Earlier constructions and restricted results

The positive cases developed along several lines. For the commutative Noetherian formulation, Auslander, Ding and Solberg treated complete local complete intersections by lifting across regular sequences (Auslander et al. 1993, Proposition 1.9). Huneke and Leuschke proved the conjecture for excellent commutative Cohen–Macaulay normal domains containing the rational numbers, among other classes (Huneke and Leuschke 2004, Theorem 0.1). Christensen and Holm established a general cohomological criterion: every left-noetherian ring satisfying Auslander’s condition satisfies the conjecture (Christensen and Holm 2010, Theorem A). That condition requires, for each fixed finitely generated first module, a common bound after which Ext vanishes for every finitely generated second module whose Ext groups vanish eventually.

Algebras with radical cube zero form another important positive class. Huneke, Şega and Vraciu treated commutative Artinian local rings (Huneke et al. 2004, Theorem 4.1(1)). For split finite-dimensional local algebras with radical cube zero, Ringel and Zhang proved the stronger assertion that every nonprojective module with no positive Ext to the algebra has a nonzero first self-extension (Ringel and Zhang 2022, Theorem 1.5). Zhang and Zhou subsequently proved the conjecture for split finite-dimensional algebras with radical cube zero (Zhang and Zhou 2026b, Theorem 1.1). Their result for Igusa–Todorov Artin algebras includes all Artin algebras with radical cube zero (Zhang and Zhou 2026a, Theorem 3.2 and Corollary 4.4). The algebra constructed here has nonzero fourth radical power, as proved in Proposition 20.

Schulz’s construction of a nonprojective module without self-extensions (Schulz 1994) is an important antecedent, but its particular skew-field example lies outside the Artin-algebra setting above; see (Diveris and Purin 2012, Introduction). A separate distinction is between all-degree and eventual vanishing: both summands in (1) must vanish in every positive degree.

Schulz also exhibited two-dimensional modules over quantum exterior algebras whose syzygies move through a parameter family and whose self-extensions vanish eventually, while the first self-extension remains nonzero (Schulz 1995, Example 7). The corresponding quantum exterior algebra occurs as a corner of our starting algebra. We use the same parameter-shifting mechanism for a resolution over a larger algebra, then remove every positive self-extension by two independent twists. For indecomposable weakly symmetric algebras over an algebraically closed field with \(J^3=0\ne J^2\), Erdmann characterizes those admitting nonprojective modules with eventually vanishing self-extensions (Erdmann 2017); eventual vanishing alone does not give (1). Xia proves the Auslander–Reiten conjecture for quantum complete intersections (Xia 2026), including the quantum exterior algebras used in Schulz’s parameter-shifting examples.

Vanishing in a finite range also differs from the hypothesis of (1). Böhmler and Marczinzik (Böhmler and Marczinzik 2026, Theorem 3.1 and the end of Section 3) construct a characteristic-two commutative self-injective example with vanishing first and second self-extensions, and nonzero third self-extension.

Tachikawa’s second conjecture is the self-injective case of (1). Enomoto proves that this conjecture for the two-fold trivial extension of an Artin algebra implies the Auslander–Reiten conjecture for that algebra (Enomoto 2026, Theorem 3.1). His reduction produces a self-injective algebra. The symmetric construction in our companion and the structural assertions of Theorem 1 are separate contributions.

How the construction removes self-extensions

The proof has two parts. First we give a conversion principle: a suitable stable map \(v:X\to F\otimes_A X\) over a symmetric algebra \(A\) produces a module over \[\Lambda=\begin{pmatrix}A&0\\F&A\end{pmatrix}.\] Its positive self-extensions are controlled by the difference between the actions of a self-extension of \(X\) on the two sides of \(v\). Total acyclicity separately gives the vanishing of Ext into \(\Lambda\). This principle is stated and proved before the example, so the subsequent construction has an explicit target.

We then build the required \(A,F,X,v\). Write \(D\) for the \(k\)-linear dual. A ten-dimensional algebra \(C\) has a simple module \(s\) whose Ext groups are especially sparse. Passing to its symmetric trivial extension \(T=C\ltimes DC\) produces \(\mathop{\mathrm{Ext}}_T^*(s,s)=k[\tau]\) with \(|\tau|=3\). An explicit Hochschild cocycle realizes \(\tau\) and supplies a homotopy compatible with twists of \(T\). For \(A=T\otimes T\) and \(X=s\otimes s\), the nonnegative self-extension algebra is \(k[\tau_1,\tau_2]\).

