Proof. We first record the extension property supplied by total acyclicity. If \(J\) is projective, then \(\mathop{\mathrm{Hom}}_G(P,J)\) is exact, since \(J\) is a summand of a finite sum of copies of \(G\). Identify \(c_jP\) with \(\mathop{\mathrm{im}}d_j^P\subset P_{j-1}\). For \(u:c_jP\to J\), the map \(u d_j^P:P_j\to J\) is a cocycle. Exactness gives \(h:P_{j-1}\to J\) with \(h d_j^P=u d_j^P\), so \(h\) extends \(u\). It follows at once that the cycles of \(\mathop{\mathrm{Hom}}_G(P,N)^a\) are maps \(c_aP\to N\) and its boundaries are exactly the maps factoring through projectives. This proves the second isomorphism, with \(N=c_0Q\).

For the first, begin in degree zero. A map \(c_0P\to c_0Q\) lifts to \(f_0:P_0\to Q_0\), and then successively to \(f_j\) for \(j>0\), using projectivity of \(P_j\) and exactness of \(Q\). To proceed downwards, \(d_j^Qf_j\) vanishes on \(\ker d_j^P=\mathop{\mathrm{im}}d_{j+1}^P\) by the chain equation already obtained in degree \(j+1\). It therefore defines a map \(c_jP=\mathop{\mathrm{im}}d_j^P\to Q_{j-1}\). The extension property gives \(f_{j-1}\). Thus every cokernel map lifts to a full chain map.

A homotopy induces a projective factorization on \(c_0\). Conversely, suppose the cokernel map of \(f\) factors as \(c_0P\to J\to c_0Q\) with \(J\) projective. Extend its first factor to \(P_{-1}\) and lift its second factor to \(Q_0\). Their composite is a homotopy component \(h_{-1}:P_{-1}\to Q_0\) whose homotopy differential removes the induced map on cokernels. It remains to nullhomotope a chain map inducing zero there. For this remaining map set \(h_{-1}=0\). For \(j\geq0\), lift \[f_j+h_{j-1}d_j^P:P_j\longrightarrow\mathop{\mathrm{im}}d_{j+1}^Q\] to \(h_j:P_j\to Q_{j+1}\); the initial image assertion is the vanishing on \(c_0\), and each later one follows from the preceding chain equation. For \(j<0\), the map \(f_j+d_{j+1}^Qh_j\) kills \(\mathop{\mathrm{im}}d_{j+1}^P\) by the equation in degree \(j+1\). It descends to \(\mathop{\mathrm{im}}d_j^P\) and extends to \(h_{j-1}:P_{j-1}\to Q_j\) by the same extension property. These equations give \(f=dh+hd\) in every degree. The product in (2) permits all these components; each equation uses only two of them.

Replace \(P\) by \(P[-a]\) to obtain degree \(a\). The construction by cokernels preserves compositions and shifts. Finally, the nonnegative tail of \(P\) resolves \(c_0P\). In every positive cohomological degree its Hom complex has the same two adjacent differentials as \(\mathop{\mathrm{Hom}}_G(P,c_0Q)\), proving the Ext assertion, including degree one. ◻

Proof. For every finite projective \(J\) the finite-dimensional adjunction gives \[D\mathop{\mathrm{Hom}}_G(J,N)\simeq\mathop{\mathrm{Hom}}_G(N,DG\otimes_GJ)\simeq\mathop{\mathrm{Hom}}_G(N,J).\] The first identification is immediate for \(J=G\), hence for finite sums and summands; the second uses a fixed symmetrizing form. Apply this termwise to a complete resolution \(P\) of \(M\). Dualizing degree-zero cohomology of \(\mathop{\mathrm{Hom}}_G(P,N)\) yields \[\frac{\mathop{\mathrm{Hom}}_G\bigl(N,\ker(P_0\to P_{-1})\bigr)} {\mathop{\mathrm{im}}\bigl(\mathop{\mathrm{Hom}}_G(N,P_1)\to \mathop{\mathrm{Hom}}_G(N,\ker(P_0\to P_{-1}))\bigr)}.\] Here \(\ker(P_0\to P_{-1})=c_1P=M[-1]\). A factorization through any projective lifts through \(P_1\twoheadrightarrow c_1P\), so the displayed quotient is precisely \(\underline{\mathop{\mathrm{Hom}}}_G(N,M[-1])\). All identifications are natural; comparison of complete resolutions also gives naturality in \(M\).

Define \(\lambda_M(g)\) by pairing \(1_M\) with \(g\). Naturality in \(N\), applied to \(f:M\to N\), identifies the pairing of \(f\) and \(g\) with \(\lambda_M(gf)\). Replacing \(N\) by \(N[a]\) and using the shift equivalences gives (7). ◻

Proof. Every chain map from \(P\) to an acyclic complex is nullhomotopic: start at the lowest nonzero degree of \(P\) and successively lift the residual map through the next differential of the target. The same argument applies to every shift. Thus the full Hom from \(P\) sends quasi-isomorphisms to quasi-isomorphisms, which proves the assertion about derived maps. This is the \(K\)-projectivity of a bounded-below projective source; it does not assert \(K\)-projectivity of a complete resolution.

For completeness, a bounded-below finite complex \(V\) admits a projective model \(\theta:Q\to V\) by the following induction. Having constructed lower terms, choose \(Q_j\) finite projective surjecting onto \[\{(x,y)\in V_j\oplus Q_{j-1}:d_Vx=\theta(y),\ d_Qy=0\}.\] Use its two components for \(\theta_j\) and \(d_j^Q\). Every cycle of the comparison cone is then a boundary, so \(\theta\) is a quasi-isomorphism. A derived isomorphism between projective models and its inverse are represented by chain maps, and their composites are homotopic to the identities by the first assertion.

