# Ext groups of modules as cohomology of spectral nerves

*Written by Claude Opus 5.5 (Anthropic) on 21 October 2026. Self-checked by the writing AI. Public domain (CC0).*

Let \(A\) be a ring of a topos and \(G\), \(M\) two \(A\)-modules. The spectrum \(A(1)^\times\) of \(A\) is a graded monoid that acts on the spectrum \(G(1)^\times\) of \(G\) ([*Graded rings and the spectra of modules*](graded-rings-and-the-spectra-of-modules.md)), and the nerve of this action is a bisimplicial diagram whose values are products of copies of the sheaves of sets underlying \(A\) and \(G\). This lesson proves that \(\operatorname{Ext}^i_A(G,M)\) is the \(i\)-th cohomology group of the total topos of this nerve, with coefficients whose values are products of copies of \(M\). The ring and module structures enter only through the maps of the diagram.

The proof has three steps. The free abelian group on the spectrum of \(G\) is a module \(X_G\) over the ring \(Q_A=\mathbb Z(C_\bullet(A(1)^\times))\), and the spectrum of \(M\) is a \(Q_A\)-module whose underlying object is constant in the sense of [*Diagonal rings and adjunction formulas*](diagonal-rings-and-adjunction-formulas.md). The comparison theorem of that lesson and Mac Lane's theorem ([*The stable ring of a ring and the Mac Lane resolution*](the-stable-ring-of-a-ring-and-the-mac-lane-resolution.md)) identify \(\operatorname{RHom}_{Q_A}(X_G,C_\bullet(M(1)))\) with \(\operatorname{RHom}_A(G,M)\). The \(Q_A\)-modules are the equivariant abelian sheaves on the pseudo-category of the action of \(A(1)^\times\), and \(X_G\) is the extension by zero of the constant sheaf from the localization at the spectrum of \(G\). The cohomology of equivariant modules ([*Cohomology of equivariant modules*](cohomology-of-equivariant-modules.md)) then passes to the total topos.

The coefficients can be taken in the derived tensor product \(A\otimes^{\mathbf L}_{\mathbb Z}\mathcal O\) with a commutative ring \(\mathcal O\), which is what deformation problems over a base that is not flat over \(\mathbb Z\) require. For every bounded complex \(M\) of modules over \(A\otimes^{\mathbf L}_{\mathbb Z}\mathcal O\), the lesson constructs an object \(\Psi(M)\) of the derived category of \(Q_A\otimes\mathcal O\)-modules whose underlying \(Q_A\)-module is the spectrum of the underlying complex of \(A\)-modules, and shows that \(\Psi\) is fully faithful on complexes with cohomology in two consecutive degrees. The construction rests on a base change form of the comparison theorem.

**Prerequisites.** [*The stable ring of a ring and the Mac Lane resolution*](the-stable-ring-of-a-ring-and-the-mac-lane-resolution.md) (Lemma 1.1, Propositions 1.2, 2.2 and 2.4, Construction 3.1, Theorem 3.2); [*Diagonal rings and adjunction formulas*](diagonal-rings-and-adjunction-formulas.md) (Propositions 1.1, 2.3, 3.4 and 3.6, Construction 2.4, Theorem 4.1, Corollary 4.2); [*Graded rings and the spectra of modules*](graded-rings-and-the-spectra-of-modules.md) (Lemma 1.1, Definitions 1.2 and 3.2, Propositions 1.3, 1.4, 2.1, 2.2, 3.1 and 3.4, Remark 3.3); [*Graded monoids and the nerves of their actions*](graded-monoids-and-the-nerves-of-their-actions.md) (conventions, 2.4, Proposition 2.3, Proposition 3.3); [*Differential graded modules in a topos*](differential-graded-modules-in-a-topos.md) (Lemma 1.1, Propositions 2.3, 2.5, 2.6 and 2.7); [*Equivariant sheaves on simplicial topoi*](equivariant-sheaves-on-simplicial-topoi.md) (Definitions 1.1 and 2.1, Section 6); [*Cohomology of equivariant modules*](cohomology-of-equivariant-modules.md) (the comparison maps of Section 1, Corollary 4.2); [*Multisimplicial objects and spectral diagrams*](multisimplicial-objects-and-spectral-diagrams.md) (Lemmas 1.1 and 1.2, Propositions 2.4, 2.5 and 3.5). From the AI-integrated Stacks project: localization of a ringed topos at a sheaf [[Stacks, Tag 04IE](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-modules.html#sites-modules-lemma-localize-ringed-topos)] and exactness of extension by zero [[Tag 03DJ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-modules.html#sites-modules-lemma-extension-by-zero-exact)]; right adjoints of exact functors preserve injectives [[Tag 015Z](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/homology.html#homology-lemma-adjoint-preserve-injectives)]; bounded below complexes of injectives are K-injective [[Tag 070J](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-bounded-below-injectives-K-injective)]; resolutions by a generating class and left derived functors through an adapted class [[Tag 05T7](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-subcategory-left-resolution), [Tag 05T9](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-subcategory-left-acyclics), [Tag 05TA](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-proposition-enough-acyclics)] and their adjunctions [[Tag 0DVC](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/derived.html#derived-lemma-pre-derived-adjoint-functors)]; bounded above complexes of flat modules are K-flat [[Tag 06YQ](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-bounded-flat-K-flat)]; K-flat resolutions [[Tag 06YS](https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/sites-cohomology.html#sites-cohomology-lemma-K-flat-resolution)].

## Conventions

\(T\) is the topos of sheaves on a site, \(\mathcal A=\operatorname{Ab}(T)\) and \(\otimes=\otimes_{\mathbb Z}\). \(A\) is a ring of \(T\), associative and unital, and \(\mathcal O\) is a commutative ring of \(T\). The notation \(\mathcal M_a\), \(c\), \((1)\), \((-1)\), \({}^{\mathrm{red}}\), \(\Delta^2\), \(\Phi=\operatorname{Tot}N\Delta^2\), \(\Lambda\) and \(\theta\) is that of the lesson on diagonal rings; \(U\) forgets every structure except the underlying object of \(\mathcal M_a\), and \(D\) denotes unbounded derived categories. For an object \(L\) of \(\mathcal M_a\) and an abelian sheaf \(W\), \(L\otimes W\) has the components \(L_q(c)\otimes W\). The hypothesis (H) of the lesson on graded monoids holds for \(S_1\), whose products are the concatenations of coordinates.

*The ring \(Q\).* \(Q=Q_A=\mathbb Z(C_\bullet(A(1)^\times))\) is the ring of type \(C_\bullet(S_1)\) of Construction 2.1 of the lesson on the stable ring; its components \(\mathbb Z(A\langle n\rangle(r))\) are flat. Further \(B_A=N\mathbb Z^{\mathrm{st}}(A)=N\Delta^2Q^{\mathrm{red}}(-1)\), with the augmentation \(\varepsilon\colon B_A\to A\), and \(P_A=t_1(B_A)\), with the projection \(\pi\colon B_A\to P_A\) and the quasi-isomorphism \(w\colon P_A\to A\) induced by \(\varepsilon\) (Proposition 2.4 (1) there), so that \(\varepsilon=w\pi\). For an \(A\)-module \(G\), \(X_G=\mathbb Z(C_\bullet(G(1)^\times))\) is the \(Q\)-module of Construction 3.1 there; \(\Phi(X_G^{\mathrm{red}}(-1))=N\mathbb Z^{\mathrm{st}}(G)\), and \(N\varepsilon_G\colon N\mathbb Z^{\mathrm{st}}(G)\to\varepsilon^*G\) is a map of \(B_A\)-modules. For a complex \(Y\) of \(Q\)-modules put \(\Phi_{\mathbb Z}(Y)=\Phi(Y^{\mathrm{red}}(-1))\), a \(B_A\)-module; this is an exact functor \(D(Q)\to D(B_A)\).

*Coefficients.* \(Q_{\mathcal O}=Q\otimes\mathcal O\), with the multiplication \((x\otimes a)(y\otimes b)=xy\otimes ab\), is a ring of type \(C_\bullet(S_1)\). Its modules are the \(Q\)-modules whose components are \(\mathcal O\)-modules, the action of \(Q\) and the structure maps being \(\mathcal O\)-linear; \(Y|_Q\) denotes the underlying \(Q\)-module. Put \(B_A^{\mathcal O}=N\mathcal O^{\mathrm{st}}(A)=B_A\otimes\mathcal O\) (Lemma 1.1), with \(\lambda\colon B_A\to B_A^{\mathcal O}\), \(b\mapsto b\otimes1\), and \(R_{\mathcal O}=t_1(B_A^{\mathcal O})=P_A\otimes\mathcal O\), with the projection \(\pi_{\mathcal O}\colon B_A^{\mathcal O}\to R_{\mathcal O}\). By Proposition 2.4 (2) of the lesson on the stable ring, \(R_{\mathcal O}\) represents \(A\otimes^{\mathbf L}_{\mathbb Z}\mathcal O\), and \(D(R_{\mathcal O})\) serves as the derived category of \(A\otimes^{\mathbf L}_{\mathbb Z}\mathcal O\); another representative gives an equivalent category (Proposition 1.2 there and Proposition 2.6 of the lesson on differential graded modules). The ring \(R_{\mathcal O}\) lies in degrees \(-1\) and \(0\), \(\mathcal O\) is central in it, and \(H^0(R_{\mathcal O})=A\otimes\mathcal O\). Let \(\bar\varepsilon\colon R_{\mathcal O}\to A\otimes\mathcal O\) be the projection, \(\varepsilon_{\mathcal O}=\bar\varepsilon\pi_{\mathcal O}=\varepsilon\otimes\mathcal O\), \(j=t_1(\lambda)\colon P_A\to R_{\mathcal O}\) and \(\iota\colon A\to A\otimes\mathcal O\), \(a\mapsto a\otimes1\). Then
\[
j\pi=\pi_{\mathcal O}\lambda,\qquad\bar\varepsilon j=\iota w:
\tag{0.1}
\]
the first because the projection onto \(t_1\) is natural, the second because both maps are \(\varepsilon\) followed by \(\iota\) in degree \(0\) and vanish in degree \(-1\). For \(M\in D(R_{\mathcal O})\), its *underlying complex of \(A\)-modules* is \(M|_A=w_!j^*M\in D(A)\), where \(w_!\) is the extension of scalars along \(w\), a quasi-inverse of \(w^*\). By (0.1) and \(w^*w_!\cong\mathrm{id}\), there are natural isomorphisms
\[
\begin{gathered}\varepsilon^*(M|_A) \cong\pi^*j^*M\\ =\lambda^*\pi_{\mathcal O}^*M,\\ (\bar\varepsilon^*H)|_A\cong\iota^*H,\end{gathered} \tag{0.2}
\]
for \(M\in D(R_{\mathcal O})\) and complexes \(H\) of \(A\otimes\mathcal O\)-modules. For a complex \(Y\) of \(Q_{\mathcal O}\)-modules, \(\Phi_{\mathcal O}(Y)=\Phi(Y^{\mathrm{red}}(-1))\) is a \(B_A^{\mathcal O}\)-module (Lemma 1.1).

