Proof. If \(\dim A=0\), then \(A=k\), \(P=Q=0\), and both sides are \(1\) with \(T=k\). Otherwise, let \(\widetilde X\) be the blowup of \(X=\operatorname{Spec}A\) at its closed point, and let \(\widetilde Y,\widetilde Z\) be the strict transforms of \(Y=\operatorname{Spec}(A/P)\) and \(Z=\operatorname{Spec}(A/Q)\). By [27], the underlying set \(S\) of \(\widetilde Y\cap\widetilde Z\) identifies with that of \(\operatorname{Proj}T\). Thus \(S\) is a finite set of closed points of the exceptional divisor, empty exactly when \(\dim T=0\). Fulton’s blowup formula and Skalit’s finite-support calculation [27] give \[\chi^A(A/P,A/Q)-e_\mathfrak{m}(A/P)e_\mathfrak{m}(A/Q) =\sum_{x\in S}[\kappa(x):k]\,\chi^{B_x}(M_x,N_x),\] where \(B_x=\mathcal O_{\widetilde X,x}\), \(M_x=\mathcal O_{\widetilde Y,x}\), and \(N_x=\mathcal O_{\widetilde Z,x}\). Each residue degree is finite and positive since the exceptional divisor is projective over \(k\). The blowup \(\widetilde X\) is regular [14]. The modules \(M_x,N_x\) are nonzero and finite over \(B_x\), with finite-length tensor product because their intersection is supported on \(S\). The local domains \(A/P\) and \(A/Q\), as well as \(A\), are formally equidimensional and universally catenary. Thus [27] gives \(\dim M_x=\dim(A/P)\), \(\dim N_x=\dim(A/Q)\), and \(\dim B_x=\dim A\). Theorem 1 therefore makes every local intersection multiplicity in the sum strictly positive. The asserted inequality follows, and equality holds exactly when \(S\) is empty. ◻

Proof. In equal characteristic, a coefficient field gives a presentation \(B=k[[X_1,\ldots,X_t]]/I\). Put \(B'=\overline{k}[[X_1,\ldots,X_t]]/I\overline{k}[[X_1,\ldots,X_t]]\). The map of power-series rings is flat, by the local flatness criterion for the variables and the flat coefficient-field extension. Its base change to the quotient has the asserted residue field and maximal ideal.

In the remaining case the residue characteristic is a prime \(p\). Cohen structure supplies a presentation \(B=V[[X_1,\ldots,X_t]]/I\), where \(V\) is a complete discrete valuation ring with uniformizer \(p\) and residue field \(k\). We construct a discrete valuation extension with unchanged uniformizer and residue field \(\overline{k}\), without assuming that \(k\) is perfect. Given a simple algebraic residue-field extension, lift its monic irreducible polynomial to \(f\in V[X]\). The algebra \(V[X]/(f)\) is finite free over \(V\), its reduction modulo \(p\) is the desired field, and all its maximal ideals lie over \((p)\). It is therefore local with maximal ideal generated by \(p\). Since \(p\) is a nonzerodivisor, it is a one-dimensional regular local ring, hence a discrete valuation ring. This argument also applies to inseparable residue extensions.

Perform this construction along a tower of simple algebraic extensions exhausting \(\overline{k}\), taking unions at limit stages. In every such union, a nonzero element is a unit times a nonnegative integral power of \(p\). Every nonzero ideal is generated by an element of least valuation, so the union is still a discrete valuation ring. Its completion \(V'\) has uniformizer \(p\) and residue field \(\overline{k}\). The map \(V\to V'\) is flat because \(V'\) is torsion-free over \(V\). Now take \[B'=V'[[X_1,\ldots,X_t]]/IV'[[X_1,\ldots,X_t]].\] The same local flatness criterion proves the required flatness. This is the usual residue-extension construction; compare [19].

For either construction, tensor a composition series of a finite-length \(B\)-module with \(B'\). Flatness preserves exactness, and every simple factor becomes \(B'/\mathfrak b'\). This proves the length assertions and, applied to \(V/\mathfrak b^aV\), the displayed identity. Equality of Hilbert–Samuel functions gives equality of module dimensions. If \(B\) is regular of dimension \(d\), its \(d\) maximal-ideal generators generate \(\mathfrak b'\), and \(\dim B'=d\), so \(B'\) is regular. Finally flat base change identifies the Tor modules after extension; their finite lengths are preserved by the preceding argument. ◻

Proof. The positive-characteristic assertion is the Cohen–Gabber normalization theorem [17]. In mixed characteristic, choose a coefficient ring \(W(k)\subseteq D\) and extend \(p\) to a parameter system \(p,t_2,\ldots,t_h\). Completeness and the parameter condition make the resulting power-series map module finite. Its kernel is zero: the source and target have dimension \(h\), whereas a nonzero ideal of the regular local domain \(W(k)[[t_2,\ldots,t_h]]\) lowers dimension. The fraction-field extension is separable in characteristic zero. ◻

Proof. In characteristic \(p\), the stages are power-series rings in the chosen roots. In mixed characteristic they are power-series rings over the complete discrete valuation ring \(W(k)[p^{1/p^n}]\). The monomials in the successive roots, with each exponent less than the relevant root degree, form bases for all transition extensions. This proves the assertions about regularity, ranks, and residue fields.

Finite presentations over the union descend to a stage. The additional relations expressing annihilation by a fixed power of \(\mathfrak m_A\) also descend after increasing that stage, because there are finitely many generators and relations to consider. The resulting finite-stage module has finite length. Formula (2) is unchanged on enlarging the stage, by flatness and the computed free ranks. The infimum and supremum extensions, including their additivity and compatibility with this formula, are Faltings’ normalized-length formalism [9], in the formulation of [6]. The same residue field implies that \(A_n\)-length and \(A\)-length agree on finite-length \(A_n\)-modules.

In mixed characteristic \(\Lambda\) is \(p\)-torsion-free, and the natural map induces \(\Lambda/p^a\Lambda\simeq A_\infty/p^aA_\infty\) for every \(a\geq1\). A module killed by \(\mathfrak m_A^a\) is killed by \(p^a\). Its submodules and finite generating sets are therefore identical over the two bases. In computing the infimum over finitely presented covers of such a module, one may replace a cover \(V\) by \(V/\mathfrak m_A^aV\); this is still finitely presented. These covers also correspond over the identical quotient rings, proving the last assertion. ◻

Proof. A finite set of elements of a direct sum is contained in a finite subsum. Additivity and (3) give the assertion for direct sums; sums and directed colimits are quotients of direct sums. For an increasing union, every finite generating set lies in one stage. Finally a finitely generated submodule of \(\varinjlim_a V_a\) is contained in the image of every sufficiently late \(V_a\), by lifting its generators to a common stage. Monotonicity bounds its length by \(\lambda_\infty(V_a)\) at all such stages. Apply (3). The common annihilator ensures throughout that these modules belong to the length category. ◻

Proof. We give the positive-characteristic construction and then specify the mixed-characteristic literature inputs.