To verify the last assertion, represent the first arrow of the triangle by \(f:P_L\to P_V\) on any projective models. Its cone represents \(R\), and the projection \(\mathop{\mathrm{Cone}}(f)\to P_L[1]\) represents \(\partial\). The cone of the deshifted projection has \(P_V\) as a subcomplex with contractible quotient \(\mathop{\mathrm{Cone}}(1_{P_L})\). It therefore represents \(V\). Replacing the projective models by homotopy equivalent ones and the map by its prescribed representative preserves this conclusion: a homotopy commuting square induces a cone map with that homotopy as its off-diagonal component, and homotopic maps have isomorphic cones. The resulting cone \(\mathop{\mathrm{Cone}}(b[-1])\) is exactly (9). ◻

Proof. Above the homology support, (4) is exact for \(P\). It identifies \((c_{n+1}P)[n+1]\) with \((c_nP)[n]\), naturally on chain maps. A homotopy changes \(c_n(f)\) by a map factoring through \(P_{n-1}\), so the construction descends to homotopy classes. Lemma 6 then makes it functorial on derived maps. The identity \(c_n(P[a])=c_{n-a}P\) gives shift compatibility. A projective resolution of a module recovers its successive syzygies, proving the assertion for modules. Restricting a lifted representative of an Ext class to high degrees gives the same successive syzygies of that map, proving the last assertion about classes.

Choose \(n\) so that all three complexes are exact in degrees at least \(n\). Their cokernels give a short exact sequence \[0\longrightarrow c_nP_L\longrightarrow c_nP_V \longrightarrow c_nP_R\longrightarrow0.\] For injectivity on the left, suppose \((x,0)=d_{n+1}^{P_V}(u,v)\). Then \(d_{n+1}^{P_R}v=0\), so \(v=d_{n+2}^{P_R}w\) for some \(w\). The equation \(d_Lb+bd_R=0\) gives \(x=d_{n+1}^{P_L}(u+b_{n+2}w)\), as required. Surjectivity and exactness in the middle follow by lifting components in the graded splitting.

There is a commutative diagram with exact rows \[\begin{tikzcd}[column sep=large] 0\arrow[r]&c_{n+1}P_R\arrow[r]\arrow[d,"\overline b_{n+1}"'] &P_{R,n}\arrow[r]\arrow[d] &c_nP_R\arrow[r]\arrow[d,equal]&0\\ 0\arrow[r]&c_nP_L\arrow[r]&c_nP_V\arrow[r]&c_nP_R\arrow[r]&0. \end{tikzcd}\] The middle vertical map sends \(z\) to the class of \((0,z)\). On an element \(d_{n+1}^{P_R}z\), its value is the class of \((b_{n+1}z,0)\), since \((b_{n+1}z,d_{n+1}^{P_R}z)\) is a boundary in \(P_V\). Thus the left vertical map is induced by \(b_{n+1}:P_{R,n+1}\to P_{L,n}\). Shifting the left vertical arrow by \([n+1]\) gives \(\langle P_R\rangle\to\langle P_L\rangle[1]\). By the description after (5), this is the connecting map of the bottom row shifted by \([n]\), and it is exactly \(\langle b\rangle\). This also gives the asserted long exact sequences. ◻

Proof. Choose bounded-below projective models \(P,Q\) and represent the map by a global chain map \(f:P\to Q\), using Lemma 6. For a sufficiently high \(n\), both complexes are exact in degrees at least \(n\) and \(c_n(f)\) factors through a finite projective \(J\): \[c_nP\xrightarrow{u}J\xrightarrow{v}c_nQ.\] Since \(J\) is injective, extend \(u\) across \(c_nP=\mathop{\mathrm{im}}d_n^P\subset P_{n-1}\). Since \(J\) is projective, lift \(v\) to \(Q_n\). The composite gives \(h_{n-1}:P_{n-1}\to Q_n\). It makes \(f_n+h_{n-1}d_n^P\) zero after projection to \(c_nQ\), so this residual map lifts through \(d_{n+1}^Q\) to \(h_n:P_n\to Q_{n+1}\). Inductively, for \(j>n\) the chain equation and the preceding homotopy equation show that \(f_j+h_{j-1}d_j^P\) lands in \(\ker d_j^Q=\mathop{\mathrm{im}}d_{j+1}^Q\). Lift it to \(h_j\). Set \(h_j=0\) for \(j<n-1\).

The homotopic map \(f'=f+dh+hd\) now vanishes in every degree \(j\geq n\). Its chain equation in degree \(n\) reads \[ f'_{n-1}d_n^P=0. \tag{11}\] Consequently its remaining components define a chain map into the actual homological subcomplex \(Q_{\leq n-1}\), obtained by retaining \(Q_j\) for \(j\leq n-1\) and setting the higher terms to zero. Equation (11) verifies the boundary condition at the cut. Thus \(f'\) factors through \(Q_{\leq n-1}\), which is bounded and projective because \(Q\) was bounded below. Composing with the chosen model maps gives the desired derived factorization. This also covers a map with a module as target: first lift it globally to a projective resolution of that module, and then apply this argument. ◻

Proof. Take a projective bimodule surjection onto \(L\). Its kernel is side-projective because the sequence splits on each side. For a positive shift, use the finite projective-injective embedding \[L\longrightarrow\mathop{\mathrm{Hom}}_k(E^e,L),\qquad z\longmapsto(a\longmapsto az), \qquad (r\varphi)(a)=\varphi(ar).\] Its target is injective by coinduction and projective by symmetry of \(E^e\). The embedding splits on each side because \(L\) is injective on that side, so its cokernel is again side-projective. Iteration supplies finite representatives for every shift.

An \(E^e\)-projective evaluates to an \(E\)-projective: this is immediate for the free bimodule \(E\otimes E\), which evaluates to \(E\otimes X\), and passes to finite sums and summands. Hence evaluation respects projective factorizations. Every short exact sequence of side-projective bimodules splits on the right, and therefore remains exact after evaluation. In particular, evaluating the sequences just used to define shifts gives the first identification in (12). Applying the projective-presentation description of connecting maps to these same sequences shows that their actual connecting maps are preserved.