*Constant type.* An object \(Y\) of \(D(Q)\) or of \(D(Q_{\mathcal O})\) is *of constant type* if \(UY\cong(cH)(1)\) in \(D(\mathcal M_1)\) for some \(H\in D(\mathcal A)\). By Proposition 3.6 (2) of the lesson on diagonal rings, an object of \(D^-\) is of constant type exactly when every cohomology object of \(UY\) is isomorphic to \(C_\bullet(H_i(1))\) for an abelian sheaf \(H_i\).

## 1. Base change of the comparison theorem

**Lemma 1.1.**

1. Let \(F\) be an additive functor between categories of abelian sheaves or of sheaves of modules, such as \(-\otimes W\) for an abelian sheaf \(W\), or \(-\otimes_{\mathcal O}W\) on \(\mathcal O\)-modules for an \(\mathcal O\)-module \(W\). Applied componentwise, \(F\) commutes with reduction, with \((-1)\) on reduced objects, with \(\Delta^2\) and with normalization, compatibly with multiplications and actions.
2. \(Q_{\mathcal O}^{\mathrm{red}}(-1)=Q^{\mathrm{red}}(-1)\otimes\mathcal O\) and \(N\Delta^2Q_{\mathcal O}^{\mathrm{red}}(-1)=B_A\otimes\mathcal O=B_A^{\mathcal O}\) as differential graded rings, and \(\Phi_{\mathbb Z}(Y|_Q)=\lambda^*\Phi_{\mathcal O}(Y)\) for every complex \(Y\) of \(Q_{\mathcal O}\)-modules.
3. For a complex \(Y\) of \(Q_{\mathcal O}\)-modules and a complex \(W\) of \(\mathcal O\)-modules, \(Y\otimes_{\mathcal O}W\), formed componentwise, is a complex of \(Q_{\mathcal O}\)-modules, and \(\Phi_{\mathcal O}(Y\otimes_{\mathcal O}W)=\Phi_{\mathcal O}(Y)\otimes_{\mathcal O}W\) as \(B_A^{\mathcal O}\)-modules, \(B_A^{\mathcal O}\) acting on the first factor.

**Proof.** (1) The normalization in each coordinate is the image of an idempotent made of integer combinations of faces and degeneracies (proof of Lemma 1.1 (2) of the lesson on graded rings), and the Dold–Kan functor \(\Gamma\) is made of finite sums and such maps. Up to the canonical isomorphisms, reduction is \(\Gamma\) applied to the restriction of the normalization to the multidegrees \(\geq\mathbf 1\), and \((-1)\) is \(\Gamma\) applied to the translate of the normalization (Lemmas 1.1 and 1.2 (3) and Propositions 2.4 and 2.5 of the lesson on multisimplicial objects); \(\Delta^2\) is made of sums and evaluations. An additive functor preserves finite sums and images of idempotents, so it commutes with all of these. The multiplications of \(L^{\mathrm{red}}\), \(L(-1)\) and \(\Delta^2L\) are induced by those of \(L\) through isomorphisms that are identities on normalizations (Lemma 1.1 of the lesson on graded rings), and the shuffle maps that define the products of normalized complexes are integer combinations of degeneracies; so the identifications respect multiplications and actions. (2) is (1) for \(F=-\otimes\mathcal O\), since the multiplication of \(Q_{\mathcal O}\) is that of \(Q\) tensored with that of \(\mathcal O\); restriction of scalars along \(Q\to Q_{\mathcal O}\) commutes with all the constructions, which gives the last formula. (3) is (1) for \(F=-\otimes_{\mathcal O}W^k\), for each term \(W^k\), followed by the total complex, which is a direct sum. \(\square\)

**Proposition 1.2 (base change).** Let \(\mathcal R\) be a ring of type \(C_\bullet(S_0)\) whose components are flat abelian sheaves, \(\mathcal B=N\Delta^2\mathcal R\), \(\mathcal R_{\mathcal O}=\mathcal R\otimes\mathcal O\) and \(\mathcal B_{\mathcal O}=N\Delta^2\mathcal R_{\mathcal O}=\mathcal B\otimes\mathcal O\) (Lemma 1.1). Let \(E\in D^-(\mathcal R_{\mathcal O})\) and \(F\in D^+(\mathcal R_{\mathcal O})\), and suppose that \(UF\cong cG\) in \(D(\mathcal M_0)\) for some \(G\in D(\mathcal A)\). Then
\[
\begin{aligned}&\theta_{E,F} \colon\operatorname{RHom}_{\mathcal R_{\mathcal O}}(E,F)\\&\quad \to\operatorname{RHom}_{\mathcal B_{\mathcal O}}(\Phi E,\Phi F)\end{aligned}
\]
is an isomorphism.

**Proof.** By Construction 2.4 of the lesson on diagonal rings, it suffices to show that \(\Phi\) induces bijections \(\operatorname{Hom}_{D(\mathcal R_{\mathcal O})}(E,F[k])\to\) \(\operatorname{Hom}_{D(\mathcal B_{\mathcal O})}(\Phi E,\Phi F[k])\) for all \(k\).

*Extension of scalars.* Let \(\mathcal P\) be the class of the \(\mathcal R\)-modules whose components are flat. The functor \(e=\mathcal O\otimes-\), from \(\mathcal R\)-modules to \(\mathcal R_{\mathcal O}\)-modules, is left adjoint to the restriction of scalars \(r\): an \(\mathcal R_{\mathcal O}\)-linear map \(f\colon\mathcal O\otimes E_1\to F_1\) corresponds to \(x\mapsto f(1\otimes x)\). Every \(\mathcal R\)-module \(M\) is a quotient of a module of \(\mathcal P\): let \(\mathbb Z(UM)\to UM\) be the sum map, \(\mathbb Z(-)\) being applied to each component; its source has flat components, and its adjoint \(\operatorname{ind}(\mathbb Z(UM))\to M\) is surjective, with \(\operatorname{ind}(\mathbb Z(UM))\in\mathcal P\) (Proposition 1.4 of the lesson on graded rings). An abelian sheaf is flat exactly when its groups of sections have no torsion (proof of Lemma 1.1 of the lesson on the stable ring), so a subsheaf of a flat sheaf is flat, \(\mathcal P\) is closed under kernels of surjections, and \(e\) carries short exact sequences of modules of \(\mathcal P\) to exact sequences. Hence \(e\) has a left derived functor on \(D^-\), computed on bounded above complexes of modules of \(\mathcal P\) [Stacks, Tags 05T9, 05TA]. Since \(r\) is exact, for \(E_1\in\mathcal P\) and \(F_1\in D(\mathcal R_{\mathcal O})\) the map \(f\mapsto r(f)\circ\eta\), with \(\eta\colon E_1\to r(\mathcal O\otimes E_1)\), \(x\mapsto1\otimes x\), is a bijection
\[
\begin{aligned}&\operatorname{Hom}_{D(\mathcal R_{\mathcal O})}(\mathcal O\otimes E_1,F_1)\\&\quad \to\operatorname{Hom}_{D(\mathcal R)}(E_1,rF_1)\end{aligned}
\]
[Stacks, Tag 0DVC]. On the differential graded side, for \(E_1\in\mathcal P\) the complex \(\Phi E_1\) is bounded above with flat terms, made of direct summands of sums of components of \(E_1\), so it is K-flat [Stacks, Tag 06YQ]; so is \(\mathcal B\). Let \(P'\to\Phi E_1\) be a quasi-isomorphism from a K-flat \(\mathcal B\)-module. For an acyclic complex \(V\) of abelian sheaves, \(P'\otimes V=P'\otimes_{\mathcal B}(\mathcal B\otimes V)\) is acyclic, so \(P'\) is K-flat as a complex of abelian sheaves. The extension of scalars of \(\Phi E_1\) along \(\mathcal B\to\mathcal B_{\mathcal O}\) is therefore \(P'\otimes_{\mathcal B}\mathcal B_{\mathcal O}=P'\otimes\mathcal O\), which maps quasi-isomorphically to \(\Phi E_1\otimes\mathcal O=\Phi(\mathcal O\otimes E_1)\): the cone of \(P'\to\Phi E_1\) is acyclic and K-flat, so it stays acyclic after tensoring with \(\mathcal O\). The unit of the extension of scalars at \(\Phi E_1\) is thus \(\Phi(\eta)\colon x\mapsto1\otimes x\), and by Proposition 2.5 (1) of the lesson on differential graded modules the map \(g\mapsto r'(g)\circ\Phi(\eta)\), with \(r'\) the restriction along \(\mathcal B\to\mathcal B_{\mathcal O}\), is a bijection
\[
\begin{aligned}&\operatorname{Hom}_{D(\mathcal B_{\mathcal O})}(\Phi(\mathcal O\otimes E_1),\Phi F_1)\\&\quad \to\operatorname{Hom}_{D(\mathcal B)}(\Phi E_1,r'\Phi F_1).\end{aligned}
\]

*The modules \(\mathcal O\otimes E_1\).* Since \(\Phi\) commutes with restriction of scalars, \(\Phi(r(f)\circ\eta)=r'(\Phi f)\circ\Phi(\eta)\). So the two bijections carry the map induced by \(\Phi\) on \(\operatorname{Hom}(\mathcal O\otimes E_1,F[k])\) to the map induced by \(\Phi\) on \(\operatorname{Hom}_{D(\mathcal R)}(E_1,rF[k])\). The latter is bijective by Theorem 4.1 of the lesson on diagonal rings, applied to \(E_1\in D^-(\mathcal R)\) and \(rF\in D^+(\mathcal R)\), for which \(U(rF)=UF\cong cG\). So \(\Phi\) is bijective on \(\operatorname{Hom}(\mathcal O\otimes E_1,F[k])\) for every \(E_1\in\mathcal P\) and every \(k\).