Assume first that \(D\) has characteristic \(p\), and put \(C_D=D_{\mathrm{perf}}=\bigcup_n D^{1/p^n}\) inside a fixed algebraic closure of its fraction field. Here \(\Lambda=A_{\mathrm{perf}}\). Write \(K=\operatorname{Frac}(A)\), \(L=\operatorname{Frac}(D)\), and \(K'=K_{\mathrm{perf}}\). For \(q=p^n\) put \[B_q=\Lambda\otimes_{A^{1/q}}D^{1/q}.\] Flatness of \(\Lambda\) over \(A^{1/q}\) and torsion-freeness of \(D^{1/q}\) embed this tensor product into \(K'\otimes_{K^{1/q}}L^{1/q}\). Generic separability makes the latter a field, by linear disjointness from a purely inseparable extension. Thus \(B_q=\Lambda D^{1/q}\subseteq C_D\), and these subrings increase to \(C_D\).

Let \(H\) be any \(\mathfrak d\)-primary ideal and set \(V_q=B_q/HB_q\) and \(W_q=\operatorname{im}(V_q\to C_D/HC_D)\). Write \(H^{[q]}=(a^q:a\in H)\) for its Frobenius power. Finite-stage normalization and the \(q\)-th-power isomorphism give \[\lambda_\infty(V_q)=q^{-h}\operatorname{length}_D(D/H^{[q]}).\] Although \(B_q\subseteq C_D\), the map on these quotients need not be injective. To recover the length of \(C_D/HC_D\) from the displayed formula, we control the kernels of \(V_q\to W_q\) by a single trace denominator. There is a nonzero \(g\in A\) with \[ gC_D\subseteq B_1,\qquad g^{1/q}C_D\subseteq B_q. \tag{7}\] Indeed, choose a \(K\)-basis \(b_1,\ldots,b_r\) of \(L\) in \(D\) and its trace-dual basis \(\beta_1,\ldots,\beta_r\). A common denominator \(g\in A\setminus\{0\}\) makes every \(g\beta_j\) lie in \(D\). The field \(LK'\) is perfect and equals \(L_{\mathrm{perf}}\). The ring \(\Lambda\) is normal, since an integral equation and the fractions occurring in it descend to a normal stage \(A^{1/q}\). For \(c\in C_D\), each \(\operatorname{Tr}_{LK'/K'}(g\beta_jc)\) is integral over \(\Lambda\) and belongs to \(K'\), hence belongs to \(\Lambda\). The trace-dual expansion of \(gc\) proves the first inclusion in (7). Taking \(q\)-th roots proves the second.

The kernel of \(V_q\to W_q\) is killed by \(g^{1/q}\), by (7). Kernel and cokernel of an endomorphism of a finite-length object have equal length. Applied to multiplication by \(g^{1/q}\), this yields \[\lambda_\infty\ker(V_q\to W_q) \leq q^{-h}\operatorname{length}_D D/(H^{[q]},g)=O(q^{-1}).\] For the estimate, if \(H\) has \(s\) generators then \(H^{s(q-1)+1}\subseteq H^{[q]}\). Thus \(H^{[q]}\) contains a power of \(\mathfrak d\) with exponent \(O(q)\), whereas \(\dim D/(g)\leq h-1\) if \(g\) is a nonunit. Hilbert–Samuel growth gives the asserted bound; for a unit \(g\) the quotient is zero. The \(W_q\) increase to \(C_D/HC_D\) and are uniformly primary-torsion. The Hilbert–Kunz limit and Lemma 6 show \[ \lambda_D(C_D/HC_D)=e_{\mathrm{HK}}(H,D). \tag{8}\] Here we use Monsky’s existence theorem for Hilbert–Kunz multiplicity [20]; see also [19].

We have identified primary-quotient lengths with Hilbert–Kunz multiplicities. For a parameter ideal, it remains to obtain the Hilbert–Samuel value and zero positive Koszul homology. If \(z\) is a parameter system of \(D\), finite-stage flatness gives \[\lambda_\infty H_i(z;B_q) =q^{-h}\operatorname{length}_D H_i(z_1^q,\ldots,z_h^q;D).\] The Koszul complex \(K(z;D)\) has finite free terms in degrees \(0,\ldots,h\) and finite-length homology. The Frobenius homology theorem of Roberts, in the form [19], says that the displayed quantity tends to zero for \(i>0\); the original result is [24]. Its hypotheses require precisely a complete characteristic-\(p\) ring and a finite-free complex of length its dimension with finite-length homology. Filtered colimits commute with homology, and these Koszul homology modules have the common annihilator \((z)\). Lemma 6 proves (4), also with \(z\) replaced by \(z^v\). Finally the Koszul Euler characteristic formula gives \[\sum_{i=0}^h(-1)^i\operatorname{length}_D H_i(z_1^q,\ldots,z_h^q;D) =q^h e((z),D).\] After division by \(q^h\), the positive-degree terms tend to zero. Together with (8) and the multiplicity power rule this proves (5).

Now suppose \(D\) has mixed characteristic. Let \(A_\infty\) be the \(p\)-adic completion of \(\Lambda\) and define \[C_D=(D\otimes_A A_\infty)_{\mathrm{perfd}},\] the perfectoidization of [6], built on Bhatt–Scholze’s construction [4]. The stated hypotheses there are exactly that \(D\) is a complete mixed-characteristic local domain with perfect residue field and the chosen finite Witt power-series base \(A\). Choose \(g\in A\setminus\{0\}\) so that \(A[1/g]\to D[1/g]\) is finite etale. Corollary 4.0.4 of [6] annihilates positive Koszul homology for every parameter system of \(A_n\otimes_A D\) by the almost ideal \((g)_{\mathrm{perfd}}\). At \(n=0\) this includes every parameter system of \(D\). Proposition 4.0.10 of that paper makes these uniformly \(\mathfrak m_A\)-power-torsion homology modules have normalized length zero. Proposition 4.0.14, again at \(n=0\), identifies the length of the parameter quotient with its Hilbert–Samuel multiplicity. These two results, applied also to \(z^v\), prove (4) and (5) for this single algebra \(C_D\). Lemma 5 permits restriction to the uncompleted tower throughout.

In either characteristic, a module killed by a power of \(\mathfrak d\) is killed by a power of \(\mathfrak m_A\), so all claimed lengths are defined. Additivity and the tower properties are those already established. Lastly the multiplicity rank formula, and equality of the residue fields, give \(e((t),D)=\operatorname{rank}_A D\,e((t),A)=\operatorname{rank}_A D\). This proves (6). ◻

Proof. For \(V=k\), the first two assertions are (4) and (6), using the Koszul resolution over regular \(A\). Induct on a composition series of \(V\). The Tor long exact sequences and length additivity discard all positive-degree terms and show that degree-zero length increases by \(r\) at each step. All modules here have bounded primary torsion.