For fixed \(L\), right projectivity makes \(L\otimes_E-\) exact, and left projectivity makes it preserve projectives. Applying it to projective resolutions and projective-injective coresolutions gives the second identification in (12). A bimodule map \(L\to L'\) gives a natural transformation on these complexes; thus evaluating it on a shifted module agrees stably with shifting its evaluation. The same reasoning proves naturality in \(X\) and compatibility with composition.

For the complex assertion, the high exact sequences \(0\to c_{n+1}P\to P_n\to c_nP\to0\) split on the right. Their evaluations remain exact, proving bounded homology of \(P\otimes_EX\); its terms are projective by the preceding argument. Right exactness gives the canonical identifications \[c_n(P\otimes_EX)=(c_nP)\otimes_EX.\] Together with (12), these identify the two sides of (13). They commute with chain maps. For a filtered complex, the cross differential in Lemma 7 evaluates to its tensor with \(X\). That lemma’s explicit diagram therefore identifies the evaluated connecting map with the connecting map after evaluation.

Finally, suppose a homotopy equivalence on one side compares \(P\) with a complex supported in degrees at most \(d\). At any \(n>d\), the homotopy between the identity and the composite through that complex gives \[1_{P_n}=d_{n+1}h_n+h_{n-1}d_n.\] Writing \(\pi_n:P_n\to c_nP\), passage to the cokernel yields \[1_{c_nP}=(\pi_nh_{n-1})\overline d_n, \qquad \overline d_n:c_nP\longrightarrow P_{n-1}.\] Thus \(c_nP\) is a summand of a projective on that side. The separate homotopy equivalence on the other side proves the other projectivity. The homotopies need not be bimodule homotopies. ◻

Proof. The usual bar resolution of \(B\) has terms \(B\otimes B^{\otimes i}\otimes B\). Its augmented complex is contracted as a complex of vector spaces by inserting a unit. Inserting a unit at one end also proves that tensoring it with a module at the other end gives a projective resolution of that module on the remaining side. Inducing the two outer actions gives \(K\), with each term a finite free \(E^e\)-module.

On the left, \(B\otimes B^{\otimes i}\otimes E\) is a projective \(B\)-resolution of \({}_BE\). By Proposition 13, this module has a projective resolution supported in degrees \(0,\ldots,4\). Comparison of projective resolutions gives a homotopy equivalence between these two resolutions. Applying \(E\otimes_B-\) preserves the homotopy equivalence, so \(K\), as a left \(E\)-complex, is homotopy equivalent to a complex of projectives supported in those degrees. The same argument using \(E_B\) works on the right. In particular \(H_i(K)=0\) for \(i>4\).

For completeness, this equivalence proves projectivity of the specific cokernel \(c_5K\). On either side, choose a homotopy from the identity of \(K\) to a map factoring through the complex ending in degree \(4\). In degree \(5\) it gives \[1_{K_5}=d_6h_5+h_4d_5.\] Modulo \(\mathop{\mathrm{im}}d_6\), the identity of \(c_5K\) therefore factors as \[c_5K\xrightarrow{\ \overline d_5\ }K_4 \xrightarrow{\ q_5h_4\ }c_5K,\] where \(q_5:K_5\to c_5K\) is the quotient map. Thus \(c_5K\) is a summand of a projective on each side. This assertion concerns the two side-module structures; the two splittings need not be bimodule splittings. The high exact sequences \(0\to c_{n+1}K\to K_n\to c_nK\to0\), for \(n\geq5\), now split on each side. Induction shows that all higher cokernels are side-projective as well.

To make the finite choice explicit, put \(N_0=c_5K\), and for \(r=0,1,2,3\) use \[0\longrightarrow N_r\longrightarrow \mathop{\mathrm{Hom}}_k(E^e,N_r)\longrightarrow N_{r+1}\longrightarrow0,\qquad n\longmapsto(a\longmapsto an).\] Here the \(E^e\)-action on the middle term is by right multiplication on its input. The middle term is finite injective, hence projective because \(E^e\) is symmetric. Since \(N_r\) is projective and therefore injective on each \(E\)-side, this sequence splits on both sides; it follows inductively that every \(N_r\) is side-projective. Take \(Y=N_4\). Its stable class is \((c_5K)[4]=\mathcal I[-1]\), as required. Lemmas 7 and 9 give the stated stable images and their evaluation compatibilities. We denote the stable map induced by \(m\) by \(\overline m:\mathcal I\to E\). ◻

Proof. The evaluated complex is \[K\otimes_E X=E\otimes B^{\otimes\bullet}\otimes X.\] Before inducing to \(E\), its bar model and \(R\otimes R\) are projective \(B\)-resolutions of \(X=s\otimes s\). They are homotopy equivalent. Induction preserves this equivalence and gives \[ K\otimes_E X\simeq (T\otimes_C R)\otimes(T\otimes_C R) =\mathcal J\otimes\mathcal J . \tag{44}\] In particular, Lemma 9 identifies its stable image with \(\mathcal IX\).

We retain the actual attaching maps in the triangle \[s[2]\longrightarrow\mathcal J\longrightarrow s \xrightarrow{\tau}s[3]\] of (30). Choose bounded-below projective models \(A_\bullet\) for \(s[2]\) and \(Q_\bullet\) for \(s\), and a chain map \(t:Q_\bullet\to A_\bullet[1]\) representing \(\tau\). The cone of \(t[-1]\) has underlying graded complex \(A_\bullet\oplus Q_\bullet\), differential \[\begin{pmatrix}d_A&t\\0&d_Q\end{pmatrix},\] and represents \(\mathcal J\), by Lemma 6. Indeed the cone construction for the rotated triangle gives this identification; comparisons of projective models lift the connecting map and give homotopy equivalences of the cones. Thus this is a homotopy equivalent projective model, with its specified subcomplex and quotient, rather than only an equality of homology groups.