*Dévissage.* Every \(\mathcal R_{\mathcal O}\)-module \(M\) is a quotient of \(\mathcal O\otimes E_1\) with \(E_1=\operatorname{ind}(\mathbb Z(UM))\in\mathcal P\), through the \(\mathcal R_{\mathcal O}\)-linear map adjoint to the surjection \(E_1\to rM\). So every \(E\in D^-(\mathcal R_{\mathcal O})\) is isomorphic to a bounded above complex of such modules [Stacks, Tag 05T7]. The dévissage in the proof of Theorem 4.1 of the lesson on diagonal rings applies without change, with these modules in place of the induced modules: if \(F\in D^{\geq b}(\mathcal R_{\mathcal O})\), then \(cG\cong UF\) gives \(G\in D^{\geq b}(\mathcal A)\), and \(\Phi F\in D^{\geq b}(\mathcal B_{\mathcal O})\) because its underlying complex is \(\Lambda UF\cong\Lambda cG\cong G\) (Propositions 2.3 (4) and 3.4 (2) there). \(\square\)

**Corollary 1.3.** Let \(Y'\in D^-(Q_{\mathcal O})\) and \(Y\in D^+(Q_{\mathcal O})\), with \(Y\) of constant type. Then the map
\[
\begin{aligned}&\operatorname{RHom}_{Q_{\mathcal O}}(Y',Y)\\&\quad \to\operatorname{RHom}_{B_A^{\mathcal O}}\bigl(\Phi_{\mathcal O}Y',\Phi_{\mathcal O}Y\bigr),\end{aligned}
\]
composed of the map induced by \(L\mapsto L^{\mathrm{red}}(-1)\) and \(\theta\) for the ring \(Q_{\mathcal O}^{\mathrm{red}}(-1)\), defined for \(Q\) before Corollary 4.2 of the lesson on diagonal rings, is an isomorphism. In particular \(\Phi_{\mathcal O}\) induces bijections \(\operatorname{Hom}_{D(Q_{\mathcal O})}(Y',Y[k])\to\) \(\operatorname{Hom}_{D(B_A^{\mathcal O})}(\Phi_{\mathcal O}Y',\Phi_{\mathcal O}Y[k])\).

**Proof.** The proof of Corollary 4.2 of the lesson on diagonal rings applies to the ring \(Q_{\mathcal O}\), with Proposition 1.2 in place of Theorem 4.1 there. Indeed, Proposition 1.1 there holds for every ring of type \(C_\bullet(S_1)\); the ring \(Q_{\mathcal O}^{\mathrm{red}}(-1)=Q^{\mathrm{red}}(-1)\otimes\mathcal O\) (Lemma 1.1) is of the form of Proposition 1.2, because the components of \(Q^{\mathrm{red}}(-1)\) are flat (proof of Corollary 4.2 there); and \(U(Y^{\mathrm{red}}(-1))=(UY)^{\mathrm{red}}(-1)\cong cH\) by Proposition 1.1 (3) there. \(\square\)

**Lemma 1.4 (objects of constant type).**

1. If \(Y\in D(Q)\), respectively \(Y\in D(Q_{\mathcal O})\), and \(UY\cong(cH)(1)\), then the underlying complex of \(\Phi_{\mathbb Z}(Y)\), respectively \(\Phi_{\mathcal O}(Y)\), is isomorphic to \(H\). In particular \(Y\) is bounded exactly when \(H\) is.
2. The objects of constant type of \(D^-(Q)\) and of \(D^-(Q_{\mathcal O})\) are closed under shifts and extensions. Those of \(D^b(Q_{\mathcal O})\) form a triangulated subcategory \(\mathcal D_c\).

**Proof.** (1) By Proposition 2.3 (4) of the lesson on diagonal rings, the underlying complex of \(\Phi(Y^{\mathrm{red}}(-1))\) is \(\Lambda U(Y^{\mathrm{red}}(-1))=\Lambda((UY)^{\mathrm{red}}(-1))\), since reduction and \((-1)\) are computed on underlying objects. By Proposition 1.1 (3) there, \(((cH)(1))^{\mathrm{red}}(-1)\cong cH\), and \(\Lambda cH\cong H\) by Proposition 3.4 (2) there. The cohomology objects of \((cH)(1)\) are the \(C_\bullet(H^i(1))\), which vanish exactly when \(H^i\) does. (2) The functor \(H\mapsto(cH)(1)\), \(D^-(\mathcal A)\to D^-(\mathcal M_1)\), is exact and fully faithful (Proposition 3.6 (2) there). Let \(Y'\to Y\to Y''\to Y'[1]\) be a distinguished triangle with \(Y'\) and \(Y''\) of constant type. Then \(UY''\to UY'[1]\) is \((ch)(1)\) for a map \(h\) of \(D^-(\mathcal A)\), and \(UY\) is isomorphic to \((c\,C)(1)\), with \(C\) the cone of \(h\) shifted by \(-1\). Extensions of bounded objects are bounded. \(\square\)

## 2. Spectra of complexes of modules and the Ext formula

**Lemma 2.1 (spectra of complexes).** Let \(M\) be a complex of \(A\)-modules.

1. The map \(s\colon\mathbb Z(A(1)^\times)\to A(1)\), \([x]\mapsto x\), is a homomorphism of \(S_1\)-graded rings, where \(A(1)=(cA)(1)\) is the reduced \(S_1\)-graded ring of Remark 3.3 of the lesson on graded rings, and \(M(1)=(cM)(1)\) is a complex of \(S_1\)-graded \(A(1)\)-modules (Proposition 2.1 (1) there). So \(C_\bullet(M(1))\) is a complex of \(Q\)-modules through the ring homomorphism \(C_\bullet(s)\colon Q\to C_\bullet(A(1))\); through \(s\), the graded monoid \(A(1)^\times\) acts on \(M(1)\) by the action of Definition 3.2 there. The functor \(M\mapsto C_\bullet(M(1))\) is exact, and \(UC_\bullet(M(1))=(cUM)(1)\); in particular \(C_\bullet(M(1))\) is of constant type.
2. There is a quasi-isomorphism of \(B_A\)-modules \(\Phi_{\mathbb Z}(C_\bullet(M(1)))\to\varepsilon^*M\), natural in \(M\).
3. If \(M\) is a complex of \(A\otimes\mathcal O\)-modules, then \(C_\bullet(M(1))\) is a complex of \(Q_{\mathcal O}\)-modules, and there is a quasi-isomorphism \(\Phi_{\mathcal O}(C_\bullet(M(1)))\to\varepsilon_{\mathcal O}^*M\) of \(B_A^{\mathcal O}\)-modules, natural in \(M\).
4. Conversely, let \(L\) be a \(Q_{\mathcal O}\)-module whose underlying object is isomorphic to \(C_\bullet(H(1))\) for an abelian sheaf \(H\). Then \(L\cong C_\bullet(H'(1))\) as \(Q_{\mathcal O}\)-modules for an \(A\otimes\mathcal O\)-module \(H'\) whose underlying abelian sheaf is \(H\).

**Proof.** (1) The first assertion is shown in the proof of Proposition 2.2 (2) of the lesson on the stable ring. The action of \(A(1)\) on \((cM)(1)\) and the action of Definition 3.2 of the lesson on graded rings both have as normalization the action \(A\otimes M\to M\) in multidegree \(\mathbf 1\), so they agree, as in Remark 3.3 there. The functors \(C_\bullet\) and \((1)\) are exact, and \(UC_\bullet(M(1))=\) \(C_\bullet((cUM)(1))=\) \((cUM)(1)\) (Proposition 2.2 (3) of the lesson on graded rings and the proof of Proposition 3.6 (2) of the lesson on diagonal rings).

(2) Since \((cM)(1)\) is reduced, \(C_\bullet(M(1))^{\mathrm{red}}(-1)\) is \(C_\bullet\) of the trivial \(S_0\)-graded module \(cM\), that is, the constant object \(cM\) of \(\mathcal M_0\). It is a module over \(Q^{\mathrm{red}}(-1)\) through the homomorphism \(\rho\colon Q^{\mathrm{red}}(-1)\to C_\bullet(cA)\) induced by \(C_\bullet(s)\), where \(C_\bullet(cA)\) is the constant ring with value \(A\) (Propositions 2.1 and 2.2 of the lesson on graded rings). By the proofs of Proposition 3.4 (2) of the lesson on diagonal rings and of Proposition 2.2 (2) of the lesson on the stable ring, \(\Delta^2C_\bullet(cA)=A\otimes\mathbb Z(N(\bar\Delta'))\), the ring of the simplicial monoid \(N(\bar\Delta')\) under the sum of strings with coefficients in \(A\), and \(\Delta^2(cM^i)=M^i\otimes\mathbb Z(N(\bar\Delta'))\), on which \(a\otimes\sigma\) acts by \(m\otimes\tau\mapsto am\otimes(\sigma+\tau)\). The augmentation \(\mathbb Z(N(\bar\Delta'))\to\mathbb Z\), \(\sigma\mapsto1\), is a homomorphism of simplicial rings and a homotopy equivalence of simplicial abelian groups (same proofs). So \(\alpha\colon A\otimes\mathbb Z(N(\bar\Delta'))\to A\) is a homomorphism of simplicial rings, and \(M^i\otimes\mathbb Z(N(\bar\Delta'))\to M^i\) is \(\alpha\)-linear and a homotopy equivalence. Normalizing and totalizing, the row filtration gives a quasi-isomorphism \(\Phi(cM)\to M\), linear over \(N\alpha\). By Proposition 2.2 (2) of the lesson on the stable ring, \(\varepsilon=N\alpha\circ N\Delta^2\rho\). Restriction of scalars along \(N\Delta^2\rho\) gives the quasi-isomorphism of \(B_A\)-modules \(\Phi_{\mathbb Z}(C_\bullet(M(1)))\to\varepsilon^*M\).