For the last assertion choose a parameter ideal \(H\) of \(D\). Tensor a composition series of \(D/H\) with \(C_D\). Right exactness gives a filtration of \(C_D/HC_D\) whose factors are quotients of \(C_D/\mathfrak dC_D\). Therefore \[0<e(H,D)=\lambda_D(C_D/HC_D) \leq\operatorname{length}_D(D/H)\lambda_D(C_D/\mathfrak dC_D).\] This proves the required strict positivity. ◻

Proof. Let \(f:X\to Y\) be finite faithfully flat. Since the schemes are Noetherian, \(f_*\mathcal O_X\) is a vector bundle of positive rank. Restriction of scalars preserves perfect coefficient modules and their supports: locally the scalar-extended coefficient algebra is a finite projective module over the original one. It therefore defines a transfer. Finite-flat base change and the projection formula give natural transformations of presheaves on \(Y\)-schemes \[\mathcal{L}\xrightarrow{f^*}\mathcal{L}_X \xrightarrow{\operatorname{tr}_f}\mathcal{L}, \qquad \mathcal{L}_X(W)=\mathcal{L}(W\times_Y X),\] whose composite is tensoring with the pullback of \(f_*\mathcal O_X\).

Work first on a Zariski open of \(Y\) on which that vector bundle is free of rank \(a>0\), and fix a trivialization there. Additivity of \(K\)-theory identifies this composite with multiplication by \(a\). The same trivialization pulls back on every \(Y\)-scheme, so this identity is natural on the whole presheaf diagram. Rationally, \(a^{-1}\operatorname{tr}_f\) makes \(\mathcal{L}\) a retract of \(\mathcal{L}_X\).

After base change to \(X\), the augmented Čech nerve of \(f\) has a section and hence an extra degeneracy. Its image under any presheaf has split descent. Thus \(\mathcal{L}_X\) satisfies descent for \(f\). The descent comparison for \(\mathcal{L}\) is a retract of this equivalence, and is itself an equivalence. Finally, both its source and its totalization are Zariski sheaves. The conclusion therefore glues over the trivializing cover of \(Y\). All constructions commute with base change. ◻

Proof. Let \(\tau\) be the topology generated by cdh covers and finite faithfully flat covers. These cover families are stable under base change. By Lemma 10 and cdh descent, \(\mathcal{L}\) is a \(\tau\)-sheaf. We show that every finite surjection \(X\to Y\) is a \(\tau\)-cover.

Reductions and finite closed decompositions are cdh-local operations. For clarity, if a reduced scheme is a union \(Y=Y_1\cup Y_2\) of reduced closed subschemes, the proper map \(Y_1\to Y\) is an isomorphism off \(Y_2\) and gives the abstract blowup square with intersection \(Y_1\cap Y_2\). This reduces closed unions to their pieces. We may consequently reduce to integral \(Y\), with induction on dimension for proper closed subsets.

Choose a reduced irreducible component \(X_0\) of \(X\) dominating \(Y\). The map \(X_0\to Y\) is finite and surjective. On a dense open \(U\subset Y\) it is finite flat of positive rank. Flattening by blowup, applied to \(\mathcal O_{X_0}\), gives a blowup \(Y'\to Y\), an isomorphism over \(U\), for which the strict transform \(X'_0\to Y'\) is flat and finitely presented [28]. The strict transform is a closed subscheme of \(X_0\times_Y Y'\), so it is finite over \(Y'\). The blowup \(Y'\) is integral, and the image of \(X'_0\) contains its generic point. Finiteness makes that image closed. Thus \(X'_0\to Y'\) is surjective, and is a finite faithfully flat cover factoring through \(X\times_Y Y'\).

Let \(Z\subsetneq Y\) be the blowup center. The family \(\{Y'\to Y,Z\to Y\}\) is a cdh cover. By induction on dimension, the finite surjection \(X\times_Y Z\to Z\) is a \(\tau\)-cover. Together with \(X'_0\to Y'\), these maps give a \(\tau\)-covering family of \(Y\) that factors through \(X\). Hence \(X\to Y\) itself generates a covering sieve. The induction starts in dimension zero, where a reduced integral base is a field and any nonzero finite algebra is finite faithfully flat. This proves the assertion, and therefore the required Čech descent. ◻

Proof of Theorem 9. Write \(X=\operatorname{Spec}T\) and \(S=\operatorname{Spec}A\). On finite-type \(S\)-schemes consider the natural comparison \[ \mathcal{L}(Y)\longrightarrow \mathcal{L}(X\times_S Y)^{hG}. \tag{15}\] Both sides are sheaves for the topology used in Lemma 11: base change preserves the cover families, and homotopy fixed points commute with limits. Applying finite-surjection descent to \(X\to S\), it suffices to prove (15) on the nonaugmented terms of its Čech nerve. Over each such term, the pulled-back \(X\)-cover admits a section, given by a projection to one of its \(X\)-coordinates. The geometric transitivity hypothesis is stable under base change. We are therefore reduced to covers admitting a section.

We prove this case by induction on the dimension of \(Y\). Nilinvariance and the closed-union squares described in the proof of Lemma 11 reduce the base to an integral reduced scheme. Indeed both sides of (15) have these descent properties; the component reduction uses finitely many closed pieces and lower-dimensional intersections.

Let \(s:Y\to X\times_S Y\) be a section. Its finitely many \(G\)-translates cover the source on points. To see this, extend the residue field at any point of \(Y\) to an algebraic closure and use transitivity on that geometric fiber. Each translate is a closed immersion, because the source is separated over \(Y\), and each factors through the reduction \[V=(X\times_S Y)_{\mathrm{red}}.\] Retain the distinct translated sections and index them by a transitive finite \(G\)-set \(I\). Two distinct sections have a proper closed coincidence locus in \(Y\): their equalizer is closed, and agreement at the generic point of integral reduced \(Y\) would imply agreement everywhere. Let \(Z\subsetneq Y\) be the reduced union of these coincidence loci, with \(Z=\varnothing\) if there is only one section.

The map \[q:W=\coprod_{i\in I}Y\longrightarrow V\] is finite and surjective. Over \(Y\setminus Z\) the section images are pairwise disjoint and cover the reduced scheme \(V\). Their disjoint union is therefore isomorphic to \(V\) there. Consequently \(q\) is a proper modification and the square \[\begin{array}{ccc} W\times_Y Z&\longrightarrow&W\\ \big\downarrow&&\big\downarrow\\ V\times_Y Z&\longrightarrow&V \end{array}\] is an abstract blowup square. It is \(G\)-equivariant.

The \(G\)-action on \(W\) merely permutes its copies of \(Y\). The canonical restriction maps give a \(G\)-equivariant equivalence \(\mathcal{L}(W)\simeq\prod_{i\in I}\mathcal{L}(Y)\). A stabilizer of one section acts identically on that copy, including on the coefficient algebra pulled back from \(Y\). This is a canonical identity of coefficient-bearing \(Y\)-schemes, so the action is the ordinary permutation action, with coherently trivial stabilizer action. The same statement holds after restriction to \(Z\). Since \(\mathcal{L}\) is rational, averaging over a finite group is exact. It follows that \[ \mathcal{L}(W)^{hG}\simeq\mathcal{L}(Y), \qquad \mathcal{L}(W\times_Y Z)^{hG}\simeq\mathcal{L}(Z). \tag{16}\] For example, if \(I=G/G_s\), the first fixed-point spectrum is \(\mathcal{L}(Y)^{hG_s}\) for the trivial \(G_s\)-action, and is \(\mathcal{L}(Y)\). This identity holds without boundedness: the trivial \(\mathbb{Q}[G_s]\)-module \(\mathbb{Q}\) is projective, so derived invariants are ordinary invariants even on unbounded rational complexes.