Tensor two copies of this model and shift by \([-4]\). Filter first by \(A_\bullet\otimes A_\bullet\), then add \(Q_\bullet\otimes A_\bullet\) and \(A_\bullet\otimes Q_\bullet\), and finally add \(Q_\bullet\otimes Q_\bullet\). Every inclusion is split in each graded degree, and the successive quotients resolve \[ X,\qquad X[-2]\oplus X[-2],\qquad X[-4]. \tag{45}\] The stable image of the full complex is \(\mathcal L=(\mathcal IX)[-4]\). With the middle terms in the displayed order, the adjacent connecting maps are the shifted row and column \[(\tau_1,\tau_2),\qquad \binom{\tau_2}{\tau_1}.\] These are the cross differentials \(t\) in the relevant factor and the identity in the other factor. Lemma 7 identifies them with the connecting maps after taking stable images.

We first compute the kernels and cokernels of these adjacent maps in every degree. On the polynomial groups the maps form the two-variable Koszul sequence; on the negative groups they form its dual. We then use the two long exact sequences of the filtration to control the full top attaching map, including its possible component into the bottom level.

Write \(\mathcal P_m=k[\tau_1,\tau_2]_m\) for polynomials of total exponent \(m\), and put \(\mathcal P_m=0\) for \(m<0\). By Proposition 16, \[\mathcal H^{3m}=\mathcal P_m,\qquad \mathcal H^{-3m-1}=D\mathcal P_m\quad(m\geq0),\] and all other \(\mathcal H^a\) vanish. Multiplication by a degree-three class on the negative groups is transpose to polynomial multiplication. For each integer \(l\), denote the adjacent maps by \[\begin{aligned} d_l:(\mathcal H^{l-2})^2&\longrightarrow\mathcal H^{l+1},& d_l(u,v)&=\tau_1u+\tau_2v,\\ d'_l:\mathcal H^{l-4}&\longrightarrow(\mathcal H^{l-1})^2,& d'_l(w)&=(\tau_2w,\tau_1w). \end{aligned}\] Products in these formulas mean shifted postcomposition. We claim \[\begin{align*} \ker d_l&=\mathop{\mathrm{im}}d'_{l-1},\tag{46}\\ \mathop{\mathrm{coker}}d_{l-1}&= \begin{cases}k,&l=0,\\0,&l\ne0,\end{cases} \qquad \ker d'_l= \begin{cases}k,&l=3,\\0,&l\ne3.\end{cases} \tag{47}\end{align*}\]

To check the middle identity, the polynomial calculation for \(m\geq0\) is the exact sequence \[ 0\longrightarrow\mathcal P_{m-1} \xrightarrow{\,(\tau_2,\tau_1)^{\mathsf t}\,} \mathcal P_m^2 \xrightarrow{\,(\tau_1,\tau_2)\,} \mathcal P_{m+1}\longrightarrow0 . \tag{48}\] The row is onto since every positive-degree monomial is divisible by a variable. If \(\tau_1u+\tau_2v=0\), coprimality implies \(u=\tau_2w\) and \(v=\tau_1w\), uniquely; the signs agree in characteristic two. When the middle group is \((D\mathcal P_m)^2\), use the dual of (48), interchanging the two middle slots: \[D\mathcal P_{m+1}\longrightarrow (D\mathcal P_m)^2\longrightarrow D\mathcal P_{m-1}.\] At \(m=0\), the last term is the actual group \(\mathcal H^2=0\), and the first map \(D\mathcal P_1\to(D\mathcal P_0)^2\) is an isomorphism. This is the seam \(l=1\). If the middle group is unsupported, both its kernel and the image into it are zero. This proves (46) for every \(l\).

For the row cokernel, a target \(\mathcal H^l=\mathcal P_m\) gives \(\mathcal P_{m-1}^2\to\mathcal P_m\), which is onto for \(m\geq1\) and has cokernel \(k\) for \(m=0\). A negative target \(\mathcal H^l=D\mathcal P_m\) gives the dual of the injective column \(\mathcal P_m\to\mathcal P_{m+1}^2\), and is onto for every \(m\geq0\). All other targets are zero. For the column kernel, a nonnegative source \(\mathcal P_m\) maps injectively into \(\mathcal P_{m+1}^2\). A source \(D\mathcal P_m\) for \(m\geq1\) is the dual of the surjective row \(\mathcal P_{m-1}^2\to\mathcal P_m\), so again the column is injective. The remaining source is \(\mathcal H^{-1}=D\mathcal P_0=k\), at \(l=3\), with zero target \((\mathcal H^2)^2\). These cases prove (47). They include the only possible multiplication from a supported negative degree across zero, namely \(\mathcal H^{-1}\to\mathcal H^2=0\).

It remains to pass from adjacent maps to the whole filtered object. Let \(\mathcal L_1\) be the stable image of the lower two levels of (45), and put \(N^l=\underline{\mathop{\mathrm{Hom}}}_E(X,\mathcal L_1[l])\). The first two levels give \[ 0\longrightarrow\mathop{\mathrm{coker}}d_{l-1}\longrightarrow N^l \xrightarrow{q_l}\ker d_l\longrightarrow0. \tag{49}\] Let \[r_l:\mathcal H^{l-4}\longrightarrow N^{l+1}\] be the actual connecting map obtained by attaching the top level. Naturality of the quotient to the middle level gives \(q_{l+1}r_l=d'_l\). Thus \(\ker r_l\subseteq\ker d'_l\), which is zero unless \(l=3\). At that index \(N^4=0\): its left term in (49) is \(\mathop{\mathrm{coker}}d_3=0\), and its right term is a kernel in \((\mathcal H^2)^2=0\). Hence \(\ker r_3=\mathcal H^{-1}=k\).

