorphism \[A\longrightarrow\widehat H,\qquad a\longmapsto\lambda_P(a)|_H\] is surjective and has \(H^\perp\) as the identity component of its kernel; the induced map \(A/H^\perp\to\widehat H\) is an isogeny. The inverse image of any torsion point is consequently a finite union of cosets of \(H^\perp\) represented by torsion points of \(A\). Here torsion points lift through a quotient of abelian varieties; this follows, for example, by choosing an isogeny complement to the kernel.

There are only countably many abelian subvarieties of \(A\), since their rational first-homology subspaces determine them. There are also only countably many torsion points. The open \(U_Z\) is therefore covered by countably many closed torsion cosets, each of dimension at most \(r\). A complex irreducible variety cannot be covered by countably many proper closed subvarieties: apply the Baire theorem in a small analytic ball in its smooth locus, and then use irreducibility. Hence one such coset contains \(D_Z\). The dimension bound forces equality, proving \(D_Z=\tau+D\) with \(\dim D=r\).

This argument also determines the subgroup generated by every component over a dense set of very general points. Let \(a\) range over \(U_Z\) minus every coset in the preceding countable collection which does not contain \(D_Z\). This complement is dense by the same Baire argument. For each fiber component \(F\) over each such \(a\), its character constraint supplies a coset of \(H_F^\perp\) of dimension at most \(r\) containing \(a\). That coset must contain \(D_Z\) and hence equal it. Thus \[H_F^\perp=D,\qquad H_F=D^\perp=K .\] The image \(s(F)\) lies in one \(K\)-coset and has dimension \(n-r=\dim K\), so it is the whole coset. Consequently the relation over every such very general \(a\) is stable under diagonal \(K\)-translation. These fibers are dense in \(Z\): every nonempty Zariski open in \(Z\) has image containing a nonempty Zariski open in \(D_Z\), which meets this set. Taking closures therefore proves this invariance on the full image \(\Gamma\subset A\times A\).

The finite map from the original relation component to \(\Gamma\) is generically one-to-one: over \(B_{\rm reg}\) there is a unique isomorphism label for a pair of points. Each component dominates \(A\), so this open is dense on it. Thus \(Z\) is also the normalization of \(\Gamma\). The morphism \(K\times Z\to\Gamma\) induced by simultaneous translation is dominant and has normal source. It therefore lifts uniquely to the normalization \(Z\). The action law, equivariance of the coordinate maps, and invariance of the difference follow by uniqueness of this lift, or by checking on the dense normal isomorphism locus and extending. ◻

Proof. Since \(\widetilde p\) is generically étale, the determinant of \(d\widetilde p\) defines an effective Cartier ramification divisor \(\operatorname{Ram}(\widetilde p)\) on \(A\), and \[\mathcal O_A(\operatorname{Ram}(\widetilde p)) \simeq \widetilde p^*K_{\mathfrak B}^{-1}.\] There is no divisorial stabilizer on \(\mathfrak B\). A sufficiently divisible power of its canonical bundle is therefore the pullback of \(\mathcal O_B(mK_B)\). Thus \[(\widetilde p^*K_{\mathfrak B}^{-1})^{\otimes m} \simeq p^*\mathcal O_B(-mK_B).\] The right side is ample because \(p\) is finite and \(-K_B\) is ample. Hence the line bundle of the total ramification divisor is ample.

Since \(D\ne A\), the group \(K\) is positive-dimensional. If every prime component of \(\operatorname{Ram}(\widetilde p)\) had proper image in \(A/K\), a general \(K\)-coset would miss the entire divisor. Its defining section would trivialize the restricted ample line bundle on that positive-dimensional projective coset, a contradiction. Thus one component \(L\) dominates \(A/K\); properness makes this map surjective. This uses ampleness of the total ramification divisor and does not assert that each of its components is ample.

The generic point of \(L\) lies over \(B_{\rm reg}\), because finiteness preserves the codimension-at-least-two bound for the inverse image of \(B_{\rm sing}\). Near a general point of \(L\), the map between smooth spaces has the transverse form \[(u_1,u_2,\ldots,u_n) \longmapsto(u_1^e,u_2,\ldots,u_n), \qquad e>1.\] Its self fiber product has, besides the identity branch, branches \(v_1=\zeta u_1,\ v_j=u_j\) for \(\zeta^e=1,\ \zeta\ne1\). They meet the identity along \(L\). There are only finitely many global irreducible components of \(\mathcal R\). Their closed intersections with \(i(L)\), excluding the identity component, contain a dense open of \(i(L)\). Irreducibility therefore gives a single residual component \(\mathcal R'\ne i(A)\) containing all of \(i(L)\), including its special points.

Suppose, for a contradiction, that \(\delta(\mathcal R')\subset D\). Lemma 79 says that this difference image is a torsion coset. It contains zero, since \(i(L)\subset\mathcal R'\), so it is an abelian subvariety \(D'\subset D\). Its polarized complement \(K'=(D')^\perp\) contains \(K\). The same lemma gives a \(K\)-action on the normalization \(Z'\) of \(\mathcal R'\).

Consider the closed set \[F=\{x\in A:i(x)\in\mathcal R'\}.\] It is proper: otherwise \(\mathcal R'\) would contain \(i(A)\), and the two irreducible \(n\)-dimensional components would be equal. For any \(x\in F\), choose an individual preimage of \(i(x)\) in \(Z'\). Its connected \(K\)-orbit remains over difference zero and remains in the inverse image of the identity part of \(\delta^{-1}(0)\), since that part is open and closed. Its image at \(k\) is therefore precisely \(i(x+k)\). Hence \(F\) is \(K\)-invariant. This reasoning does not require a global lift of \(i(L)\) to \(Z'\).