(3) The same argument applies with \(s\otimes\mathcal O\colon\) \(\mathbb Z(A(1)^\times)\otimes\mathcal O\to\) \(A(1)\otimes\mathcal O=\) \((A\otimes\mathcal O)(1)\), the spectrum being additive, and with \(\varepsilon_{\mathcal O}=\varepsilon\otimes\mathcal O\) (Lemma 1.1).

(4) The vertex maps \(L_q\to L_0\circ C(v_0)\) of \(L\) are those of its underlying object, which are isomorphisms (Proposition 2.3 (2) of the lesson on graded monoids). By Proposition 1.3 of the lesson on graded rings, applied to the \(S_1\)-graded ring \(\mathbb Z(A(1)^\times)\otimes\mathcal O\), whose \(C_\bullet\) is \(Q_{\mathcal O}\), \(L\cong C_\bullet(L')\) for an \(S_1\)-graded module \(L'\) over that ring, and the underlying \(S_1\)-graded abelian sheaf of \(L'\) is isomorphic to \(H(1)\), by the full faithfulness of \(C_\bullet\) (Proposition 2.3 (2) of the lesson on graded monoids). Transport the module structure to \(H(1)\). Each local section of \(\mathcal O\) acts by an endomorphism of the spectral diagram \(H(1)\), which is \(\varphi(1)\) for a unique endomorphism \(\varphi\) of \(H\) (Proposition 3.5 of the lesson on multisimplicial objects); this makes \(H\) an \(\mathcal O\)-module, \(\mathcal O\) acting on \(H(1)\) through it. The action of \(\mathbb Z(A(1)^\times)\) by additive maps is a morphism \(A(1)^\times\times H(1)^\times\to H(1)^\times\) of type \(\pi\), so it is \(f(1,1)^\times\) for a unique bilinear map \(f\colon A\times H\to H\) (Proposition 3.1 of the lesson on graded rings), and the associativity and unit laws make \(f\) an action of \(A\), as in the proof of Proposition 3.4 there. The two actions commute, since they commute on \(H(1)\) at the index of dimension \(0\). Let \(H'\) be \(H\) with these actions. Then \(L'\cong H'(1)\) as modules, by the formula for the actions on spectra, and \(L\cong C_\bullet(H'(1))\). \(\square\)

**Lemma 2.2 (Mac Lane's theorem in Hom form).** For an \(A\)-module \(G\) and \(M\in D(A)\), the map
\[
\begin{aligned}&\mu_{G,M} \colon\operatorname{RHom}_A(G,M)\\&\quad \to\operatorname{RHom}_{B_A}\bigl(N\mathbb Z^{\mathrm{st}}(G),\varepsilon^*M\bigr),\end{aligned}
\]
given by restriction of scalars along \(\varepsilon\) followed by composition with \(N\varepsilon_G\colon N\mathbb Z^{\mathrm{st}}(G)\to\varepsilon^*G\), is an isomorphism, natural in \(G\) and \(M\). For a fixed \(M\) it is represented by a map of complexes natural in \(G\).

**Proof.** Let \(M\to I\) be a K-injective resolution of \(A\)-modules and \(\varepsilon^*I\to J\) a K-injective resolution of \(B_A\)-modules. The map of complexes
\[
\begin{gathered}\operatorname{Hom}^\bullet_A(G,I)\\ \to\operatorname{Hom}^\bullet_{B_A}\bigl(N\mathbb Z^{\mathrm{st}}(G),J\bigr),\\ g\\ \longmapsto(\varepsilon^*I\to J)\circ\varepsilon^*(g)\circ N\varepsilon_G,\end{gathered}
\]
is natural in \(G\) and represents \(\mu_{G,M}\). On \(H^k\) it is the map \(\operatorname{Hom}_{D(A)}(G,M[k])\to\) \(\operatorname{Hom}_{D(B_A)}(N\mathbb Z^{\mathrm{st}}(G),\varepsilon^*M[k])\), \(g\mapsto\varepsilon^*(g)\circ N\varepsilon_G\) (Proposition 2.3 (2) of the lesson on differential graded modules). Compose it with the adjunction bijection of Proposition 2.5 (1) there, from \(\operatorname{Hom}_{D(B_A)}(N\mathbb Z^{\mathrm{st}}(G),\varepsilon^*M[k])\) to \(\operatorname{Hom}_{D(A)}(A\otimes^{\mathbf L}_{B_A}N\mathbb Z^{\mathrm{st}}(G),M[k])\). This bijection is natural in the second variable and sends \(N\varepsilon_G\) to \(c_G\) (Construction 3.1 of the lesson on the stable ring), so it sends \(\varepsilon^*(g)\circ N\varepsilon_G\) to \(g\circ c_G\). Since \(c_G\) is an isomorphism (Theorem 3.2 there), \(g\mapsto g\circ c_G\) is bijective, and so is the map on \(H^k\). \(\square\)

**Theorem 2.3.** Let \(G\) be an \(A\)-module, \(Y\in D^+(Q_{\mathcal O})\), \(M\in D(R_{\mathcal O})\), and \(\psi\colon\Phi_{\mathcal O}(Y)\to\pi_{\mathcal O}^*M\) an isomorphism of \(D(B_A^{\mathcal O})\). Consider the maps
\[
\begin{aligned}&\operatorname{RHom}_Q(X_G,Y|_Q)\\ &\quad \xrightarrow{\ \kappa\ }\operatorname{RHom}_{B_A}\bigl(N\mathbb Z^{\mathrm{st}}(G),\Phi_{\mathbb Z}(Y|_Q)\bigr)\\ &\quad \xrightarrow{\ \sim\ }\operatorname{RHom}_{B_A}\bigl(N\mathbb Z^{\mathrm{st}}(G),\varepsilon^*(M|_A)\bigr)\\ &\quad \xleftarrow{\ \mu\ }\operatorname{RHom}_A(G,M|_A).\end{aligned}
\]
Here \(\kappa\) is the map of Corollary 4.2 of the lesson on diagonal rings, with \(\Phi(X_G^{\mathrm{red}}(-1))=N\mathbb Z^{\mathrm{st}}(G)\); the middle isomorphism is \(\lambda^*\psi\), through \(\Phi_{\mathbb Z}(Y|_Q)=\lambda^*\Phi_{\mathcal O}(Y)\) (Lemma 1.1), followed by (0.2); and \(\mu=\mu_{G,M|_A}\). Then \(\mu\) is an isomorphism, and \(\kappa\) is an isomorphism if \(Y\) is of constant type. In that case
\[
\begin{aligned}&\operatorname{RHom}_Q(X_G,Y|_Q)\\&\quad \cong\operatorname{RHom}_A(G,M|_A),\end{aligned} \tag{2.1}
\]
naturally in \(G\) and in \((Y,M,\psi)\), and for fixed \(Y\), \(M\) and \(\psi\) this isomorphism is represented by a zigzag of quasi-isomorphisms of complexes natural in \(G\).

**Proof.** \(\mu\) is an isomorphism by Lemma 2.2. The components of \(Q\) are flat, \(X_G\) is a module, \(Y|_Q\in D^+(Q)\), and \(U(Y|_Q)=UY\) is of the form \((cH)(1)\); so \(\kappa\) is an isomorphism by Corollary 4.2 of the lesson on diagonal rings. For fixed resolutions of \(Y|_Q\), of \((Y|_Q)^{\mathrm{red}}(-1)\) and of \(\Phi_{\mathbb Z}(Y|_Q)\), the map \(\kappa\) is given by Construction 2.4 there followed by composition with the resolutions, so it is represented by a map of complexes natural in \(G\). The middle isomorphism is composition with a fixed homotopy equivalence between K-injective \(B_A\)-modules representing \(\Phi_{\mathbb Z}(Y|_Q)\) and \(\varepsilon^*(M|_A)\), and \(\mu\) is represented as in Lemma 2.2. \(\square\)

**Corollary 2.4 (the Ext formula).** For an \(A\)-module \(G\) and \(M\in D^+(A)\) there is an isomorphism
\[
\begin{aligned}&\operatorname{RHom}_Q\bigl(X_G,C_\bullet(M(1))\bigr)\\&\quad \cong\operatorname{RHom}_A(G,M),\end{aligned}
\]
natural in \(G\) and \(M\), and represented for a fixed \(M\) by a zigzag of quasi-isomorphisms of complexes natural in \(G\). In particular \(\operatorname{Hom}_{D(Q)}(X_G,C_\bullet(M(1))[i])\cong\) \(\operatorname{Ext}^i_A(G,M)\) for all \(i\).