Moreover, \[q_l(\mathop{\mathrm{im}}r_{l-1})=\mathop{\mathrm{im}}d'_{l-1}=\ker d_l.\] For \(l\ne0\), \(q_l\) is an isomorphism, so \(r_{l-1}\) is onto. At \(l=0\), the vanishing \(\mathcal H^{-2}=0\) gives \(N^0=k\), while the whole incoming source of \(r_{-1}\) is \(\mathcal H^{-5}=0\). Consequently \(\mathop{\mathrm{coker}}r_{l-1}\) is \(k\) at \(l=0\) and zero elsewhere.

The final long exact sequence gives \[0\longrightarrow\mathop{\mathrm{coker}}r_{l-1} \longrightarrow\underline{\mathop{\mathrm{Hom}}}_E(X,\mathcal L[l]) \longrightarrow\ker r_l\longrightarrow0.\] The two scalar groups therefore survive precisely at \(l=0,3\). This argument permits an additional component of the top attaching map into the bottom level: \(N^4=0\) and \(\mathcal H^{-5}=0\) already force the asserted kernel and cokernel for the full \(r_l\). Finally \(YX=(\mathcal IX)[-1]=\mathcal L[3]\), so \(l=a+3\). This gives exactly (43). ◻

Proof. Represent the perfect intermediate object by a bounded complex \(P\) of finite projective bimodules. We first replace it by a bounded complex \(P'\) whose terms belong to \(\mathop{\mathrm{add}}(DB\otimes B)\).

For a finite bimodule \(L\), define \[J_B(L)=\mathop{\mathrm{Hom}}_k(B,L),\qquad (a\phi b)(c)=\phi(ca)b,\qquad j_L(z)(c)=cz.\] The map \(j_L:L\to J_B(L)\) is a natural bimodule monomorphism. Set \(C^0(L)=L\) and \(C^{r+1}(L)=\mathop{\mathrm{coker}}j_{C^r(L)}\). These are additive functors and preserve split retracts. For a fixed \(d=4\) we have the functorial exact sequence \[ 0\longrightarrow L\longrightarrow J_B(C^0(L)) \longrightarrow\cdots\longrightarrow J_B(C^{d-1}(L)) \longrightarrow C^d(L)\longrightarrow0. \tag{50}\]

For the free bimodule \(L=B\otimes B\), this construction is the left-module coinduced coresolution of \(B\), tensored over \(k\) with the right regular module \(B\). This follows from \(\mathop{\mathrm{Hom}}_k(B,B\otimes B)=\mathop{\mathrm{Hom}}_k(B,B)\otimes B\) and from exactness of tensoring vector spaces. Each coinduced left-module term is a finite injective, hence belongs to \(\mathop{\mathrm{add}}(DB)\). The last left-module cokernel is also injective: by Proposition 13, \(\mathop{\mathrm{id}}({}_BB)\leq4\), and dimension shifting gives, for every finite left module \(V\), \[\mathop{\mathrm{Ext}}_B^1(V,C^d(B)) \cong\mathop{\mathrm{Ext}}_B^{d+1}(V,B)=0.\] Here \(C^d(B)\) denotes the same coresolution construction on the left regular module. Thus every term on the right of \(L\) in (50), including the last cokernel, lies in \(\mathop{\mathrm{add}}(DB\otimes B)\) when \(L=B\otimes B\). The assertion extends to every finite projective bimodule \(L\), since the functors in (50) preserve finite sums and split retracts.

Apply this one fixed finite coresolution to each term of \(P\). Naturality provides the horizontal maps, giving a bounded double complex. Put the first coinduced term in homological degree \(0\) relative to \(L\), and the last cokernel in degree \(-d\). Its totalization \(P'\) is bounded and has all terms in \(\mathop{\mathrm{add}}(DB\otimes B)\). The augmentation \(P\to P'\) is a quasi-isomorphism: filtering its cone by the finitely many original degrees has exact associated complexes (50). Thus the given derived factorization can be written \(B\to P'\to DB\).

For \(V\in\mathop{\mathrm{add}}(DB\otimes B)\) and \(i>0\), we next prove \[ \mathop{\mathrm{Ext}}_{B^e}^i(B,V)=0,\qquad \mathop{\mathrm{Ext}}_{B^e}^i(V,DB)=0. \tag{51}\] It suffices to take \(V=DB\otimes B\). Coinduction identifies, for a bimodule \(Q\), \[\mathop{\mathrm{Hom}}_{B^e}(Q,DB\otimes B) \cong\mathop{\mathrm{Hom}}_{B^{\mathrm{op}}}(\mathop{\mathrm{Res}}Q,B).\] A projective bimodule resolution of \(B\) restricts to a right-projective resolution of the free right module \(B\). This proves the first vanishing. For the second, tensor a left-projective resolution \(Q_\bullet\) of \(DB\) over \(k\) with the right regular module \(B\). It is a projective bimodule resolution of \(DB\otimes B\), and \[\mathop{\mathrm{Hom}}_{B^e}(Q_\bullet\otimes B,DB) \cong\mathop{\mathrm{Hom}}_B(Q_\bullet,DB).\] Its positive cohomology vanishes since \({}_BDB\) is injective. Both assertions pass to finite sums and summands.