The divisor \(L\subset F\) is an irreducible component of \(F\), because \(F\) is proper in the \(n\)-fold \(A\). A connected group preserves each irreducible component of a closed invariant set with finitely many components. Thus \(K+L=L\). But \(L\to A/K\) is surjective, which gives \(K+L=A\). This contradiction proves the required assertion about \(\delta(\mathcal R')\). ◻

Proof. The action of \(K\) supplied by Lemma 79 preserves difference zero. Connectedness and the open-and-closed identity partition preserve the nonidentity condition. Since \(L\to A/K\) is surjective, a translate of the first coordinate of \(z_0\) belongs to \(L\). Choose such a \(k\), and put \(e_0=k\cdot z_0\) and \(x_0=s(e_0)=t(e_0)\). This is the translated normalization point retained below.

The divisor \(L\) is Cartier because \(A\) is smooth. Its local equation pulls back to a nonzero nonunit in the integral local ring \(\mathcal O_{Z,e_0}\). A minimal prime over that equation has height one, by the principal ideal theorem. Taking its closure with reduced structure gives a prime divisor \(E\) through \(e_0\) and contained in \(s^{-1}(L)\). The finite map \(s:Z\to A\) preserves dimension, so its closed image \(s(E)\) has dimension \(n-1\) and equals the irreducible divisor \(L\).

The image of the proper action morphism \(K\times E\to Z\) is closed and irreducible. Its first-coordinate image is \(K+L=A\). Since \(Z\to A\) is finite, this image has dimension \(n\) and hence is all of \(Z\). As \(\delta_Z\) is \(K\)-invariant, \[\delta_Z(E) =\delta_Z(K\cdot E) =\delta_Z(Z)=D .\] These are equalities of actual images; projectivity ensures that no special point is lost by taking closures. ◻

Completion of the proof of Proposition 74. Apply Lemmas 80 and 81 to the fixed \(D,K,Z,z_*\). They provide a divisor \(L\subset A\), a residual component \(\mathcal R'\subset\mathcal R\) containing all of \(i(L)\), and a reduced prime divisor \(E\subset Z\) through the translated point \(e_0\), with \[\delta_Z(E)=D,\qquad \delta(\mathcal R')\not\subset D.\] Write \(x_0=s(e_0)=t(e_0)\in L\) and \(\alpha_0=\nu(e_0)\). The arrow \(\alpha_0\) is nonidentity and has difference zero. It is this translated arrow that composition will retain. Form the space of composable arrows \[\mathcal T =\mathcal R'\times_{\,t,A,s} Z .\] The inclusion of all of \(i(L)\) in the actual, possibly nonnormal component \(\mathcal R'\) gives a global morphism \(i|_L:L\to\mathcal R'\): both \(L\) and \(\mathcal R'\) carry their reduced structures. Hence the map \[\iota:E\longrightarrow\mathcal T,\qquad e\longmapsto \bigl(i(s(e)),e\bigr)\] is a closed embedding, the graph of this morphism over the closed subscheme \(E\subset Z\). Since \(E\) is irreducible, one irreducible component \(\mathcal T_0\) contains its entire image. Give \(\mathcal T_0\) its reduced structure. Since \(E\) is reduced, \(\iota\) factors as a closed embedding into \(\mathcal T_0\); in particular, this component contains the retained marked point. Figure 2 records this component choice and the subsequent composition.

This same component dominates both factors. Indeed, \(\mathcal R'\times Z\) has pure dimension \(2n\). The equations for \(\mathcal T\) are the pullback of the diagonal in \(A\times A\), which is locally cut out by \(n\) equations. The dimension inequality for an ideal with \(n\) generators implies \(\dim\mathcal T_0\ge n\). Both projections from \(\mathcal T\) to its factors are finite, so \(\dim\mathcal T_0\le n\). Its images in both irreducible \(n\)-dimensional factors are closed of dimension \(n\), and thus are the whole factors. The dimension estimate uses smoothness of \(A\) through the equations of its diagonal.

Composition of the two labelled arrows gives \(\mu:\mathcal T_0\to\mathcal R\). Both \(\mathcal T_0\) and \(\mathcal R\) are finite over the outer source coordinate in \(A\), so \(\mu\) is finite. Its closed irreducible image has dimension \(n\), since that source coordinate is surjective. Consequently its image is an irreducible component \(\mathcal R''\) of \(\mathcal R\). On the embedded \(E\), composition with the identity is exactly \(\nu(e)\), with its label. Thus \(\mathcal R''\) contains the marked nonidentity arrow \(\alpha_0=\nu(e_0)\) over difference zero, and its difference image contains all of \(\delta_Z(E)=D\).

It also contains a point outside \(D\). In fact, dominance of \(\mathcal T_0\to\mathcal R'\) allows us to choose composable arrows \[z\longrightarrow x\longrightarrow y\] with \(x-z\notin D\), whereas \(y-x\in D\), because the second arrow comes from \(Z\). Since \(D\) is a subgroup, \(y-z=(y-x)+(x-z)\notin D\). The closed irreducible difference image of \(\mathcal R''\) therefore contains \(D\) properly, and has dimension greater than \(r\). It still meets nonidentity inertia over zero, contradicting the choice of \(r\).

Thus every stabilizer is trivial. A Deligne–Mumford stack with trivial inertia is an algebraic space, and its coarse morphism is then an isomorphism. Therefore \(\mathfrak B=B\), and \(B\) is smooth. ◻