**Proof.** By Lemma 2.1, \(Y=C_\bullet(M(1))\) is an object of \(D^+(Q)\) of constant type, with a natural isomorphism \(\Phi_{\mathbb Z}(Y)\cong\varepsilon^*M\). So \(\kappa\colon\) \(\operatorname{RHom}_Q(X_G,Y)\to\) \(\operatorname{RHom}_{B_A}(N\mathbb Z^{\mathrm{st}}(G),\Phi_{\mathbb Z}(Y))\) is an isomorphism (Corollary 4.2 of the lesson on diagonal rings), and composing with this isomorphism and with \(\mu_{G,M}^{-1}\) (Lemma 2.2) gives the statement, as in the proof of Theorem 2.3. \(\square\)

**Remark 2.5 (degree \(0\)).** For an \(A\)-module \(M\), Corollary 2.4 in degree \(0\) can be seen directly. A \(Q\)-linear map \(X_G\to C_\bullet(M(1))\) is a map of the underlying objects \(C_\bullet(G(1)^\times)\to C_\bullet(M(1)^\times)\) compatible with the actions of \(C_\bullet(A(1)^\times)\), because \(X_G\) is the free abelian object on \(C_\bullet(G(1)^\times)\) and \(Q\) acts on \(C_\bullet(M(1))\) through the action of \(C_\bullet(A(1)^\times)\). By Proposition 2.3 (2) of the lesson on graded monoids, these maps are the \(C_\bullet(v)\) for the equivariant maps \(v\colon G(1)^\times\to M(1)^\times\), and by Proposition 3.4 of the lesson on graded rings these are the \(u(1)\) for the \(A\)-linear maps \(u\colon G\to M\). So \(\operatorname{Hom}_Q(X_G,C_\bullet(M(1)))=\operatorname{Hom}_A(G,M)\).

## 3. Spectra of complexes over the derived tensor product

Let \(\mathcal D_c\subseteq D^b(Q_{\mathcal O})\) be the triangulated subcategory of the objects of constant type (Lemma 1.4 (2)).

**Proposition 3.1.**

1. \(\Phi_{\mathcal O}\) maps \(\mathcal D_c\) into \(D^b(B_A^{\mathcal O})\), and its restriction to \(\mathcal D_c\) is exact and fully faithful. Its essential image \(\mathcal T\) is a triangulated subcategory of \(D^b(B_A^{\mathcal O})\), and \(\Phi_{\mathcal O}\colon\mathcal D_c\to\mathcal T\) is an equivalence.
2. \(\pi_{\mathcal O}^*\) maps \(D^b(R_{\mathcal O})\) into \(\mathcal T\).

**Proof.** (1) The values are bounded by Lemma 1.4 (1), and full faithfulness is Corollary 1.3 on \(H^0\). The essential image of a fully faithful exact functor is closed under shifts and, up to isomorphism, under cones: a map \(\Phi_{\mathcal O}Y'\to\Phi_{\mathcal O}Y\) is \(\Phi_{\mathcal O}(g)\) for a map \(g\) of \(\mathcal D_c\), and its cone is isomorphic to \(\Phi_{\mathcal O}\) of the cone of \(g\), which lies in \(\mathcal D_c\). (2) Let \(M\in D^b(R_{\mathcal O})\). Since \(R_{\mathcal O}\) lies in nonpositive degrees, \(M\) has the truncations of Lemma 1.1 of the lesson on differential graded modules, and it is an iterated extension of the objects \(H^i(M)[-i]\), each represented by a module concentrated in one degree. On such a module \(R_{\mathcal O}^{-1}\) acts by \(0\) for degree reasons, and so does \(dR_{\mathcal O}^{-1}\), since \((dr)x=d(rx)+r\,dx=0\). So it is \(\bar\varepsilon^*H\) for an \(A\otimes\mathcal O\)-module \(H\), and \(\pi_{\mathcal O}^*\bar\varepsilon^*H=\varepsilon_{\mathcal O}^*H\cong\Phi_{\mathcal O}(C_\bullet(H(1)))\) by Lemma 2.1 (3), with \(C_\bullet(H(1))\in\mathcal D_c\) by Lemma 2.1 (1). Since \(\pi_{\mathcal O}^*\) is exact and \(\mathcal T\) is closed under extensions, \(\pi_{\mathcal O}^*M\in\mathcal T\). \(\square\)

**Definition 3.2.** For \(M\in D^b(R_{\mathcal O})\), let \(\Psi(M)\) be an object of \(\mathcal D_c\) with an isomorphism \(\psi_M\colon\Phi_{\mathcal O}(\Psi(M))\to\pi_{\mathcal O}^*M\). By Proposition 3.1 it exists, and the pair is unique up to unique isomorphism. By full faithfulness, the maps of \(D^b(R_{\mathcal O})\) act on these objects, and \(\Psi\colon D^b(R_{\mathcal O})\to D^b(Q_{\mathcal O})\) is a functor with a natural isomorphism \(\psi\colon\Phi_{\mathcal O}\Psi\cong\pi_{\mathcal O}^*\). It is exact, as the composite of the exact functor \(\pi_{\mathcal O}^*\) and a quasi-inverse of the exact equivalence \(\Phi_{\mathcal O}\colon\mathcal D_c\to\mathcal T\).

**Theorem 3.3.**

1. \(\Psi\) maps \(D^{[a,b]}(R_{\mathcal O})\) into \(D^{[a,b]}(Q_{\mathcal O})\) for all \(a\leq b\), and \(U\Psi(M)\cong(cH)(1)\) with \(H\) the underlying complex of abelian sheaves of \(M\).
2. For a bounded complex \(H\) of \(A\otimes\mathcal O\)-modules there is an isomorphism \(\Psi(\bar\varepsilon^*H)\cong C_\bullet(H(1))\), natural in \(H\).
3. For \(M\in D^b(R_{\mathcal O})\) there is an isomorphism \(\Psi(M)|_Q\cong C_\bullet((M|_A)(1))\) in \(D(Q)\), natural in \(M\).
4. For an \(A\)-module \(G\) and \(M\in D^b(R_{\mathcal O})\) there is an isomorphism, natural in \(G\) and \(M\),
\[
\begin{aligned}&\operatorname{RHom}_Q\bigl(X_G,\Psi(M)|_Q\bigr)\\&\quad \cong\operatorname{RHom}_A(G,M|_A).\end{aligned}
\]

**Proof.** (1) \(\Psi(M)\) is of constant type, \(U\Psi(M)\cong(cH)(1)\), and by Lemma 1.4 (1) \(H\) is the underlying complex of \(\Phi_{\mathcal O}\Psi(M)\cong\pi_{\mathcal O}^*M\), which is that of \(M\). The cohomology objects of \((cH)(1)\) are the \(C_\bullet(H^i(1))\), which vanish exactly when \(H^i\) does. (2) \(C_\bullet(H(1))\) lies in \(\mathcal D_c\) by Lemma 2.1 (1), and Lemma 2.1 (3) gives a natural isomorphism \(\Phi_{\mathcal O}(C_\bullet(H(1)))\cong\varepsilon_{\mathcal O}^*H=\pi_{\mathcal O}^*\bar\varepsilon^*H\). By the uniqueness in Definition 3.2, \(C_\bullet(H(1))\cong\Psi(\bar\varepsilon^*H)\). (3) Both objects lie in \(D^b(Q)\) and are of constant type, by (1) and Lemma 2.1 (1). By Lemma 1.1 (2), \(\psi_M\), (0.2) and Lemma 2.1 (2),
\[
\begin{aligned}&\Phi_{\mathbb Z}(\Psi(M)|_Q) =\lambda^*\Phi_{\mathcal O}\Psi(M)\\ &\quad \cong\lambda^*\pi_{\mathcal O}^*M\\ &\quad \cong\varepsilon^*(M|_A)\\ &\quad \cong\Phi_{\mathbb Z}\bigl(C_\bullet((M|_A)(1))\bigr),\end{aligned}
\]
and \(\Phi_{\mathbb Z}\) is fully faithful on the objects of constant type of \(D^b(Q)\) (Corollary 4.2 of the lesson on diagonal rings, on \(H^0\)). So there is a unique isomorphism inducing this one. (4) is Theorem 2.3 for \(Y=\Psi(M)\) and \(\psi=\psi_M\). \(\square\)

By (2) and (3), \(\Psi(M)\) extends the spectrum \(C_\bullet(M(1))\) of complexes of \(A\otimes\mathcal O\)-modules, and its underlying \(Q\)-module is the spectrum of the underlying complex of \(A\)-modules. Exercise 5.3 shows that \(\Psi(M)\) carries more information than this underlying \(Q\)-module.

**Proposition 3.4 (two consecutive degrees).** For every integer \(n\), the restriction of \(\Psi\) to \(D^{[n,n+1]}(R_{\mathcal O})\) is fully faithful. Its essential image consists of the \(Y\in D(Q_{\mathcal O})\) such that \(UY\cong(cH)(1)\) for some \(H\in D^{[n,n+1]}(\mathcal A)\).

**Proof.** By Proposition 2.4 (3) of the lesson on the stable ring, with \(B=\mathcal O\), restriction along \(\pi_{\mathcal O}\) induces an equivalence between the full subcategories of \(D(R_{\mathcal O})\) and of \(D(B_A^{\mathcal O})\) formed by the objects with cohomology in \([n,n+1]\). On \(D^{[n,n+1]}(R_{\mathcal O})\), \(\Psi\) is the composite of this equivalence and a quasi-inverse of \(\Phi_{\mathcal O}\colon\mathcal D_c\to\mathcal T\), so it is fully faithful, and by Theorem 3.3 (1) its values have the stated form. Conversely, let \(UY\cong(cH)(1)\) with \(H\in D^{[n,n+1]}(\mathcal A)\). Then \(Y\in\mathcal D_c\), and \(\Phi_{\mathcal O}(Y)\) has cohomology in \([n,n+1]\) by Lemma 1.4 (1). So \(\Phi_{\mathcal O}(Y)\cong\pi_{\mathcal O}^*M\) for some \(M\in D^{[n,n+1]}(R_{\mathcal O})\), and \(Y\cong\Psi(M)\) by full faithfulness. \(\square\)

On all of \(D^b(R_{\mathcal O})\), \(\Psi\) is not fully faithful: for \(A=\mathcal O=\mathbb Z\) in the topos of sets, \(\operatorname{Hom}(\Psi(R_{\mathcal O}),\Psi(R_{\mathcal O})[4])\cong\mathbb Z/2\) while \(\operatorname{Hom}(R_{\mathcal O},R_{\mathcal O}[4])=0\), as shown in [*Connectivity of powers of simplicial ideals*](connectivity-of-powers-of-simplicial-ideals.md) (Proposition 2.8) from the stable homology of the free abelian group functor.