Finally choose a projective resolution \(Q_B\to B\) and an injective coresolution \(DB\to I_{DB}\), regarded as homological complexes. There are natural quasi-isomorphisms of Hom complexes \[ \begin{split} \mathop{\mathrm{Hom}}^\bullet_{B^e}(B,P')&\longrightarrow \mathop{\mathrm{Hom}}^\bullet_{B^e}(Q_B,P'),\\ \mathop{\mathrm{Hom}}^\bullet_{B^e}(P',DB)&\longrightarrow \mathop{\mathrm{Hom}}^\bullet_{B^e}(P',I_{DB}). \end{split} \tag{52}\] To verify them, filter by the finitely many terms of \(P'\). For each successive quotient the claim is one of (51), together with equality in degree zero and vanishing of negative Ext between modules. A shift of a term merely shifts this comparison. Hence terms in negative degrees cause no exception, and induction on this finite filtration proves (52).

Taking degree-zero cohomology shows that both derived arrows have actual chain-map representatives \(\alpha:B[0]\to P'\) and \(\beta:P'\to DB[0]\). Their composite is the ordinary map \(\beta_0\alpha_0\), so it factors through \(P'_0\in\mathop{\mathrm{add}}(DB\otimes B)\). It equals \(v\) as an ordinary bimodule map, because \(\mathop{\mathrm{Hom}}_{\mathrm D(B^e)}(B,DB)=\mathop{\mathrm{Hom}}_{B^e}(B,DB)\). ◻

Proof. The character identities make \(\eta\) a bimodule map. By Lemma 22, a perfect factorization would give an ordinary factorization through \(\mathop{\mathrm{add}}(DB\otimes B)\). We show that every such factorization \(B\to DB\) sends \(1\) to a functional vanishing on \(p_f\).

First consider maps \[B\xrightarrow{\alpha}DB\otimes B\xrightarrow{\beta}DB.\] Identify \(DB\otimes B\) with \(\mathop{\mathrm{Hom}}_k(B,B)\) by \(\lambda\otimes c\mapsto[x\mapsto\lambda(x)c]\). The element \(\alpha(1)\) satisfies \(a\alpha(1)=\alpha(1)a\). Under this identification, its corresponding linear map \(f_0\) therefore satisfies \[f_0(xa)=f_0(x)a\quad(x,a\in B).\] Thus \(f_0(x)=d_0x\), where \(d_0=f_0(1)\). If \((c_i)\) is any vector-space basis of \(B\), with dual basis \((c_i^*)\), then \[\alpha(1)=\sum_i c_i^*\otimes d_0c_i.\] Right linearity of \(\beta\) reduces it to the left-module endomorphism \(\lambda\mapsto\beta(\lambda\otimes1)\) of \(DB\). Duality identifies these endomorphisms with endomorphisms of the right regular module \(B\), which are left multiplications. Thus the endomorphism of \(DB\) is right multiplication by some \(b'\in B\). Consequently \[\beta(\lambda\otimes c)=\lambda b'c,\qquad (\beta\alpha(1))(p_f) =\sum_i c_i^*(b'd_0c_ip_f) =\mathop{\mathrm{tr}}_k\bigl(L_{b'd_0}R_{p_f}\bigr).\] Here \(L_a\) and \(R_a\) are left and right multiplication on the vector space \(B\). The order \(b'd_0\) follows from the dual action \((\lambda b'c)(p_f)=\lambda(b'cp_f)\).

Write \(b'd_0=r+\sum_v a_vp_v\), where \(r\in\mathop{\mathrm{rad}}B\) and the \(p_v\) are its four vertex idempotents. Left and right multiplications commute, and \(p_f^2=p_f\), so \[(L_rR_{p_f})^n=L_{r^n}R_{p_f}=0\] for sufficiently large \(n\). The radical part has trace zero. For each \(v\), the operator \(L_{p_v}R_{p_f}\) is a projection onto \(p_vBp_f\). The corner dimensions of \(B\), obtained from the corner table (23), are even; in particular \(\dim_k(p_vBp_f)=4\) for every \(v\). Each projection therefore has trace zero in characteristic two. Linearity gives \((\beta\alpha(1))(p_f)=0\).

A factorization through a finite sum of \(DB\otimes B\) is a sum of such composites. A factorization through a summand extends to that finite sum by its split inclusion and projection. The same vanishing thus holds throughout \(\mathop{\mathrm{add}}(DB\otimes B)\). But \(\eta(1)(p_f)=\chi_B(p_f)=1\), a contradiction. ◻

Proof. The ordinary adjoint of \(f:L\to S\) is \(\widehat f(l\otimes x)=f(l)(x)\). A factorization of \(f\) through a finite \(E^e\)-projective evaluates to a factorization through an \(E\)-projective.

Conversely embed \(L\) in a finite projective-injective \(E^e\)-module \(J\). The embedding splits as a right \(E\)-module map because \(L_E\) is projective, hence injective over symmetric \(E\). Thus \(LX\to JX\) is injective. Suppose \(\widehat f\) factors as \(LX\to P\to X\) with \(P\) finite \(E\)-projective. Injectivity of \(P\) extends its first factor to \(JX\to P\). Composing with \(P\to X\) and applying ordinary adjunction gives \(J\to S\) whose restriction to \(L\) is \(f\). This is a bimodule projective factorization. The two factorization ideals therefore correspond exactly. Naturality is inherited from ordinary adjunction. ◻

Proof. Write \(U=U_H\) and \(\phi=\pi\overline m:\mathcal I\to S\). All shifts of \(U\) and \(\mathcal I\) have finite side-projective representatives by Lemma 9. Consider the finite-dimensional linear map \[\phi_*:\underline{\mathop{\mathrm{Hom}}}_{E^e}(U[1],\mathcal I) \longrightarrow\underline{\mathop{\mathrm{Hom}}}_{E^e}(U[1],S).\] Symmetric duality, Lemma 5 applied over \(E^e\), identifies its transpose with \[ \underline{\mathop{\mathrm{Hom}}}_{E^e}(S,U)\longrightarrow\underline{\mathop{\mathrm{Hom}}}_{E^e}(\mathcal I,U), \qquad b\longmapsto b\phi. \tag{54}\] Indeed the fixed first argument is \(U[1]\), and its duality shift is \(U[1][-1]=U\); naturality in the second argument gives the displayed precomposition.