Write \(\mathcal C\) for the bimodule obtained from the two cones of these classes. In the stable category, maps factoring through projective modules are identified with zero, and the shift \([1]\) is inverse syzygy. The stable maps from \(X\) to \((\mathcal C\otimes_A X)[a]\) form a line for \(a=0\) and \(a=3\) and vanish in every other integer degree. This profile supplies a common target for two differently twisted maps.

For a nonzero scalar \(H\), let \(h_H\) fix \(C\) and scale \(DC\) by \(H\), and let \(U_H\) be \(A\) with right action twisted by \(h_H\otimes h_H\). Its tensor functor fixes \(X\) but multiplies its degree-\(3m\) self-extensions by \(H^{-m}\). Using the symmetrizing trace and the cocycle’s boundary identity, we construct maps \(U_H\to\mathcal C[3]\) whose evaluations on \(X\) are nonzero. After rescaling the maps for \(H_1\) and \(H_2\), their evaluations are the same nonzero element of that degree-three line. Their finite fiber \(F\) then has a single diagonal stable map \(v:X\to F\otimes_A X\) in degree zero, with both projections equal to \(\mathop{\mathrm{id}}_X\). In positive degrees, projection identifies the stable maps from \(X\) to \((F\otimes_A X)[a]\) with two copies of the self-extensions of \(X\). In degree \(3m>0\), the comparison takes two independently chosen self-extension classes \(g,j\) to \(F(g)v+v[3m]j\), where \(F(g)\) denotes tensoring with \(F\). Under these projections its matrix is \[\begin{pmatrix}H_1^{-m}&1\\H_2^{-m}&1\end{pmatrix} \otimes\mathop{\mathrm{id}}_{k[\tau_1,\tau_2]_{3m}}.\] This matrix is invertible for every \(m>0\). In degree zero, the comparison is surjective with a nonzero kernel. The conversion principle therefore removes all positive self-extensions while keeping a nonzero stable endomorphism, which proves nonprojectivity.

The main construction-specific ingredients are the explicit cocycle with its twisted boundary identity, the lift through both cones, and the resulting uniform comparison of the two twist actions. The surrounding tools have a substantial history. Trivial extensions and their Hochschild complexes are studied in (Cibils et al. 2003); Hochschild evaluation as a characteristic action is treated in (Lowen 2008, sec. 4.3). Bimodules attached to Hochschild classes and their iterated tensor products occur in the support-variety constructions of Erdmann, Holloway, Snashall, Solberg and Taillefer (Erdmann et al. 2004, sec. 2). We use the natural symmetric stable duality of (Linckelmann 2013, sec. 2) and the complete-resolution methods of (Veliche 2006). Gorenstein-projective modules over triangular matrix rings are studied in (Eshraghi et al. 2014). We supply the particular comparison and finiteness arguments needed here.

For reference, the objects and their roles are summarized in Table 1. The finite cocycle table appears in Appendix 10; its identities are finite algebra calculations. The resolution, duality, cone, and complete-resolution arguments establish the assertions in all degrees.

Section 2 fixes the stable-category conventions, and Section 3 proves the conversion principle. Sections 4 and 5 give the starting algebra and its self-extension calculation. Section 6 computes the two-cone profile; Sections 7 and 8 construct the lifts and the finite fiber. Section 9 proves the structural and scalar-extension assertions.

The conversion principle

For a finite bimodule \(F\) projective on both sides, tensoring with \(F\) is exact and preserves projectives. Over a symmetric algebra it therefore induces an exact functor on the stable category and commutes with shifts. In the following statement we write \(FX=F\otimes_A X\) and use these shift identifications when writing \(F(g)\).

For the triangular algebra \(\Lambda=\begin{pmatrix}A&0\\F&A\end{pmatrix}\), we write a left module as a triple \((M,N,\eta)\): here \(M,N\) are left \(A\)-modules and \(\eta:F\otimes_A M\to N\) is \(A\)-linear. The action is \[\begin{pmatrix}a&0\\f&b\end{pmatrix}(m,n) =(am,\eta(f\otimes m)+bn).\] A morphism \((\alpha,\beta)\) satisfies \(\beta\eta=\eta'(F\otimes\alpha)\).