**Proposition 3.5 (tensor products).** For \(Y\in D(Q_{\mathcal O})\) and \(I\in D^-(\mathcal O)\), let \(Y\otimes^{\mathbf L}_{\mathcal O}I\) be \(Y\otimes_{\mathcal O}I'\), formed componentwise, for a quasi-isomorphism \(I'\to I\) from a K-flat complex of \(\mathcal O\)-modules [Stacks, Tag 06YS]; as in Proposition 2.7 of the lesson on differential graded modules, this is a well defined functor, and on \(D(R_{\mathcal O})\) the functor \(\otimes^{\mathbf L}_{\mathcal O}\) is that of Proposition 2.7, for the central homomorphism \(\mathcal O\to R_{\mathcal O}\). Let \(M\in D^b(R_{\mathcal O})\) and \(I\in D^-(\mathcal O)\) be such that \(M\otimes^{\mathbf L}_{\mathcal O}I\) is bounded. Then \(\Psi(M)\otimes^{\mathbf L}_{\mathcal O}I\) is bounded, and there is an isomorphism, natural in \(M\) and \(I\),
\[
\Psi(M)\otimes^{\mathbf L}_{\mathcal O}I\cong\Psi\bigl(M\otimes^{\mathbf L}_{\mathcal O}I\bigr).
\]

**Proof.** By Lemma 1.1 (3), \(\Phi_{\mathcal O}(Y\otimes_{\mathcal O}I')=\Phi_{\mathcal O}(Y)\otimes_{\mathcal O}I'\), and restriction of scalars along \(\pi_{\mathcal O}\) commutes with \(-\otimes_{\mathcal O}I'\). So
\[
\begin{aligned}&\Phi_{\mathcal O}\bigl(\Psi(M)\otimes^{\mathbf L}_{\mathcal O}I\bigr)\\ &\quad \cong\Phi_{\mathcal O}\Psi(M)\otimes^{\mathbf L}_{\mathcal O}I\\ &\quad \cong\pi_{\mathcal O}^*M\otimes^{\mathbf L}_{\mathcal O}I\\ &\quad =\pi_{\mathcal O}^*\bigl(M\otimes^{\mathbf L}_{\mathcal O}I\bigr)\\ &\quad \cong\Phi_{\mathcal O}\Psi\bigl(M\otimes^{\mathbf L}_{\mathcal O}I\bigr),\end{aligned}
\]
naturally. It remains to see that \(Y=\Psi(M)\otimes^{\mathbf L}_{\mathcal O}I\) lies in \(\mathcal D_c\); full faithfulness on \(\mathcal D_c\) (Proposition 3.1) then gives the isomorphism. The object \(\Psi(M)\in D^b(Q_{\mathcal O})\) is an iterated extension of shifts of its cohomology modules. Their underlying objects are the \(C_\bullet(H^i(1))\) of Theorem 3.3 (1), so by Lemma 2.1 (4) they are isomorphic to modules \(C_\bullet(H'(1))\) with \(H'\) an \(A\otimes\mathcal O\)-module. For these, \(C_\bullet(H'(1))\otimes_{\mathcal O}I'=C_\bullet((H'\otimes_{\mathcal O}I')(1))\), because the spectrum of an abelian sheaf consists of finite products of copies of it with maps given by integer matrices; this is of constant type by Lemma 2.1 (1). The functor \(-\otimes^{\mathbf L}_{\mathcal O}I\) is exact, so by Lemma 1.4 (2), \(Y\in D^-(Q_{\mathcal O})\) is of constant type, \(UY\cong(cH'')(1)\). By Lemma 1.4 (1), \(H''\) is the underlying complex of \(\Phi_{\mathcal O}(Y)\cong\pi_{\mathcal O}^*(M\otimes^{\mathbf L}_{\mathcal O}I)\), which is bounded. So \(Y\) is bounded and lies in \(\mathcal D_c\). \(\square\)

## 4. Ext groups as cohomology of the pseudo-category of the spectral action

For an \(S_1\)-graded monoid \(G'\) of \(T\) and an \(S_1\)-graded \(G'\)-object \(X\), let \(\mathcal X(G',X)\) be the simplicial topos of Proposition 3.3 of the lesson on graded monoids. Its level \(\mathcal X_n\) is \(T^{\mathfrak K_n}/F_n\), where \(\mathfrak K_n\) is the total category of \(q\mapsto C_q(S_1)^{n+1}\) and \(F_n\) is the simplicial object \(q\mapsto C_q(G')^n\times C_q(X)\) of \(\operatorname{Diagr}(T)\); the transition morphisms are induced by the simplicial structure of the nerve. Every level carries the constant ring \(\mathbb Z\), or \(\mathcal O\), with the identity structure, so the modules of \(B\mathcal X(G',X)\) are defined (Section 6 of the lesson on equivariant sheaves). Put
\[
\begin{gathered}\mathcal X_A=\mathcal X(A(1)^\times,e_{S_1}),\\ BA=B\mathcal X_A,\\ \mathcal X_G=\mathcal X(A(1)^\times,G(1)^\times)\end{gathered}
\]
for an \(A\)-module \(G\).

**Proposition 4.1.** Let \(\mathcal X=\mathcal X(G',X)\).

1. An object of \(B\mathcal X\) is an object \(L\) of \(T^{\mathfrak K_0}\) with a map \(L\to C_\bullet(X)\) and maps \(C_q(G')(k_1)\times L(q,k)\to\) \(L(q,k_1\otimes k)\), natural in \((q,(k_1,k))\in\mathfrak K_1\), associative and unital, and compatible with the projections to \(C_\bullet(X)\) and the action of \(C_\bullet(G')\) on \(C_\bullet(X)\). Its morphisms are the equivariant maps over \(C_\bullet(X)\).
2. A module of \(B\mathcal X\) is such an object whose map to \(C_\bullet(X)\) is an abelian group object of \(T^{\mathfrak K_0}/C_\bullet(X)\), the action being additive on the fibres. The modules of \(BA\) are the \(Q\)-modules of type \(C_\bullet(S_1)\), and the modules of \(BA\) for the ring \(\mathcal O\) are the \(Q_{\mathcal O}\)-modules.
3. \(\mathcal X\) is a good two-sided pseudo-category satisfying condition (I) of the lesson on cohomology of equivariant modules.

**Proof.** (1) The morphisms of topoi induced by morphisms of diagrams are those of the conventions of the lesson on graded monoids. The source and the target of \(\mathcal X\) are induced by the vertex maps \((g,x)\mapsto x\) and \((g,x)\mapsto gx\) of the nerve, whose types are \((k_1,k)\mapsto k\) and \((k_1,k)\mapsto k_1\otimes k\). So for \(L\to C_\bullet(X)\) in \(\mathcal X_0\), \(s^*L\) has the value \(C_q(G')(k_1)\times L(q,k)\) at \((q,(k_1,k))\), and \(t^*L\) the value
\[
\begin{gathered}L(q,k_1\otimes k)\\ \times_{C_q(X)(k_1\otimes k)}\\ \bigl(C_q(G')(k_1)\times C_q(X)(k)\bigr),\end{gathered}
\]
the fibre product being formed with the action. A morphism \(s^*L\to t^*L\) over \(F_1\) is therefore a family of maps \((g,l)\mapsto gl\) as in the statement, natural on \(\mathfrak K_1\), with \(gl\) lying over \(g\) times the image of \(l\). The degeneracy \(F_0\to F_1\), \(x\mapsto(1,x)\), pulls this morphism back to \(l\mapsto1\cdot l\), so condition 1 of Definition 1.1 of the lesson on equivariant sheaves is the unit law. The edges \(e_{01}\), \(e_{12}\) and \(e_{02}\) send \((g_1,g_2,x)\) to \((g_1,x)\), \((g_2,g_1x)\) and \((g_2g_1,x)\), so condition 2 is associativity. Morphisms of \(B\mathcal X\) are the maps of \(\mathcal X_0\) compatible with the structures.

(2) The ring \(\mathbb Z\) of \(\mathcal X_0\) is the constant sheaf over \(C_\bullet(X)\), so a module of \(\mathcal X_0\) is an abelian group object over \(C_\bullet(X)\), and a module structure is a structure that is a morphism of abelian sheaves on \(\mathcal X_1\), that is, additive on the fibres. For \(X=e_{S_1}\), maps \(\mathbb Z(C_q(G')(k_1))\otimes L(q,k)\to L(q,k_1\otimes k)\) are the same as maps \(C_q(G')(k_1)\times L(q,k)\to\) \(L(q,k_1\otimes k)\) additive in the second variable, by the universal property of the free abelian sheaf, and the axioms correspond. By Definition 1.2 of the lesson on graded rings, the modules of \(BA\) are therefore the modules over \(\mathbb Z(C_\bullet(A(1)^\times))=Q\). With \(\mathcal O\) in place of \(\mathbb Z\), the structures are \(\mathcal O\)-linear, which gives the \(Q_{\mathcal O}\)-modules.