Let \(\zeta=f^*\otimes f^*\in E\), as in Lemma 17. Both its left and right actions are the character \(\chi\). The map \[b_0:S\longrightarrow U,\qquad b_0(1)=\zeta\] is a bimodule map even for the twisted right action, because \[\zeta\cdot b=\zeta\sigma_H(b) =\chi(\sigma_H(b))\zeta=\chi(b)\zeta.\] We will prove that \(b_0\phi\ne0\) stably.

Suppose otherwise. The derived map \(b_0\pi m:K\to U\) then has zero stable image and factors through a perfect \(E^e\)-complex by Lemma 8. To apply that lemma with projective models, one may take a bounded-below projective resolution \(Q\to U\) with finite terms and lift the global chain map \(K\to U\) to \(K\to Q\); its stable image is the same map \(b_0\phi\).

Restriction of this factorization to \(B^e\) still has a perfect intermediate object. Indeed an \(E^e\)-free module restricts to the outer bimodule \(E\otimes E\). Tensoring the left and right \(B\)-projective resolutions of \(E\) from Proposition 13 gives a \(B^e\)-projective resolution of length at most \(8\). Finite sums and summands have the same bound. A bounded complex of these restricted modules is perfect by its finite term filtration.

Let \(\operatorname{Bar}(B)\to B\) denote the bar resolution. Inclusion of both outer \(B\)-factors in \(E\) defines a \(B^e\)-chain map \[\iota:\operatorname{Bar}(B)\longrightarrow\mathop{\mathrm{Res}}_{B^e}K,\] and hence a derived map \(B\to\mathop{\mathrm{Res}}_{B^e}K\). Since \(\sigma_H\) fixes \(B\), the decomposition \[\mathop{\mathrm{Res}}_{B^e}U =B\oplus(DC\otimes C)\oplus(C\otimes DC)\oplus DB\] is a decomposition of \(B\)-bimodules. Let \(\rho\) be projection to the last summand. The composite \[\operatorname{Bar}(B)\xrightarrow{\iota}\mathop{\mathrm{Res}}K \xrightarrow{\,b_0\pi m\,}\mathop{\mathrm{Res}}U\xrightarrow{\rho}DB\] is exactly the character map \(\eta\) of (53) composed with the bar augmentation: in degree zero it sends \(a\otimes c\) to \(\chi_B(ac)\chi_B\), since \(\rho(\zeta)=\chi_B\), and in higher degrees it is zero. Thus our supposed factorization would make \(\eta\) factor through a perfect \(B^e\)-complex. Lemma 23 excludes this possibility. We have proved \(b_0\phi\ne0\).

It follows from (54) that \(\phi_*\) is nonzero. Choose \(\beta:U[1]\to\mathcal I\) with \(\phi\beta\ne0\). Lemma 24, applied only to the side-projective source \(U[1]\), identifies this nonzero class with the composite \[U[1]X\xrightarrow{\beta X}\mathcal IX \xrightarrow{\overline m X}EX\cong X .\] This identity follows from ordinary adjunction and from the evaluation of the representation \(\pi\). The evaluated maps involving \(U[1],\mathcal I,E\) are defined stably by Lemma 9; no stable evaluation functor on the nonside-projective module \(S\) is needed. The composite is nonzero, so \(\beta X\) itself is nonzero. The canonical shift comparison identifies \(U[1]X\) with \((UX)[1]=X[1]\), proving the assertion. ◻

Proof of Proposition 19. Take the fixed finite side-projective \(Y\) constructed in Lemma 20. Lemma 21 proves its required Hom profile. For any \(H\in k^\times\), shift the map of Lemma 25 by \([-1]\) and identify \(\mathcal I[-1]\) with \(Y\) to obtain a stable map \(U_H\to Y\). By the shift and evaluation compatibilities of Lemma 9, its evaluation is a nonzero class \(X\to YX\). A stable map between the fixed finite modules is represented by an ordinary bimodule map; choose one and call it \(g_H\). All Hom spaces, dualities, and selections used above are over the original field \(k\), so this requires no scalar extension. The construction of \(Y\) did not depend on \(H\). ◻

Proof. The projection \(p^a\) in (56) is an isomorphism for \(a>0\). For \(a\geq2\) this follows from the support of \(W\); at \(a=1\) it also uses the surjectivity of \((\mathcal H^0)^2\to W^0\). The same support argument applies for \(a\leq-4\). At \(a=-3\), both \(W^{-4}\) and \(\mathcal H^{-3}\) are zero, so \(\mathcal V^{-3}=0\) and \(p^{-3}\) is again an isomorphism. The remaining relevant low degrees can be read directly from (56): \[ \mathcal V^{-2}=k, \qquad \mathcal V^0=kv_0, \qquad \mathcal V^1=0. \tag{58}\]

Naturality of the bimodule projections \(F\to U_i\) gives \[ p^a\Delta^a(g,h)= \bigl(U_1(g)+h,\ U_2(g)+h\bigr). \tag{59}\] Indeed each projection sends \(v_0\) to the identity of \(X\), and evaluation of a bimodule map commutes with the specified shift comparisons. By Proposition 18, at \(a=3m>0\) the map in (59) is the scalar matrix \[\begin{pmatrix}H_1^{-m}&1\\H_2^{-m}&1\end{pmatrix} \otimes 1_{\mathcal P_m},\] and at \(a=-3m-1\leq-4\) it is \[\begin{pmatrix}H_1^{m+2}&1\\H_2^{m+2}&1\end{pmatrix} \otimes 1_{D\mathcal P_m}.\] In both cases \(m\geq1\). Algebraic independence of \(H_1,H_2\) implies that each determinant is nonzero in \(k\), for every such \(m\). In all other degrees in the asserted ranges, the source and target are zero by Proposition 16 and the already proved isomorphism \(p^a\).