Comparison of complete resolutions

For homological complexes \(P_\bullet,Q_\bullet\), use the full Hom complex \[ \mathop{\mathrm{Hom}}_R^a(P_\bullet,Q_\bullet) =\prod_{j\in\mathbb Z}\mathop{\mathrm{Hom}}_R(P_j,Q_{j-a}), \qquad (\partial f)_j=d^Q_{j-a}f_j+f_{j-1}d^P_j. \tag{5}\] This product convention is also used in (Veliche 2006, sec. 1.1.1). The product permits comparison maps with infinitely many nonzero components. Each component of the differential uses only two maps. Its degree-\(a\) cocycles are chain maps \(P_\bullet\to Q_\bullet[a]\), and its boundaries are nullhomotopic chain maps. Throughout this section the field has characteristic two, so the displayed signs apply to shifts and cones as well.

Proof of the conversion principle

Proposition 2 reduces the construction to a finite bimodule \(F\) projective on both sides and a stable map \(v\) with the stated two action properties. Surjectivity at degree zero is needed separately: it is the term that removes a possible self-extension in degree one in (11).

The ten-dimensional algebra and its trivial extension

Let \(C\) be the vector space with the following basis, arranged by corners. The notation \(iCj\) specifies the left idempotent first. \[ \begin{array}{c|c@{\qquad}c|c} \text{corner}&\text{basis}&\text{corner}&\text{basis}\\ \hline eCe&e,x,y,z&eCf&u,v\\ fCe&t,j&fCf&f,n \end{array} \tag{12}\] The elements \(e,f\) are orthogonal idempotents with sum \(1\). Besides their actions specified by the corners, the complete list of nonzero products is \[ \begin{gathered} xy=qz,\qquad yx=z,\qquad xu=yu=v,\qquad tx=j,\qquad ty=q^2j,\\ ut=y+qx,\qquad vt=qz,\qquad uj=z,\qquad tu=(1+q)n,\qquad nt=qj,\qquad un=v. \end{gathered} \tag{13}\] All products not specified by these rules are zero.

Put \(L=DC\) and equip it with the dual bimodule actions \[ (a\phi)(c)=\phi(ca),\qquad (\phi a)(c)=\phi(ac) \quad(a,c\in C,\ \phi\in L). \tag{14}\] Trivial extensions and their cohomology are studied in (Cibils et al. 2003). Here the trivial extension is \[ T=C\ltimes L, \qquad (a,\phi)(b,\psi)=(ab,a\psi+\phi b). \tag{15}\] For every lower-case basis letter \(a\) of \(C\), its upper-case letter \(a^*\) denotes the dual basis vector in \(L\); for example, \(j^*=J\) and \(f^*=F\). We also set \((a^*)^*=a\). Thus a capital letter \(J\) is a basis vector, whereas the radical below is denoted by \(\mathfrak r\).

Let \(s\) be the one-dimensional left \(T\)-module on which \(f\) acts as \(1\) and every other basis letter acts as zero. It is inflated from the corresponding \(C\)-simple. Write \(s^{\mathrm r}\) for the right simple defined by the same character.

The bar class and the finite identities

Set \(I=ke\oplus kf\). The normalized relative bar resolution of \(T\) has terms \[ P_n=T\otimes_I\mathfrak r^{\otimes_I n}\otimes_I T \quad(n\ge0),\qquad P_n=0\quad(n<0). \tag{16}\] We write \(a_0[a_1|\cdots|a_n]a_{n+1}\) for an elementary tensor. The differential multiplies each adjacent pair of factors and sums the results; the augmentation \(P_0\to T\) is multiplication. All signs are \(+\) because the characteristic is two. Splitting by idempotents expresses each term as a finite sum of projective bimodules \(Ti\otimes jT\) with \(i,j\in\{e,f\}\). Exactness follows from a contraction: replace the first coefficient \(a_0\) by its projection to \(\mathfrak r\), insert that projection as a new first bar, and put \(1\) in the first coefficient. In augmented degree send \(a\) to \(1[]a\). Writing \(\sigma\) for this contraction, the identity \(d\sigma+\sigma d=\mathop{\mathrm{id}}\) follows by canceling successive multiplications; the \(I\)-component of \(a_0\) is balanced over \(I\). This contraction is right \(T\)-linear, so \(P\otimes_T s\) is a projective resolution of \(s\) as well.