Apply \(\mathcal{L}\) to the abstract blowup square and take homotopy \(G\)-invariants. The inductive hypothesis over \(Z\) identifies \[\mathcal{L}(V\times_Y Z)^{hG}\simeq\mathcal{L}(Z).\] Here nilinvariance allows one to pass between \(V\times_Y Z\) and the corresponding reduced or unreduced pullback of the original cover. Combining this with (16), the pullback square gives \[\mathcal{L}(V)^{hG} \simeq\mathcal{L}(Y)\mathop{\times}_{\mathcal{L}(Z)}\mathcal{L}(Z) \simeq\mathcal{L}(Y).\] Nilinvariance restores \(X\times_S Y\) in place of \(V\). In dimension zero there are no proper nonempty coincidence loci, and the same proof is simply the calculation for a disjoint union of section copies. This starts the induction and proves (15).

Taking \(Y=S\) and using (14) yields (9). Finally, exact rational invariants give \[\pi_0\Bigl(\bigl(K^J(H\otimes_A^{\mathbf L}T)_\mathbb{Q}\bigr)^{hG}\Bigr) =\bigl(K_0^J(H\otimes_A^{\mathbf L}T)\otimes\mathbb{Q}\bigr)^G,\] which proves (10), in fact as an isomorphism. ◻

Proof. The Koszul resolution on \(z_1,\ldots,z_e\) resolves \(k\) over \(A\), so (17) with \(v=1\) proves the assertion for \(V=k\). Induct on the length of \(V\) using a composition series. In the long exact Tor sequence associated to a short exact sequence with quotient \(k\), nonnegativity and additivity show that every positive Tor term has length zero. The image of the connecting map into degree zero also has length zero. Additivity in degree zero then gives the asserted factor \(r\). This simultaneously proves finiteness at every step. ◻

Proof. We first establish finiteness and additivity, then compare the evaluations on pullbacks before applying descent to the norm class. Represent \(P\) by a bounded finite free \(T\)-complex. Its ordinary homology is finite and supported at the closed point, hence has finite length. For the \(C\)-evaluation, choose a common \(v\) such that each \(z_j^v\) acts nullhomotopically on \(P\). This is possible by the support criterion above. Set \(K_v=K(z_1^v,\ldots,z_e^v;T)\). Successively splitting the cones of these nullhomotopic actions gives \[ K_v\otimes_T P\otimes_T C \simeq \bigoplus_{j=0}^e (P\otimes_T C)[j]^{\oplus\binom ej}. \tag{20}\] The homotopies may be tensored with \(C\) and with the other Koszul factors, so this splitting respects the module structure on which \(\lambda\) is defined.

In the opposite order, taking Koszul homology first gives a finite spectral sequence whose terms are finite direct sums of \(H_j(z_1^v,\ldots,z_e^v;C)\). Their lengths are finite by (17). Hence every homology module on the left side of (20) has finite length. The splitting and nonnegativity then give finiteness for every \(H_i(P\otimes_T C)\). The homology modules and spectral-sequence terms to which \(\lambda\) is applied are annihilated by fixed powers of \(\mathfrak{m}_A\). The Euler sums are therefore defined, and the long exact homology sequence of a triangle proves their additivity.

Let \(Q\) now be a perfect \(A\)-complex supported at \(\mathfrak{m}_A\). Its homology modules have finite length. The finite hyper-Tor spectral sequence for \(Q\otimes_A^{\mathbf L}C\), together with Lemma 12, gives \[ \chi_C(Q\otimes_A^{\mathbf L}T) =r\sum_j(-1)^j\operatorname{length}_A H_j(Q). \tag{21}\] Indeed every positive Tor row has length zero and the zeroth row has the displayed scaled lengths.

For the ordinary evaluation, choose a bounded finite free \(A\)-resolution \(L_\bullet\) of \(T\), possible because \(A\) is regular. Its alternating rank is \(r\), as is seen after tensoring with the fraction field of \(A\). For every finite-length \(A\)-module \(V\), the finite complex \(V\otimes_A L_\bullet\) therefore gives \[\sum_i(-1)^i\operatorname{length}_A\operatorname{Tor}_i^A(V,T) =\sum_i(-1)^i\operatorname{rank}_A(L_i)\operatorname{length}_A(V) =r\operatorname{length}_A(V).\] The hyper-Tor spectral sequence for \(Q\otimes_A^{\mathbf L}T\) consequently yields \[ \chi_T(Q\otimes_A^{\mathbf L}T) =r\sum_j(-1)^j\operatorname{length}_A H_j(Q). \tag{22}\] We have used that finite-length \(T\)-modules have the same \(A\)-length and \(T\)-length, because both residue fields are \(k\). Equations (21) and (22) prove agreement on classes pulled back from \(A\).

Apply Theorem 9 with \(H=A\) and \(J=\mathfrak{m}_A\). Every invariant class in \(K_0^{\mathfrak{m}_A}(T)\otimes\mathbb{Q}\) is the pullback of a rational class downstairs. Extending the two Euler homomorphisms \(\mathbb{Q}\)-linearly, they therefore agree on that invariant subspace. The class \(\sum_{g\in G}[g^*P]\) is invariant, which gives (19). ◻

Proof. First let \(X\in\mathcal S_S\), \(Y\in\mathcal N_S\), and \(f:X\to Y\). Choose \(v\) such that \(z_j^v\operatorname{id}_X=0\) for every \(j\). Centrality gives \(z_j^vf=0\). The exact Hom sequence of the triangle \[\operatorname{fib}(z_j^v:Y\to Y)\longrightarrow Y \xrightarrow{z_j^v}Y\] therefore lifts \(f\) through the first object. Repeat this operation for all the parameters, always using the zero action on the original source \(X\). The result factors \(f\) through \(K_v\otimes_A Y[-e]\). This object belongs to \(\mathcal N_S\) by thickness, and to \(\mathcal S_S\) by the usual Koszul homotopies. Thus every map from a supported object to a small object factors through a supported small object.

Here is the resulting roof argument. A morphism between \(X,Y\) in \(\mathcal S_S\), computed in \(\mathcal D_S/\mathcal N_S\), has a representative \[X\xleftarrow{s}Z\xrightarrow{f}Y, \qquad \operatorname{cofib}(s)=N_0\in\mathcal N_S.\] In the triangle \(Z\to X\xrightarrow{u}N_0\), factor \(u\) through \(N_1\in\mathcal S_S\cap\mathcal N_S\) by the preceding paragraph. Let \(Z_1\) be the fiber of \(X\to N_1\). The induced morphism of triangles gives \(c:Z_1\to Z\) and \(s_1:Z_1\to X\) with \(sc=s_1\). Now \(Z_1\in\mathcal S_S\) and \(\operatorname{cofib}(s_1)=N_1\), so \(X\xleftarrow{s_1}Z_1\xrightarrow{fc}Y\) is a representative entirely in the supported subquotient. This proves fullness.