Finally, \(\Delta^0(c,d)=(c+d)v_0\) is surjective. Since \(\mathcal H^{-2}=0\), the map \(\Delta^{-2}:0\to\mathcal V^{-2}=k\) is injective. Its surjectivity is neither claimed nor needed. ◻

Proof. Both column complexes have projective terms and are exact. Since \(\Lambda=\mathsf D_0(E)\oplus\mathsf D_1(E)\), (61)–(62) identify their Hom complexes into \(\Lambda\) with sums of \[\mathop{\mathrm{Hom}}_E(P,E),\qquad\mathop{\mathrm{Hom}}_E(P,F),\qquad0.\] These are exact: \(P\) is totally acyclic and \(F\) is finite projective on the left. Hence both column complexes are totally acyclic, and so is their cone.

For an explicit finite description, \(Z\) is the triple with upper component \(X\) and lower component \[ N_Z=(FX\oplus Ep_f) \big/\{(\widetilde v(x),i_X(x)):x\in X\}, \tag{64}\] whose structure map \(FX\to N_Z\) is induced by the first summand. To verify this formula, first quotient the degree-zero lower component by the image of the differential in \(\mathsf D_0(P)\). This gives \(FX\) in its first summand. The remaining relations arise from \(P_0\) in the shifted second column. Its two components induce \(\widetilde v\epsilon\) and \(i_X\epsilon\), respectively, yielding (64).

The upper-component functor sends the projective generators \(\mathsf D_0(E)\) and \(\mathsf D_1(E)\) to \(E\) and \(0\), and therefore sends every finite projective \(\Lambda\)-module to an \(E\)-projective. But \(X\) is nonprojective, since its degree-three self-extension group is nonzero by Proposition 16. Thus \(Z\) is nonprojective. ◻

Proof. Write \(A_0=\mathsf D_0(P)\) and \(B_0=\mathsf D_1(P)\), so that the underlying graded complex of \(P_Z\) is \(A_0\oplus B_0[1]\). By (62), the block \(\mathop{\mathrm{Hom}}_\Lambda^\bullet(A_0,B_0[1])\) is zero. Projection onto the two diagonal blocks therefore gives a short exact sequence of full Hom complexes \[ \begin{split} 0\longrightarrow\mathcal C^\bullet \longrightarrow\mathop{\mathrm{End}}_\Lambda^\bullet(P_Z) \longrightarrow \mathop{\mathrm{End}}_\Lambda^\bullet(A_0)\oplus \mathop{\mathrm{End}}_\Lambda^\bullet(B_0[1]) \longrightarrow0,\\[-2pt] \mathcal C^\bullet=\mathop{\mathrm{Hom}}_\Lambda^\bullet(B_0[1],A_0). \end{split} \tag{65}\] The underlying degreewise Hom spaces are products as in Lemma 4; the block decomposition has only four blocks. In particular, it does not replace any product by a direct sum.

The diagonal cohomology is \((\mathcal H^a)^2\). To fix the cross-block index explicitly, in degree \(a\) we have \[\begin{align*} \mathcal C^a &=\prod_j\mathop{\mathrm{Hom}}_E(P_{j-1},FP_{j-a})\\ &=\prod_i\mathop{\mathrm{Hom}}_E(P_i,FP_{i-(a-1)}) =\mathop{\mathrm{Hom}}_E^{a-1}(P,FP). \end{align*}\] The differentials agree under this reindexing in characteristic two. Lemma 4 consequently gives \(H^a\mathcal C^\bullet=\mathcal V^{a-1}\).

The connecting map in (65), from diagonal cohomology in degree \(a\) to \(H^{a+1}\mathcal C^\bullet=\mathcal V^a\), is precisely \(\Delta^a\). Indeed, lift two diagonal cocycles to the diagonal blocks of an endomorphism of the cone and take its cochain differential. The cross component is the sum of its two compositions with \(V\). The first diagonal acts downstairs by \(F\) of the corresponding map, while the second acts by precomposition. Under the composition-compatible comparison in Lemma 4, this gives \(F(g)v_0+v_0[a]h\), as in (57). Thus the long exact sequence gives \[ 0\longrightarrow\mathop{\mathrm{coker}}\Delta^{a-1} \longrightarrow H^a\mathop{\mathrm{End}}_\Lambda^\bullet(P_Z) \longrightarrow\ker\Delta^a\longrightarrow0. \tag{66}\]

For \(a>1\) or \(a\leq-3\), both outer terms vanish by Lemma 26. At \(a=1\), they vanish by surjectivity of \(\Delta^0\) and injectivity of \(\Delta^1\). At \(a=-2\), they vanish by the isomorphism \(\Delta^{-3}\) and injectivity of \(\Delta^{-2}\). Finally, apply Lemma 4 to the totally acyclic complex \(P_Z\) to identify this cohomology with \(H^a\mathop{\mathrm{Hom}}_\Lambda(P_Z,Z)\). ◻

Proof of Theorem 1. Fix the finite bimodule maps in (55), the complete resolution \(P\), and the lifted map used in (63). These choices exist over \(k\) by the preceding propositions. The algebra \(\Lambda\) in (60) and its module \(Z\) in (64) are finite-dimensional. By Lemma 27, \(Z\) is nonprojective and has the totally acyclic resolution \(P_Z\). Proposition 28 supplies both hypotheses on its complete self-Hom groups in Proposition 11. Apply that proposition with \(R=\Lambda\) to obtain \[A=\Lambda\ltimes D\Lambda, \qquad M=A\otimes_\Lambda Z.\] It proves that \(A\) is finite-dimensional and symmetric, that \(M\) is finite-dimensional and nonprojective, and that \(\mathop{\mathrm{Ext}}_A^i(M,M)=0\) for every \(i>0\). These are all the assertions of Theorem 1. ◻