Give a lower-case letter weight zero and an upper-case letter weight one. This weight records the effect of scaling the square-zero ideal \(L=DC\); it is distinct from the positive grading in Lemma 6. The complete table in Appendix 10 defines an \(I\)-bimodule map \[p:\mathfrak r\otimes_I\mathfrak r\otimes_I\mathfrak r\longrightarrow T.\] Every unlisted value is zero. The following records exactly the properties of this finite table that the proof needs.

The evaluation of a Hochschild class on modules is the characteristic action; see (Lowen 2008, sec. 4.3). We use the same letter \(p\) for its comparison map \(P\to P[3]\): for \(n\ge3\) apply the cochain to the last three bars and multiply its output into the right coefficient, \[ a_0[a_1|\cdots|a_n]a_{n+1}\longmapsto a_0[a_1|\cdots|a_{n-3}] p(a_{n-2},a_{n-1},a_n)a_{n+1}; \tag{19}\] the map is zero for \(n<3\). This is a chain map: strict-prefix multiplications cancel, and the remaining equation on the last four bars is precisely the Hochschild cocycle equation. At the bottom index the target differential is zero. Composing its evaluation on \(s\) with the augmentation represents the class in Lemma 7.

We have now obtained the finite algebra and the specific degree-three class. The next step is to compute all self-extension degrees of \(s\), so that this nonzero class can be identified as a polynomial generator.

The resolution over \(C\)

Let \(s^{\mathrm r}\) denote the one-dimensional right \(f\)-simple. For every integer \(i\geq0\), set \[\ell_i=x+q^i y.\] The transcendence of \(q\) implies \(q\ne0\) and \(1+q^m\ne0\) for every positive integer \(m\). We will use these conditions for arbitrarily large indices.

The trivial-extension triangle and its product

Write \(L=DC\). The duality calculation in Lemma 8 determines \(L\otimes_C^{\mathbf L}s\). The extension \(0\to L\to T\to C\to0\) then provides a connecting morphism over \(T\). We identify its products explicitly so that the calculation controls every positive degree.

Two independent degree-three generators

We now pass to the symmetric algebra and simple used in the remaining construction: \[A=T\otimes_{k}T,\qquad X=s\otimes_{k}s.\] The tensor product of the symmetrizing forms makes \(A\) symmetric, and \(\dim_{k}A=400\). Let \(p_1,p_2\) be the external copies of the bimodule class \(p\) on the two factors, and write \(\tau_i=p_i(X)\) for their evaluations.

We have obtained two polynomial classes acting on \(X\), together with specific bimodule representatives for them. The next step uses their cones to leave only two stable Hom groups; their degrees will determine the shift in the subsequent fiber construction.

Negative degrees and multiplication

We first record the symmetric stable duality, including the compatibility with multiplication that will be needed below; see also (Linckelmann 2013, sec. 2). For finite modules \(M,N\) over a symmetric algebra \(G\), it says that \[ D\underline{\mathop{\mathrm{Hom}}}_{G}(M,N)\simeq\underline{\mathop{\mathrm{Hom}}}_{G}(N,M[-1]), \qquad D=\mathop{\mathrm{Hom}}_{k}(-,k), \tag{23}\] naturally in \(N\). Here is a direct proof with our indexing conventions. Choose a complete resolution \(Q\) of \(M\), so that \(M=\mathop{\mathrm{coker}}(Q_1\to Q_0)\). For a finite projective \(Q_i\), symmetry gives a natural isomorphism \[D\mathop{\mathrm{Hom}}_G(Q_i,N)\simeq\mathop{\mathrm{Hom}}_G(N,Q_i).\] It suffices to check this for \(Q_i=G\), where it is the isomorphism \(DG\simeq G\), and then to pass to finite sums and summands. Dualizing the cochain complex \(\mathop{\mathrm{Hom}}_G(Q,N)\) therefore gives the chain complex \(\mathop{\mathrm{Hom}}_G(N,Q)\). Its homology at zero consists of maps into \(\ker(Q_0\to Q_{-1})=\Omega M\), modulo maps through \(Q_1\). These are exactly the projective factorizations, since maps from a projective to \(\Omega M\) lift through \(Q_1\twoheadrightarrow\Omega M\). Vector-space duality is exact, proving (23) and its naturality.