For faithfulness, suppose a map \(f:Z\to Y\) between supported objects becomes zero in \(\mathcal D_S/\mathcal N_S\). The calculus of Verdier roofs gives a map \(t:W\to Z\) with small cone and \(ft=0\). The triangle of \(t\) makes \(f\) factor through that small cone. Factoring its first leg through a supported small object as above shows that \(f\) is already zero in the supported subquotient. Apply this to the numerator of any roof to obtain faithfulness.

Finally take \(X\in\mathcal P_S^J\). It is a retract of \(q_S(Y)\) for some \(Y\in\mathcal D_S\). Choose \(v\) with all \(z_j^v\) acting as zero on \(X\). The cone of each such zero map splits, so \(X\) is a retract of \(K_v\otimes_A X\). The latter is a retract of \(q_S(K_v\otimes_A Y)\), whose lift is uniformly supported. Hence \(X\) belongs to the idempotent completion of the fully faithful supported subquotient. The reverse inclusion follows because support is closed under retracts. ◻

Proof. We first bound homology on \(\mathcal S_T\), then show that it kills \(\mathcal S_T\cap\mathcal N_T\), and finally extend it to the idempotent-completed quotient.

Begin with a bounded free sequence \(F\) in \(\mathcal S_T\). Choose uniform powers annihilating its identity. Splitting their cones gives an isomorphism in the homotopy category \[ K_v\otimes_A F\simeq \bigoplus_{j=0}^e F[j]^{\oplus\binom ej}. \tag{25}\] For ordinary length, every Koszul homology \(H_j(K_v\otimes_A T)\) has finite length. Taking Koszul homology first in the bounded double complex bounds the homology lengths of \(K_v\otimes_A F_n\) by fixed constants times the ranks of the terms of \(F_n\). They are therefore \(O(s_n)\). For \(C\), the same spectral sequence uses Equation (23): positive Koszul rows have length zero and the zero row has finite length. Additivity and nonnegativity give the same bound. The splitting in Equation (25) bounds the homologies of \(F_n\) themselves. They have a common primary annihilator, because a power of every \(z_j\) annihilates them.

A general object \(X\in\mathcal S_T\) may be a retract in \(\mathcal D_T\) of a bounded free sequence \(F\) that is not supported. Choose \(v\) killing the parameter actions on \(X\). Then \(X\) is a retract of \(K_v\otimes_A X\), which is a retract of the uniformly supported bounded free sequence \(K_v\otimes_A F\). The preceding bounds apply to this sequence and hence to the homology summands defining \(X\). This is a retract before taking the rank quotient. Thus homology on \(\mathcal S_T\) takes values in the indicated abelian length quotients.

Fix \(v\). Koszul tensoring a rank-small free generator has homology of normalized length zero by the same bounds. The class of objects \(Y\in\mathcal D_T\) with this property for \(K_v\otimes_A Y\) is thick: use homology long exact sequences, nonnegativity, and retracts. It therefore contains \(\mathcal N_T\). If \(X\in\mathcal S_T\cap\mathcal N_T\), choose \(v\) killing its parameter actions. Then \(X\) is a retract of \(K_v\otimes_A X\), so its homology is zero in both length quotients.

The homology functors consequently invert maps whose cones are supported and small. They descend to \(\mathcal S_T/(\mathcal S_T\cap\mathcal N_T)\), preserving homology long exact sequences. Now consider summands after quotienting. By Lemma 16, every object of \(\mathcal P_T^J\) is a summand of an object of this subcategory. Extend a homology functor to a summand \((X,a)\) by taking \(\operatorname{im}\mathsf H_i(a)\) in its abelian target. This does not require a sequence of chain projectors lifting \(a\). Distinguished triangles in an idempotent completion are summands of distinguished triangles in the original triangulated category [1]. Taking images of compatible idempotents preserves exact sequences in an abelian category, so the extended functors are still homological.

Each object is a summand of a fixed bounded uniformly supported Koszul lift. Its homology objects therefore vanish outside a finite range, and their lengths are bounded by those of that lift. This proves finiteness, nonnegativity, and boundedness of the \(b_i\). The finite long exact sequences prove additivity of the Euler sums.

For the last assertion, the splitting in Equation (25) now takes place in \(\mathcal P_T\), since the chosen powers act as zero there. Applying the extended homological functors gives the left side of Equation (24). Its Koszul tensor has a uniformly supported lift, whose actual homology computes the right side. This proves the identity without defining its coefficients by formal division. ◻

Proof. The Koszul and composition-series proof of Lemma 12 applies here: it uses regularity and the length hypothesis, but not completeness. A power of \(J\) annihilating \(V\) also annihilates its derived tensor homology. ◻

Proof. The two evaluations recover the factor \(r\) differently: the ordinary one uses the alternating rank of a resolution of \(T\), while the normalized one uses the vanishing of positive Tor lengths. To make both calculations on an arbitrary summand, we descend their finite spectral sequences through supported Verdier roofs before taking idempotent images.

Let \(Y\) be a uniformly supported bounded free sequence over \(A\) and let \(a\) be an idempotent endomorphism of its image in \(\mathcal S_A/(\mathcal S_A\cap\mathcal N_A)\). Every supported object of \(\mathcal P_A\) has this form as a summand, by Lemma 16: in its proof one may enlarge the initial lift to a bounded free sequence before taking the Koszul tensor. Put \[V_b=\operatorname{im}\mathsf H_b^A(a) \quad\text{in }\overline{\mathcal A}_A.\] These images exist in the abelian quotient even when \(a\) has no chain-level projector representing it.

Choose a fixed bounded finite free \(A\)-resolution \(P_\bullet\) of \(T\); such a resolution exists because \(A\) is regular. The double complex \(Y\otimes_A P_\bullet\) computes \(Y\otimes_A^{\mathbf L}T\). Filtering by the degree of \(P\) gives a finite spectral sequence with first page \[ E^1_{u,b}=H_b(Y)\otimes_A P_u. \tag{26}\] It is natural in homotopy-category maps from this page onward: homotopies in \(Y\) preserve the filtration and act trivially on the first page. On retracts in the prequotient category the same construction is defined by taking the corresponding idempotent images on every page and on the abutment filtration. Its terms and finite abutment filtration can be regarded as \(A\)-modules. The abutment’s ordinary \(A\)-length equals its \(T\)-length, since the residue fields agree.

A map with supported small cone is an isomorphism on \(H_b(Y)\) in \(\overline{\mathcal A}_A\). It is thus an isomorphism on Equation (26), on every subsequent page, and on the finite filtered abutment in the length quotient. The spectral sequence therefore descends through all Verdier roofs in the supported subquotient. The natural augmentation \(\operatorname{Tot}(Y\otimes_A P_\bullet)\to Y\otimes_A T\) identifies its abutment with the homology of scalar extension, and this identification also descends through those roofs. Thus the induced action of \(a\) on the abutment is precisely the scalar-extended idempotent, not a separately chosen action.