(3) \(\mathcal X\) is a good two-sided pseudo-category by Proposition 3.3 (2) of the lesson on graded monoids. By Proposition 3.3 (1) there, each \(\mathcal X(v_n)\) is, up to equivalence, the localization of \(\mathcal X_0\) at an object \(V_n\). The module inverse image along a localization has the exact left adjoint extension by zero [Stacks, Tags 04IE, 03DJ], so it carries injective modules to injective modules [Stacks, Tag 015Z]. This is condition (I). \(\square\)

**Proposition 4.2 (localization at the spectrum of a module).** Let \(G\) be an \(A\)-module and \(C_G=C_\bullet(G(1)^\times)\), an object of \(BA\) by 2.4 of the lesson on graded monoids. The morphisms of diagrams \(C_q(A(1)^\times)^n\times C_q(G(1)^\times)\to\) \(C_q(A(1)^\times)^n\times C_q(e_{S_1})\) have identity type and induce a morphism \(p\colon\mathcal X_G\to\mathcal X_A\) of simplicial topoi whose components are localizations.

1. By Proposition 4.1, \(B\mathcal X_G\) is the category \(BA/C_G\) of objects of \(BA\) over \(C_G\). The functor \(p^*\colon\operatorname{Mod}(BA)\to\operatorname{Mod}(B\mathcal X_G)\), \(L\mapsto L\times C_G\) with the diagonal action, is exact and has an exact left adjoint \(p_!\), and \(p_!\mathbb Z\cong X_G\).
2. \(p^*\) carries injective modules to injective modules. For \(Y\in D^+(Q)\) there is an isomorphism \(R\Gamma(B\mathcal X_G,p^*Y)\cong\operatorname{RHom}_Q(X_G,Y)\), natural in \(Y\): for a bounded below complex \(I\) of injective \(Q\)-modules representing \(Y\), both sides are \(\operatorname{Hom}^\bullet(\mathbb Z,p^*I)=\operatorname{Hom}^\bullet_Q(X_G,I)\).
3. For a module \(L\) of \(BA\), \(\operatorname{ner}^{\mathrm o}(p^*L)\cong p^*\operatorname{ner}^{\mathrm o}(L)\) and \(\operatorname{ner}(p^*L)\cong p^*\operatorname{ner}(L)\), naturally, where \(p\) also denotes the morphisms of total topoi.

**Proof.** (1) An object of \(B\mathcal X_G\) is an object of \(T^{\mathfrak K_0}\) over \(C_G\) with a compatible action, that is, an object of \(BA\) over \(C_G\). The inverse image along the localization \(p_0\) is \(L\mapsto L\times C_G\), and the inverse image of the structure of \(L\) is the diagonal action. For a module \(N\) of \(B\mathcal X_G\), let \(p_!N\) be the cokernel of the map
\[
\begin{gathered}\mathbb Z(N\times_{C_G}N)\to\mathbb Z(N),\\ [(n,n')]\\ \longmapsto[n+n']-[n]-[n'],\end{gathered}
\]
where \(\mathbb Z(-)\) is the free abelian object of \(T^{\mathfrak K_0}\), formed componentwise. The free abelian object on an object of \(BA\) is a \(Q\)-module by Proposition 4.1 (2), and this map is \(Q\)-linear, because the action on \(N\) is additive on the fibres. A \(Q\)-linear map \(p_!N\to L\) is an equivariant map \(f\colon N\to L\) of objects with \(f(n+n')=f(n)+f(n')\) on the fibres, which is the same as a morphism \(N\to L\times C_G\) of modules of \(B\mathcal X_G\). So \(p_!\) is left adjoint to \(p^*\). Forgetting the actions, the same argument shows that \(p_!N\) is the extension by zero of \(N\) along the localization of \(T^{\mathfrak K_0}\) at \(C_G\), both being left adjoint to \(L\mapsto L\times C_G\); this is exact [Stacks, Tags 04IE, 03DJ], so \(p_!\) is exact. A morphism \(\mathbb Z\to L\times C_G\) of modules of \(B\mathcal X_G\), where \(\mathbb Z\) is the constant sheaf over \(C_G\), is an equivariant section of \(L\times C_G\to C_G\), that is, an equivariant map \(C_G\to L\), and these are the \(Q\)-linear maps \(\mathbb Z(C_G)=X_G\to L\). So \(p_!\mathbb Z\cong X_G\) by the Yoneda lemma.

(2) A right adjoint of an exact functor carries injectives to injectives [Stacks, Tag 015Z]. So \(p^*I\) is a bounded below complex of injectives representing \(p^*Y\), hence K-injective [Stacks, Tag 070J]. The global sections of a module \(F\) of \(B\mathcal X_G\) are \(\operatorname{Hom}(\mathbb Z,F)\) (Section 4 of the lesson on cohomology of equivariant modules), so \(R\Gamma(B\mathcal X_G,p^*Y)\) is \(\operatorname{Hom}^\bullet(\mathbb Z,p^*I)=\) \(\operatorname{Hom}^\bullet_Q(p_!\mathbb Z,I)=\) \(\operatorname{Hom}^\bullet_Q(X_G,I)\).

(3) The second nerve of a module \(L\) of \(B\mathcal X\) has the levels \(\mathcal X(v_0)^*L\), its transition maps being inverse images of the structure of \(L\); the first nerve uses the last vertices (Definition 2.1 of the lesson on equivariant sheaves). The components of \(p\) commute with the transition morphisms of \(\mathcal X_G\) and \(\mathcal X_A\) up to the canonical isomorphisms between iterated fibre products (conventions of the lesson on graded monoids). So \(p_n^*\mathcal X_A(v_0)^*L\cong\mathcal X_G(v_0)^*p_0^*L\), compatibly with the transition maps, and the same holds for the last vertices. \(\square\)

**Theorem 4.3 (Ext groups as cohomology).** Let \(G\) be an \(A\)-module and \(Y\in D^+(Q)\) an object of constant type with an isomorphism \(\Phi_{\mathbb Z}(Y)\cong\varepsilon^*M\) for some \(M\in D(A)\): for instance \(Y=C_\bullet(M(1))\) with \(M\in D^+(A)\) (Lemma 2.1), or \(Y=\Psi(M')|_Q\) with \(M'\in D^b(R_{\mathcal O})\) and \(M=M'|_A\) (Theorem 3.3). Then there are isomorphisms, natural in \(G\),
\[
\begin{aligned}&R\Gamma\bigl(\operatorname{Top}^{\mathrm o}(\mathcal X_G),p^*\operatorname{ner}^{\mathrm o}(Y)\bigr)\\ &\quad \cong R\Gamma(B\mathcal X_G,p^*Y)\\ &\quad \cong\operatorname{RHom}_Q(X_G,Y)\\ &\quad \cong\operatorname{RHom}_A(G,M),\end{aligned}
\]
and the same with the first total topos \(\operatorname{Top}(\mathcal X_G)\) and the first nerve. There is a spectral sequence
\[
\begin{aligned}&E_1^{a,b} =H^b\bigl((\mathcal X_G)_a,\operatorname{ner}_a(p^*Y)\bigr)\\&\quad \Longrightarrow\operatorname{Ext}^{a+b}_A(G,M).\end{aligned}
\]

**Proof.** The first isomorphism is Proposition 4.2 (3) together with Corollary 4.2 of the lesson on cohomology of equivariant modules, which applies to \(\mathcal X_G\) and \(p^*Y\in D^+(B\mathcal X_G)\) by Proposition 4.1 (3); that corollary also gives the version with the first total topos and the spectral sequence. The second isomorphism is Proposition 4.2 (2). The third is the composite of \(\kappa\), an isomorphism by Corollary 4.2 of the lesson on diagonal rings, the given isomorphism, and \(\mu_{G,M}^{-1}\) (Lemma 2.2). \(\square\)

**Proposition 4.4 (relative cohomology).** Let \(u\colon G\to H\) be a homomorphism of \(A\)-modules, \(\operatorname{cone}(u)\) the complex \(G\xrightarrow{u}H\) with \(H\) in degree \(0\), \(\mathcal Y_G=\operatorname{Top}^{\mathrm o}(\mathcal X_G)\) and \(\mathcal Y_H=\operatorname{Top}^{\mathrm o}(\mathcal X_H)\). The morphism \(f\colon\mathcal Y_G\to\mathcal Y_H\) induced by \(u\) is a localization, so \(f^*\) has the exact left adjoint \(f_!\). For \(F\) in the derived category of bounded below complexes of abelian sheaves on \(\mathcal Y_H\), define the relative cohomology
\[
\begin{aligned}&R\Gamma(\mathcal Y_H/\mathcal Y_G,F)\\&\quad =\operatorname{RHom}\bigl(\operatorname{cone}(f_!\mathbb Z\to\mathbb Z),F\bigr).\end{aligned}
\]
Then for \(Y\) and \(M\) as in Theorem 4.3, with \(q\colon\mathcal X_H\to\mathcal X_A\) the morphism of Proposition 4.2 for \(H\),
\[
\begin{aligned}&R\Gamma\bigl(\mathcal Y_H/\mathcal Y_G,q^*\operatorname{ner}^{\mathrm o}(Y)\bigr)\\&\quad \cong\operatorname{RHom}_A(\operatorname{cone}(u),M).\end{aligned}
\]

**Proof.** By the conventions of the lesson on graded monoids, \(\mathcal Y_H=T^{\mathfrak L}/F_H\), where \(\mathfrak L\) is the total category of the bisimplicial shape \((a,q)\mapsto C_q(S_1)^{a+1}\) and \(F_H\) is given by \(\operatorname{Ner}(C_\bullet(A(1)^\times),C_\bullet(H(1)^\times))\); likewise for \(G\). The morphism \(F_G\to F_H\) induced by \(u\) has identity type, so \(f\) is the localization of \(\mathcal Y_H\) at it, and \(f_!\) is exact [Stacks, Tags 04IE, 03DJ]. Let \(J\) be a bounded below complex of injectives representing \(F\). Then \(f^*J\) is a bounded below complex of injectives representing \(f^*F\) [Stacks, Tag 015Z], and
\[
\begin{aligned}&\operatorname{Hom}^\bullet\bigl(\operatorname{cone}(f_!\mathbb Z\to\mathbb Z),J\bigr)\\&\quad=\operatorname{cone}\bigl(\Gamma(\mathcal Y_H,J)\\&\qquad\to\Gamma(\mathcal Y_G,f^*J)\bigr)[-1],\end{aligned}
\]
the map being restriction. So the relative cohomology is the cone, shifted by \(-1\), of the restriction map.