In particular, \[ DH^a\simeq H^{-a-1}. \tag{24}\] For \(m\geq0\), let \(\mathcal P_m\) denote the degree-\(m\) part of the polynomial ring \(k[\tau_1,\tau_2]\) when each variable is assigned degree one. Thus \[ H^{3m}=\mathcal P_m,\qquad H^{-3m-1}=D\mathcal P_m\quad(m\geq0), \tag{25}\] and all other groups vanish.

These identifications also determine the action of the two classes in negative degrees. Postcomposition by \(\tau_i[a]\) gives \(H^a\to H^{a+3}\). By naturality of (23), its transpose, after shifting the source back to \(X\), is \[H^{-a-4}\longrightarrow H^{-a-1},\qquad g\longmapsto g[3]\circ\tau_i.\] When \(a\leq-4\), the last map is multiplication by \(\tau_i\) in the polynomial ring (22). The action within negative degrees is therefore the dual of polynomial multiplication. There is no nonzero product crossing from negative to nonnegative degree: the only nonzero possible source is \(H^{-1}\), and its target is \(H^2=0\).

Finite bimodules from projective tails

We will form the two cones as ordinary complexes before passing to stable bimodules. We give the passage explicitly to retain finite representatives and exact tensor evaluation.

Bimodules attached to Hochschild classes and their iterated tensor products have a precursor in the support-variety constructions of Erdmann, Holloway, Snashall, Solberg and Taillefer (Erdmann et al. 2004, sec. 2). Here we compute the full stable Hom profile directly, including negative degrees and the top attaching map.

If \(V\) is a bounded below complex of finite projective \(R\)-modules that is exact in sufficiently high degrees, put \[ \langle V\rangle= \bigl(\mathop{\mathrm{coker}}(V_{N+1}\longrightarrow V_N)\bigr)[N] \tag{26}\] in the stable category, with \(N\) sufficiently large. This does not depend on \(N\): writing \(V_N^{\mathrm c}\) for this cokernel, exactness gives \[0\longrightarrow V_{N+1}^{\mathrm c} \longrightarrow V_N\longrightarrow V_N^{\mathrm c} \longrightarrow0.\] Thus \(V_{N+1}^{\mathrm c}\simeq V_N^{\mathrm c}[-1]\) stably. Chain maps defined in sufficiently high degrees induce maps on these cokernels, compatibly with increasing \(N\) and with shifts. A chain homotopy changes the bottom induced map by a factorization through a projective, so homotopic tail maps induce the same stable map.

For a degreewise split short exact sequence of such complexes \(0\to B\to V\to D\to0\), their cokernels at \(N\) form a short exact sequence once \(H_{N+1}(D)=0\). Indeed, a possible extra relation in the \(B\)-cokernel is obtained by lifting a cycle in \(D_{N+1}\). That cycle is a boundary; subtracting a lifted boundary leaves a relation already in \(B\). The resulting stable sequence has the usual long exact stable Hom sequence: apply \(\mathop{\mathrm{Hom}}\) from a complete projective resolution to the short exact sequence and use its cohomology description of stable Hom. If the cross term of the differential is \(r:D_j\to B_{j-1}\), the connecting stable map \(\langle D\rangle\to\langle B\rangle[1]\) is induced by \(r\). This follows by lifting along the degreewise splitting and taking the \(B\)-component of the differential, which is precisely \(r\).

Call a bimodule side-projective if it is projective separately as a left and as a right \(A\)-module. Finite stable shifts of a finite side-projective bimodule have finite side-projective representatives. For a syzygy, take a finite \(R\)-projective surjection; its kernel sequence splits on each side because its quotient is side-projective. For a cosyzygy, take a finite \(R\)-projective-injective embedding; its cokernel sequence splits on each side because its submodule is side-injective, by symmetry of \(A\). Likewise a stable map \(M\to N\) between such bimodules has a finite side-projective cone: represent the map by \(f\), embed \(M\) into a finite \(R\)-projective \(I\), and take the cokernel of \((f,i):M\to N\oplus I\). Again the sequence splits on each side.