In particular \(a\) acts as an actual idempotent on every page and on that filtered abutment. The differentials commute with this action. Taking its images commutes with passage to homology, since an idempotent decomposes a complex in an abelian category into its image and complementary image. It also preserves the finite exact sequences defining the abutment filtration: for a filtration preserved by an idempotent \(a\), one has \(\operatorname{im}(a)\cap F_u=a(F_u)\) and hence \(\operatorname{gr}_u\operatorname{im}(a) =\operatorname{im}(\operatorname{gr}_u a)\). Consequently the Euler length on the scalar-extended summand is the Euler length of the images on the first page. They are \(\operatorname{rank}_A(P_u)\) copies of \(V_b\), and hence this value is \[ \left(\sum_u(-1)^u\operatorname{rank}_A P_u\right) \sum_b(-1)^b\overline\operatorname{length}_A V_b =r\sum_b(-1)^b\overline\operatorname{length}_A V_b. \tag{27}\] The equality of alternating ranks with \(r\) follows by tensoring the resolution with the fraction field of \(A\).

For \(C\), use the finite hyper-Tor spectral sequence \[ {}^CE^2_{u,b}=\operatorname{Tor}_u^A(H_b(Y),C) \ \Longrightarrow\ H_{u+b}(Y\otimes_A^{\mathbf L}C). \tag{28}\] It can be constructed from the functorial finite Postnikov filtration of \(Y\). It has bounded \(b\) and \(0\leq u\leq e\), so there is no unbounded convergence issue. Lemma 18 makes every positive Tor term zero in \(\overline{\mathcal A}_C\). It also implies that \(V\mapsto V\otimes_A C\) induces an exact functor \[\overline{\mathcal A}_A\longrightarrow \overline{\mathcal A}_C\] that multiplies length by \(r\). Indeed, on a short exact sequence its possible failure of left exactness is the image of a positive Tor module of zero length. Negligible-length sequences map to negligible-length sequences by the same lemma. This explains both exactness after the Serre quotient and descent of the functor to that quotient.

It follows that a supported-small-cone map induces isomorphisms on the second page of Equation (28) and its finite filtered abutment in the length quotient. Thus this spectral sequence too is functorial through the supported Verdier roofs. Taking the images of the same idempotent \(a\) gives only the zero Tor row, namely the images of \(V_b\) under this exact tensor functor. Their Euler length is \(r\sum_b(-1)^b\overline\operatorname{length}_A V_b\), exactly the value in Equation (27). Finally this last sum is \(\chi_A(Y,a)\) by construction. This proves both equalities for every summand and hence for every \(X\in\mathcal P_A^J\). ◻

Proof. The sequence \(E_S=(S^{s_n})_n\) is an additive generator. Indeed, if \(a_n=O(s_n)\), there is a single integer \(c\) such that \(a_n\leq cs_n\) for every \(n\), after increasing \(c\) to handle the finitely many exceptional indices. Coordinate inclusions and projections make \((S^{a_n})_n\) a summand of \(E_S^c\). Consequently \[\mathcal D_S\simeq\operatorname{Perf}(B_S), \qquad B_S=\prod_n M_{s_n}(S).\]

The \(A\)-module underlying \(B_A\) is flat. One can check this by the ideal criterion: every finitely generated ideal of the Noetherian ring \(A\) is finitely presented, so its tensor product with this product of finite free modules is the corresponding product of its tensor products; each required injection is then a product of injections. Moreover, \(T\) is finitely presented over \(A\), so tensoring with \(T\) commutes with that product. Thus \[ B_A\otimes_A^{\mathbf L}T \simeq B_A\otimes_A T \cong B_T. \tag{29}\]

The same product argument makes every morphism module between the original free sequences flat over \(A\). Hence the explicit dg quotient of this degree-zero category by its full rank-small subcategory computes the derived quotient [7]. Its morphisms are sums of paths formed from tensor products of the original morphism modules and contracting homotopies in homological degree \(1\). They are concentrated in nonnegative degrees and degreewise \(A\)-flat, hence are \(K\)-flat. Passing to the perfect hull and idempotent completion kills exactly \(\mathcal N_A\). The image of \(E_A\) remains a generator, and its endomorphism algebra \(H=\operatorname{End}_{\mathcal P_A}(q_AE_A)\) is connective, giving \(\mathcal P_A\simeq\operatorname{Perf}(H)\). This assertion concerns the chosen generator, not arbitrary shifts of perfect objects.

Since \(T\) is finitely presented over \(A\), tensoring each original morphism module with \(T\) gives the corresponding module of morphisms between free \(T\)-sequences. Tensoring the path formula therefore gives exactly the \(T\)-linear dg quotient by the rank-small free sequences. The same flatness criterion applies over the Noetherian ring \(T\). The upstairs endomorphism algebra is consequently \(H\otimes_A T\simeq H\otimes_A^{\mathbf L}T\), and its perfect hull and idempotent completion are \(\mathcal P_T\). This also agrees with the preservation of exact quotients under scalar extension [18]. All constructions are centrally \(A\)-linear and preserve central localization. ◻

Proof. Lemma 20 identifies the two supported categories with the supported perfect categories of \(H\) and \(H\otimes_A^{\mathbf L}T\), where \(H\) is connective. The identification over \(T\) is compatible with the \(G\)-action: scalar twisting acts on the \(T\) factor in the coefficient algebra. Theorem 9 therefore gives a surjection \[K_0(\mathcal P_A^J)\otimes\mathbb{Q} \longrightarrow \bigl(K_0(\mathcal P_T^J)\otimes\mathbb{Q}\bigr)^G.\] The norm class \(\sum_g[gX]\) is invariant. The homomorphism \(\chi_{\mathrm{ord}}-\chi_C\), extended rationally to the Grothendieck group, vanishes on the image from \(A\) by Proposition 19. Applying it to a preimage of this norm class proves the assertion. ◻

Proof. The equicharacteristic case is Serre’s positivity theorem [25]. We may therefore assume that the residue characteristic is positive. Completion, followed by the extension in Lemma 3, reduces to a complete regular local ring with algebraically closed residue field. For clarity, a flat local extension \(R\to R_1\) with \(\mathfrak{m}_RR_1=\mathfrak{m}_{R_1}\) preserves the length of every finite-length module: a composition series becomes a filtration with factors \(R_1/\mathfrak{m}_{R_1}\). Flat base change identifies the Tor modules after extension. The same argument applied to \(M/\mathfrak{m}_R^nM\) preserves the Hilbert–Samuel function, and hence the dimension of \(M\); it applies equally to \(N\). It also preserves regularity, since it preserves the dimension and the minimal number of generators of the maximal ideal.