Forming the external tensor of ordinary complexes gives all four cells of (27) as projective resolutions of shifts of the regular \(A\)-bimodule. We pass to their stable representatives only after this construction.

Evaluation on a finite left module means tensoring over \(A\) on the right. Tensoring with a side-projective bimodule is an exact functor of left \(A\)-modules and preserves projectives. Every \(R\)-projective evaluates to an \(A\)-projective, since a free bimodule evaluates to a module of the form \(A\otimes_{k}V\). For evaluation at a fixed module, the right-split sequences used above remain exact. Their evaluations therefore preserve the stable shifts and long exact Hom sequences in the construction. Accordingly the filtration (27), after evaluation on \(X\), has stable cells \[X,\qquad X[-2]^{\oplus2},\qquad X[-4].\] Its adjacent connecting maps are the shifted actions of \(\tau_1,\tau_2\): they are induced by the original bimodule comparison maps tensored with \(X\). This specifies the attaching maps whose Hom groups we now compute.

The Hom profile and its top projection

The two exact sequences in the proof account for the complete filtration, including any attaching map not determined by its adjacent component. We have therefore obtained the required two-scalar profile together with a concrete way to detect its degree-three class: after shifting the cone by three, it suffices to project to \(X[-1]\). The next construction supplies maps with a nonzero projection there.

A twisted trace-dual tensor

For a basis letter \(w\) of \(T\), write \(w^*\) for its trace-dual letter. In \(P_0=T\otimes_I T\), set \[ \xi_H=\sum_{w\text{ in the basis of }T}h(w)\otimes_I w^*. \tag{33}\] The trace-dual coevaluation identity is standard; see (Broué 2009, Proposition 3.3). The associative symmetric trace gives the ordinary coevaluation identity \(a\sum_w w\otimes w^*=\sum_w w\otimes w^*a\). Applying \(h\) to the first factor and projecting to \(\otimes_I\) gives \[h(a)\xi_H=\xi_Ha.\] Consequently \(b:U\to P_0\), defined by \(b(x)=x\xi_H\), is a bimodule map.

We need the stronger fact that \(b\) takes values in \(\ker\pi\), where \(\pi:P_0\to T\) is multiplication. For a lower-case basis letter \(w\) of \(C\), the product \(ww^*\) is the dual of its left vertex idempotent, and \(w^*w\) is the dual of its right vertex idempotent. Indeed \((ww^*)(c)=w^*(cw)\); the positive grading makes this zero for \(c\in\mathop{\mathrm{rad}}C\), and the idempotent values determine the assertion. The row dimensions of \(C\) are \(6,4\), and its column dimensions are also \(6,4\). Separating the lower-case and upper-case letters therefore gives \[\pi\xi_H=\sum_{w\text{ lower-case}}ww^* +H\sum_{w\text{ lower-case}}w^*w=0\] in characteristic two. The same calculation on each left vertex \(r\in\{e,f\}\) gives \[ \sum_{\substack{w\in\mathfrak r\text{ a basis letter}\\ \mathrm{left}(w)=r}}h(w)w^*=r^*. \tag{34}\] Here the full sum is zero and the omitted idempotent term is \(rr^*=r^*\).

Taking the external square, define \[ \beta_H=b\otimes_kb: U_H\longrightarrow\ker(P^{\mathrm{tot}}_0\to A)=\Omega A, \qquad P^{\mathrm{tot}}=P\otimes_kP. \tag{35}\] Its image lies in the indicated kernel because \(\pi b=0\). The remaining task is to lift this map through the top-cell projection.

Two homotopies on the bar resolution

Let \(D_b:Q\to P\) be \(b\epsilon\) in degree zero and zero in every positive degree. It is a chain map. A homotopy for \(D_b\) will supply a comparison representing \(\beta_H\) in the top cell. Define maps \(B_i:Q_i\to P_{i+1}\) for every \(i\geq0\) by \[ B_i([a_1|\cdots|a_i]) =\sum_{w\in\mathfrak r\text{ a basis letter}} [a_1|\cdots|a_i|h(w)]w^*, \tag{36}\] retaining only composable terms and extending by (32).

We now use \(B\) to represent the prescribed top projection. In \(K[3]\), the cell indexed by \(\{1,2\}\) is \(P^{\mathrm{tot}}[-1]\). Put \(V=B\otimes D_b\), which raises the underlying total degree by one, as this shift requires. Lemma 14 gives \[dV+Vd=(b\otimes b)\epsilon^{\mathrm{tot}},\] where the right side is supported in degree zero. Thus \(dV_0=\beta_H\epsilon^{\mathrm{tot}}_0\), and at positive indices \(V\) satisfies the comparison equation. The complex with terms \(P^{\mathrm{tot}}_{i+1}\) and augmentation to \(\Omega A\) resolves \(\Omega A\). Consequently \(V\) is a comparison lifting \(\beta_H\).

To extend \(V\) from the top cell to \(K[3]\), we must account for the two attaching maps. Deleting slot 1 contributes \(pB\otimes D_b\) to the cell indexed by \(\{2\}\). The suffix comparison \(p:P\to P[3]\) from the finite cochain data lowers the underlying bar degree by three, so \(pD_b=0\). The contribution from deleting slot 2 is therefore zero. It remains to cancel \(pB\otimes D_b\) by a component in cell \(\{2\}\). Lemma 14 shows that \(pB:Q\to P[2]\) is a chain map. The additional boundary identity (17) supplies its nullhomotopy; cocycle closure by itself would not suffice.

The tail map and its evaluation

The homotopy \(G\) supplies the required correction in the singleton cell \(\{2\}\), whose shift in \(K[3]\) is \(1\). Together with the top component \(V\), it defines a graded map \(\Phi:Q^{\mathrm{tot}}\to K[3]\): \[ \Phi_{\{1,2\}}=B\otimes D_b,\qquad \Phi_{\{2\}}=G\otimes D_b,\qquad \Phi_{\{1\}}=\Phi_{\varnothing}=0. \tag{39}\] The second component lowers underlying degree by one, as the singleton shift requires; the first is the comparison \(V\) constructed above.

In the top cell, the chain defect \(d\Phi+\Phi d\) is \(D_b\otimes D_b\). In cell \(\{2\}\), deletion of slot 1 gives \(pB\otimes D_b\), canceled by \((dG+Gd)\otimes D_b\). Deleting slot 2 from either nonzero component gives zero because \(pD_b=0\). Thus the only defect is \(D_b\otimes D_b\), supported in total source degree zero. The map therefore commutes with the differential in every positive total degree. Passing to high cokernels, at a common index at least \(4\) as in Lemma 11, gives \(f_H:U_H\to\mathcal C[3]\).

The top component is still \(V\), so the resulting stable projection is \(\beta_H\), as identified above.

By Proposition 12, the top-cell projection is an isomorphism on the degree-three group of the cone calculation. Lemma 16 therefore proves that \(f_HX\) is nonzero, completing Proposition 13. We have obtained the two required maps by taking \(H=H_1,H_2\); the next section combines them into a finite bimodule.

A finite representative and an exact sequence

Write \(R=A^e\). Lemma 11 provides the finite bimodule \[C_1=\mathop{\mathrm{coker}}(K_2\longrightarrow K_1), \qquad \mathcal C=C_1[1]\] in the stable category of \(R\). It is projective on both \(A\)-sides. To fix actual finite representatives for its shifts, for a finite left \(R\)-module \(M\) use the injection \[ j_M:M\longrightarrow I_R(M):=\mathop{\mathrm{Hom}}_{k}(R,M), \qquad j_M(m)(t)=tm, \qquad (r\varphi)(t)=\varphi(tr). \tag{40}\] Evaluation at \(1\) proves injectivity. Since \(R\) is symmetric, \(I_R(M)\cong DR\otimes_{k}M\cong R\otimes_{k}M\) as left \(R\)-modules, so this is a finite projective module. Put \(\Sigma_RM=\mathop{\mathrm{coker}}j_M\). When \(M\) is projective on each \(A\)-side, the sequence defining \(\Sigma_RM\) splits on each side: \(A\) is self-injective, so \(M\) is injective on either side. Thus \(\Sigma_RM\) remains projective on both sides and represents \(M[1]\).