We use the vanishing theorem of Roberts and Gillet–Soulé in its general regular local form [22, 11]: \[ \chi^R(V,W)=0 \quad\text{if}\quad \operatorname{length}_R(V\otimes_R W)<\infty \quad\text{and}\quad \dim V+\dim W<d. \tag{30}\] Take prime filtrations of \(M\) and \(N\). All pairs of factors still have finite-length tensor product, since their supports are contained in the respective original supports. Additivity of the Euler characteristic and Equation (30) discard every pair except those having both original dimensions. There are top-dimensional factors in each filtration, so at least one such pair remains. Proving strict positivity for each remaining pair proves it for \(M,N\). Their primes satisfy \(\sqrt{P+Q}=\mathfrak{m}_R\).

Finally, a zero-dimensional prime quotient is \(k\), and its complementary \(d\)-dimensional prime quotient is \(R\), since a regular local ring is a domain. For that pair the characteristic equals \(\operatorname{length}_R(k)=1\). Thus we may assume \(h,l>0\). ◻

Proof. Take a finite \(R\)-free resolution of \(E'\) and extend it to \(D'\). The resulting bounded free complex is supported at the closed point: after localizing its homology away from \(\mathfrak{m}_R\), the supports of \(D'\) and \(E'\) are disjoint. The same holds after every twist, whose kernel on \(R\) is still \(P\). Lemma 13 supplies finite normalized homology lengths and equality of the ordinary and normalized Euler evaluations after summing all conjugates.

Applying \(\sigma\) entrywise gives a semilinear isomorphism from the untwisted complex to its conjugate, so all ordinary Euler characteristics in that sum are equal to \(\chi^R(D',E')\). Ordinary lengths over \(D'\) and over \(R\) agree here because both residue fields are \(k\).

Choose an injection \(D^{a_D}\to D'\) with torsion cokernel. The cokernel has dimension less than \(h\), and therefore its intersection characteristic with \(E'\) vanishes by Equation (30). Additivity gives \[\chi^R(D',E')=a_D\chi^R(D,E').\] The analogous argument for \(E\subset E'\) gives \(\chi^R(D,E')=a_E\chi^R(D,E)\). The norm comparison now yields Equation (32). ◻

Proof. The complexes in question have compatible \(C\)-actions. The Koszul homology \(H_q(x;C)\) is annihilated by \(x\) and by \(P\), and is therefore bounded-primary-torsion. For \(q>0\) its length is zero by Equation (34). A free \(B\)-resolution of \(L\), with each term of finite rank, gives \[\lambda\bigl(\operatorname{Tor}_p^B(H_q(x;C),L)\bigr)=0\quad(q>0).\] Indeed every such homology is a subquotient of a finite direct sum of a length-zero module. For \(q=0\) these lengths are finite because \(\lambda(U)<\infty\). The change-of-rings spectral sequence \[\operatorname{Tor}_p^B\bigl(\operatorname{Tor}_q^R(C,B),L\bigr) \ \Longrightarrow\ \operatorname{Tor}_{p+q}^R(C,L)\] now gives the asserted equality of lengths. Only finitely many terms enter each total degree.

For the sharper bound, choose an \(R\)-free resolution \(P_L\) of \(L\) in degrees \(0,\ldots,d\) and consider \[K(x;R)\otimes_R C\otimes_R P_L.\] Taking Koszul homology first, all positive rows have length zero, and the zero row is \(U\otimes_R P_L\). Consequently the total homology has length zero in degrees greater than \(d\). On the other hand, \(xL=0\), so in the derived category of \(R\)-modules \[K(x;R)\otimes_R^{\mathbf L}L \simeq\bigoplus_{j=0}^h L[j]^{\binom hj}.\] Tensoring this identity derivedly with \(C\) preserves the splitting as a statement about \(C\)-complexes. The total homology in degree \(i+h\) therefore contains \(\operatorname{Tor}_i^R(C,L)\) as a direct summand. If \(i>l=d-h\), that total degree exceeds \(d\), proving the bound. This argument also shows directly that all the displayed lengths are finite. ◻

Proof. The algebra \(S\) is finitely presented as a \(\Lambda\)-module and has finite normalized length. Every \(S_j\) is finite over the twisted image of \(B\): \(D'\) is finite over the twisted image of \(R/P\), and \(A_j\) is finite free over \(A\). Equation (35) annihilates \(S_j\) and \(S\). Moreover, \(S_j\) is Artinian local with residue field \(k\). To see locality, the tower parameter roots are nilpotent over the Artinian quotient \(D'/I_\sigma\), and reduction leaves the common residue field \(k\). Thus finite \(S_j\)-modules have finite lengths over both \(B\) and \(A_j\), and these lengths agree: a composition series over \(S_j\) has factors \(k\), of length one for either ring. This argument is unchanged if a residue-field identification is composed with an automorphism.

Given \(\epsilon>0\), choose a finitely generated \(S\)-submodule \(V\subset U\) with \(\lambda(U/V)<\epsilon\). This follows from the supremum property of normalized length: the \(S\)-span of a finitely generated \(\Lambda\)-submodule is still finitely generated over \(\Lambda\). To obtain a finite presentation as well, choose a surjection \(S^r\to V\) with kernel \(K\). Its kernel has finite length, bounded by \(r\lambda(S)\). Choose a finitely generated \(S\)-submodule \(K_0\subset K\) such that \(\lambda(K/K_0)<\epsilon\). Then \[P_\epsilon=S^r/K_0\longrightarrow U\] has kernel and cokernel of length less than \(\epsilon\). It is finitely presented over \(S\) and over \(\Lambda\), and is still annihilated by \(\mathfrak{m}_B^a\).

Fix \(B\)-resolutions of \(k\) and \(E'\) with finite free terms, and write \(r_i(L)\) for the rank in degree \(i\), with \(r_{-1}(L)=0\). If \(K_\epsilon\) and \(W_\epsilon\) are the kernel and cokernel of \(P_\epsilon\to U\), the two long exact homology sequences give \[\begin{align*} &\left|\lambda\operatorname{Tor}_i^B(P_\epsilon,L) -\lambda\operatorname{Tor}_i^B(U,L)\right| \\ &\quad\le \bigl(r_i(L)+r_{i-1}(L)\bigr)\lambda(K_\epsilon) +\bigl(r_{i+1}(L)+r_i(L)\bigr)\lambda(W_\epsilon). \tag{41}\end{align*}\] For example, each homology of the complex obtained by tensoring the resolution with \(K_\epsilon\) has length at most the rank of the corresponding term times \(\lambda(K_\epsilon)\); the same holds for \(W_\epsilon\). All these quantities are finite. Also \(|\lambda(P_\epsilon)-\lambda(U)|<2\epsilon\).

A finite \(S\)-presentation of \(P_\epsilon\) descends to an \(S_j\)-module \(P_j\) for some \(j\), with \(P_\epsilon=\Lambda\otimes_{A_j}P_j\). Flatness of \(\Lambda/A_j\) gives, for every \(i\), \[\operatorname{Tor}_i^B(P_\epsilon,L) =\Lambda\otimes_{A_j}\operatorname{Tor}_i^B(P_j,L).\] The finite-stage homology on the right is finite over \(S_j\). The normalization formula in Theorem 7, and the length comparison above, therefore give exactly \[\lambda\operatorname{Tor}_i^B(P_\epsilon,L) =\frac{\operatorname{length}_{A_j}\operatorname{Tor}_i^B(P_j,L)}{s(j)} =\frac{\operatorname{length}_B\operatorname{Tor}_i^B(P_j,L)}{s(j)}.\] The same formula holds for \(P_\epsilon\) itself.

At step \(n\), choose \(\epsilon\) so that all errors in Equation (41), for \(i\le n\) and \(L=k,E'\), and the length error are less than \(1/n\). Descend this presentation and enlarge its stage if necessary so that \(j\ge n\). Further flat tower extension leaves every normalized value unchanged. Take \(M_n=P_j\) and \(s_n=s(j)\). This proves Equations (38) and (39), including the common annihilator and \(s_n\to\infty\).

The case \(i=0,L=k\) gives the limit in Equation (40). A minimal generating set and the annihilator give a surjection \((B/\mathfrak{m}_B^a)^{\beta_0(M_n)}\to M_n\). Consequently \[\operatorname{length}_B(M_n) \le\beta_0(M_n)\operatorname{length}_B(B/\mathfrak{m}_B^a).\] Divide by \(s_n\) and take limits to obtain the stated positive lower bound. ◻

Proof. Every element \(b\in\mathfrak{m}_B^a\) acts nullhomotopically on \(F_n^\infty\), because it annihilates \(M_n\). Restrict a chosen homotopy to the brutal truncation. Its only remaining error is in degree \(l\), where it is \[ F_{n,l}\xrightarrow{h_l}F_{n,l+1}^\infty \xrightarrow{d_{l+1}}F_{n,l}. \tag{43}\] This is a factorization by chain maps through \(F_{n,l+1}^\infty[l]\): the map induced by \(d_{l+1}\) is a chain map because \(d_ld_{l+1}=0\), and the map induced by \(h_l\) is a chain map because the truncation has no degree \(l+1\) term. The middle module has rank \(o(s_n)\) by Equation (42), and is therefore killed in the asymptotic quotient. These identities persist after extension to \(E'\), without a flatness assumption over \(B\).

The ideal \(\mathfrak{m}_BE'\) for each indicated structural map is primary to \(\mathfrak{m}_{E'}\). Choose \(v\) so large that every \(z_j^v\) belongs to the extension of \(\mathfrak{m}_B^a\) for all \(\gamma\). This is one fixed choice, since the group is finite. Expressing \(z_j^v\) as a finite \(E'\)-linear combination of elements of \(\mathfrak{m}_B^a\) proves that it acts by zero on the sequence in the quotient. The same combinations give genuine nullhomotopies on the full extended resolutions. This proves support.

Apply the ordinary version of Equation (24). In its right-hand side, one may replace the truncated extended resolutions by the full ones in each fixed homological degree. Indeed, the quotient of the full resolution by its brutal truncation has only terms of degrees at least \(l+1\), all of rank \(o(s_n)\) in each fixed degree. After tensoring that tail with \(K(z^v;E')\), taking Koszul homology first bounds each fixed-degree homology by finite sums of \[\operatorname{rank}_B F_{n,i}^\infty\, \operatorname{length}_{E'} H_q(z^v;E')=o(s_n).\] The Koszul homology lengths are fixed and finite, and only finitely many terms contribute to each total degree. The long exact homology sequence thus proves the replacement assertion.

On each full extended resolution the powers \(z_j^v\) act nullhomotopically, as established above. Koszul tensoring therefore gives the direct sum of shifts with multiplicities \(\binom lj\). Its underlying homology has ordinary lengths \(\operatorname{length}_B\operatorname{Tor}_i^B(M_n,E')\), unchanged by conjugating the \(E'\)-action: applying \(\gamma\) entrywise is a semilinear isomorphism. Equations (39) and (36), followed by coefficient comparison in Equation (24), now identify the ordinary homology-length coefficients with \[\tau_i(E')=\lambda\operatorname{Tor}_i^R(C,E').\] Their alternating sum is exactly \(c_\sigma\). ◻

Proof. Let \(b_i\) be the nonnegative homology-length coefficients of this evaluation in Equation (24). Consider the finite double complex \[K(z^v;E')\otimes_{E'}F_{n,\gamma}' \otimes_{E'}C_E.\] Taking Koszul homology first, Theorem 7 makes all positive Koszul rows have length zero. The remaining row has complex degrees \(0,\ldots,l\). Thus the normalized homology lengths of this double complex vanish in total degrees greater than \(l\), as well as in negative degrees. Equation (24) says that their polynomial is \((1+t)^l\sum_i b_it^i\). All \(b_i\) are nonnegative. Boundedness and comparison of the least degree rule out negative-degree coefficients. Any \(b_i>0\) with \(i>0\) would then give a positive coefficient in degree \(i+l>l\), a contradiction. Consequently only \(b_0\) can be nonzero, and the Euler evaluation equals \(b_0\).

In degree zero, the displayed double complex has homology \[C_E^{\beta_0(M_n)}\big/ \left(\operatorname{im}(d_1)+(z_1^v,\ldots,z_l^v) C_E^{\beta_0(M_n)}\right).\] Since \(l>0\), \(d_1\) is the degree-one presentation differential of \(M_n\). Minimality over \(B\) puts all its entries in \(\mathfrak{m}_B\), whose image under any twist is contained in \(\mathfrak{m}_{E'}\). The base parameter powers are also in \(\mathfrak{m}_{E'}\). There is therefore a surjection from this degree-zero homology onto \[(C_E/\mathfrak{m}_{E'}C_E)^{\beta_0(M_n)}.\] Its normalized length is at least \(\beta_0(M_n)\eta_E/s_n\). The degree-zero coefficient in Equation (24) is \(b_0\), so taking the ultralimit and using the ordinary limit in Equation (40) gives \(b_0\ge\nu_\sigma\eta_E\). Equations (40) and (45) give the explicit remaining inequalities in Equation (46). Define \(b_{\sigma,\gamma}=b_0\). ◻

Proof of Theorem 1. By Lemma 22, use the domain pair fixed above. For each \(\sigma\in G_D\), carry out the construction with its specified \(R\)-action. Apply Theorem 21 to the supported sequence of Lemma 26, over \(A_E\subset E'\). It equates the sums of the ordinary and normalized evaluations of all \(G_E\)-conjugates. The former evaluation is \(c_\sigma\) for every conjugate, whereas the latter is \(b_{\sigma,\gamma}\). Hence \[ |G_E|c_\sigma =\sum_{\gamma\in G_E}b_{\sigma,\gamma}>0 \tag{47}\] by Lemma 27. Finally, Lemma 23 gives \[|G_D|a_Da_E\,\chi^R(D,E) =\frac{1}{|G_E|} \sum_{\sigma\in G_D}\sum_{\gamma\in G_E} b_{\sigma,\gamma}>0.\] Both ranks and both group orders are positive. Thus \(\chi^R(D,E)>0\), and the reduction proves the theorem. Neither norm comparison requires a Galois action on \(C_D\) or \(C_E\), nor invariance of either normalized length under a twist. ◻