In particular, fix the actual finite bimodule \[ \mathcal Y=\Sigma_R^4C_1. \tag{41}\] It is projective on both sides and represents \(\mathcal C[3]\). This fixes one target for both lifts of Proposition 13. Concretely, the tail map there at index \(4\) induces \(\Omega_R^4U_H\to C_1\), because \[\mathop{\mathrm{coker}}\bigl(K[3]_5\longrightarrow K[3]_4\bigr) =\mathop{\mathrm{coker}}(K_2\longrightarrow K_1).\] Taking four cosyzygies gives the stable map \(U_H\to \mathcal Y\), using \(\Sigma_R^4\Omega_R^4U_H\cong U_H\) stably. It therefore has an actual bimodule-map representative between these finite modules.

For \(i=1,2\), put \[\sigma_i=h_{H_i}\otimes h_{H_i},\qquad U_i=A_{\sigma_i}.\] Thus \(U_i\) has its usual left action and right action \(u\cdot a=u\sigma_i(a)\). Each \(U_i\) is free of rank one on both \(A\)-sides. The canonical identification \(U_i\otimes_A X=X\) uses the fact that \(\sigma_i\) fixes the simple character of \(X\).

By Lemma 16, the top projections of the evaluated lifts are \(H_i^2\beta_0\) for one fixed nonzero class \(\beta_0\in\underline{\mathop{\mathrm{Hom}}}_{A}(X,X[-1])\). By Proposition 12, the top projection \[W^3=\underline{\mathop{\mathrm{Hom}}}_{A}(X,\mathcal C X[3]) \longrightarrow \underline{\mathop{\mathrm{Hom}}}_{A}(X,X[-1])\] is an isomorphism. Rescale the two lifts by \(H_i^{-2}\), and choose actual representatives \(g_i:U_i\to \mathcal Y\) of these rescaled maps. Their evaluations are the same nonzero element \(w\in W^3\).

Take the finite free \(R\)-module \(Q=R\otimes_{k}\mathcal Y\) and its surjection \(\pi:Q\to \mathcal Y\), \(\pi(r\otimes y)=ry\). Define \(F\) by the exact sequence \[ 0\longrightarrow F\longrightarrow U_1\oplus U_2\oplus Q \xrightarrow{\ (g_1,g_2,\pi)\ }\mathcal Y\longrightarrow0, \tag{42}\] and let \(\rho_i:F\to U_i\) be the two projections. This is a construction of finite modules: \(C_1\) is a cokernel of two finite terms, four finite cokernels give \(\mathcal Y\), and one finite kernel gives \(F\). Since \(\mathcal Y\) is projective on each side, (42) splits on each side. Consequently \(F\) is projective as both a left and a right \(A\)-module.

Tensoring (42) with \(X\) preserves exactness. Furthermore, \(Q\otimes_AX\) is projective over \(A\): evaluation of a free \(A^e\)-module is a free \(A\)-module, and evaluation preserves direct summands. Thus the evaluated stable triangle is \[ FX\longrightarrow X\oplus X\xrightarrow{\ (w,w)\ } \mathcal C X[3]\longrightarrow FX[1]. \tag{43}\] Here and below \(FM\) means \(F\otimes_AM\). Tensoring with \(F\) is exact because \(F\) is right projective; it preserves projectives because \(F\) is left projective. It therefore acts on stable maps compatibly with shifts. The actual maps \(\rho_i\) give natural transformations from this functor to tensoring with \(U_i\).

We have obtained an actual finite bimodule \(F\) and a nonzero stable map \(v:X\to FX\). It remains to compare tensoring an extension with \(F\) and composing that extension with \(v\).

The twist eigenvalues and the comparison map

The hypotheses of Proposition 2 are now verified for the finite symmetric algebra \(A\), its finite module \(X\), and the actual finite bimodule \(F\) and stable map \(v\). Applying that proposition to \[\Lambda=\begin{pmatrix}A&0\\ F&A\end{pmatrix}\] produces the required finite nonprojective Gorenstein-projective left module, with both positive Ext vanishings in every degree. In particular, the degree-one assertion uses the surjectivity at degree zero established in Proposition 19, in addition to the positive-degree invertibility.