The same holds for the categories of equivariant sheaves. The homomorphism \(u\) induces a morphism \(\mathcal X_G\to\mathcal X_H\) of simplicial topoi, whose components are the localizations along the maps induced by \(u\). On equivariant sheaves its inverse image is \(L\mapsto L\times_{C_H}C_G\), from \(BA/C_H\) to \(BA/C_G\), and its composite with \(q^*\) is \(p^*\). By Proposition 4.2 (2), for a bounded below complex \(I\) of injective \(Q\)-modules representing \(Y\), the restriction \(\Gamma(B\mathcal X_H,q^*I)\to\Gamma(B\mathcal X_G,p^*I)\) is the map \(\operatorname{Hom}^\bullet_Q(X_H,I)\to\operatorname{Hom}^\bullet_Q(X_G,I)\) given by composition with \(X_u\). Let \(J\) be an injective resolution of \(\operatorname{ner}^{\mathrm o}(q^*I)\cong q^*\operatorname{ner}^{\mathrm o}(I)\), and use \(f^*J\) for \(\mathcal Y_G\); then \(f^*\operatorname{ner}^{\mathrm o}(q^*I)=\operatorname{ner}^{\mathrm o}(p^*I)\), as in Proposition 4.2 (3). The comparison maps of Section 1 of the lesson on cohomology of equivariant modules, \(\Gamma(B\mathcal X_H,q^*I)\to\Gamma(\mathcal Y_H,J)\) and \(\Gamma(B\mathcal X_G,p^*I)\to\Gamma(\mathcal Y_G,f^*J)\), commute with the restrictions, and they are quasi-isomorphisms (Corollary 4.2 there). So the relative cohomology is quasi-isomorphic to
\[
\begin{aligned}&\operatorname{cone}\bigl(\operatorname{Hom}^\bullet_Q(X_H,I)\\&\qquad\to\operatorname{Hom}^\bullet_Q(X_G,I)\bigr)[-1]\\&\quad=\operatorname{Hom}^\bullet_Q\bigl(\operatorname{cone}(X_u),I\bigr).\end{aligned}
\]
As in the proof of Theorem 2.3, the isomorphisms \(\operatorname{RHom}_Q(X_G,Y)\cong\operatorname{RHom}_A(G,M)\) of Theorem 4.3 are represented by zigzags of quasi-isomorphisms of complexes natural in \(G\): \(\kappa\) is given by Construction 2.4 of the lesson on diagonal rings, the given isomorphism by a fixed homotopy equivalence of K-injective \(B_A\)-modules, and \(\mu\) as in Lemma 2.2. Applied to \(u\), they give quasi-isomorphisms between the shifted cones, and for a K-injective resolution \(M\to I_M\) the cone
\[
\begin{aligned}&\operatorname{cone}\bigl(\operatorname{Hom}^\bullet_A(H,I_M)\\&\qquad\to\operatorname{Hom}^\bullet_A(G,I_M)\bigr)[-1]\\&\quad=\operatorname{Hom}^\bullet_A\bigl(\operatorname{cone}(u),I_M\bigr)\end{aligned}
\]
computes \(\operatorname{RHom}_A(\operatorname{cone}(u),M)\). \(\square\)

**Remark 4.5 (the explicit form).** By the conventions of the lesson on graded monoids, \(\operatorname{Top}^{\mathrm o}(\mathcal X_G)\) is the topos of bisimplicial diagrams of \(T\) of type \(\operatorname{Ner}(C_\bullet(S_1),C_\bullet(S_1))\) over \(\operatorname{Ner}(C_\bullet(A(1)^\times),C_\bullet(G(1)^\times))\). The value of this nerve in bidegree \((a,q)\), at an object \((k_1,\ldots,k_a,k)\) of \(C_q(S_1)^{a+1}\), is
\[
\begin{aligned}&C_q(A(1)^\times)(k_1) \times\cdots\\&\quad \times C_q(A(1)^\times)(k_a)\\&\quad \times C_q(G(1)^\times)(k),\end{aligned}
\]
a product of sheaves \(A\langle n\rangle(r)=A^P\) and \(G\langle n\rangle(r)=G^P\), that is, a product of copies of \(A\) and of \(G\). The sheaf \(p^*\operatorname{ner}^{\mathrm o}(C_\bullet(M(1)))\) has at the same place the value \(M\langle n\rangle(r)=M^P\), with the index set \(P\) of the factor \(G^P\), pulled back to it. So Theorem 4.3 expresses \(\operatorname{Ext}^i_A(G,M)\) as the cohomology of a bisimplicial diagram of products of copies of \(A\) and \(G\), with coefficients in products of copies of \(M\). The ring and module structures enter only through the maps of the diagram: the faces of the nerve use the multiplication of \(A\) and its action on \(G\), and the multisimplicial maps use the additions.

## 5. Exercises

**5.1.** Let \(\mathbb 1\) be the object of \(\mathcal M_1\) with the value \(\mathbb Z\) at the objects \((q,(o_q,()))\), where \(o_q\) is the string of identities of \([-1]\), with identity maps between them, and \(0\) elsewhere. Show that \(\mathbb 1\) is a functor, that \(Q\cong\operatorname{ind}(\mathbb 1)\) and \(X_A\cong Q\) as \(Q\)-modules, and that \(\operatorname{Hom}_Q(Q,L)\cong\Gamma(T,L_0(o_0))\) for every \(Q\)-module \(L\). Deduce that \(\operatorname{Hom}_Q(X_A,C_\bullet(M(1)))\cong\Gamma(T,M)\) for an \(A\)-module \(M\), and compare with Corollary 2.4.

**5.2.** Let \(M\) be an \(A\)-module. Show that every equivariant map from the final object of \(BA\) to \(C_\bullet(M(1))\), that is, every global section of the module \(C_\bullet(M(1))\) of \(BA\), vanishes. Show moreover that \(R\Gamma(BA,C_\bullet(M(1)))=0\) and \(R\Gamma(\operatorname{Top}^{\mathrm o}(\mathcal X_A),\operatorname{ner}^{\mathrm o}C_\bullet(M(1)))=0\).

**5.3.** Let \(T\) be the topos of sets, \(p\) a prime, and \(A=\mathcal O=\mathbb F_p\).

(a) Show that \(\mathbb F_p\otimes(p\mathbb Z\to\mathbb Z)\), the presentation of Lemma 1.1 of the lesson on the stable ring for \(\mathbb Z\to\mathbb F_p\) tensored with \(A\), is the differential graded ring \(\mathbb F_p[e]=\mathbb F_p\oplus\mathbb F_pe\), with \(e\) in degree \(-1\), \(e^2=0\) and \(d=0\).

(b) Let \(E=\mathbb F_p[e]\), a module over itself, and \(E'=\mathbb F_p\oplus\mathbb F_p[1]\) with \(e\) acting by \(0\). Show that \(E\) and \(E'\) are isomorphic as complexes of \(\mathbb F_p\)-vector spaces, but not in \(D(\mathbb F_p[e])\).

(c) Deduce that there are \(M,M'\in D^{[-1,0]}(R_{\mathcal O})\) with \(M|_A\cong M'|_A\) and \(\Psi(M)|_Q\cong\Psi(M')|_Q\), but \(\Psi(M)\not\cong\Psi(M')\).

## 6. Complete solutions

**5.1.** An arrow of \(\int C_\bullet(S_1)\) with source \((q,(o_q,()))\) lies over some \(\alpha\colon[m]\to[q]\), and \(C(\alpha)(o_q,())=(o_m,())\). The only object of \(C_m(S_1)\) that receives an arrow from \((o_m,())\) is \((o_m,())\) itself, because \((\Delta^{\mathrm{op}})^0\), the grading at \([-1]\), is a point and there are no arrows between different strings. So the arrows out of these objects stay among them, and \(\mathbb 1\) is a functor, arrows from other objects into them acting by \(0\). By Proposition 1.4 (2) of the lesson on graded rings, \(\operatorname{ind}(\mathbb 1)_q(c)\) is the sum over the factorizations \(c=c_1\otimes(o_q,())\); since \((o_q,())\) is the unit of \(C_q(S_1)\), the only one is \(c_1=c\). So \(\operatorname{ind}(\mathbb 1)_q(c)=Q_q(c)\), with \(Q\) acting by multiplication, and \(\operatorname{ind}(\mathbb 1)=Q\). The module \(X_A\) is \(\mathbb Z(C_\bullet(A(1)^\times))\) with \(C_\bullet(A(1)^\times)\) acting on itself by multiplication, which is \(Q\). By adjunction, \(\operatorname{Hom}_Q(Q,L)=\operatorname{Hom}_{\mathcal M_1}(\mathbb 1,UL)\), the families of global sections \(\ell_q\) of \(L_q(o_q)\) with \(L(\alpha)\ell_q=\ell_m\) for all \(\alpha\colon[m]\to[q]\). The map \(\alpha_q\colon[q]\to[0]\) is unique, so such a family is determined by \(\ell_0\), through \(\ell_q=L(\alpha_q)\ell_0\); conversely every \(\ell_0\) gives a family, because \(\alpha_q\beta=\alpha_m\) for \(\beta\colon[m]\to[q]\). So \(\operatorname{Hom}_Q(Q,L)=\Gamma(T,L_0(o_0))\). For \(L=C_\bullet(M(1))\), \(L_0(o_0)=M\langle0\rangle=M\), and \(\operatorname{Hom}_Q(X_A,C_\bullet(M(1)))=\) \(\Gamma(T,M)=\) \(\operatorname{Hom}_A(A,M)\), which is \(H^0\) of \(\operatorname{RHom}_A(A,M)\), as Corollary 2.4 asserts for \(G=A\).

**5.2.** Let \(\ell\) be such a map, with components \(\ell_c\in\Gamma(T,C_q(M(1))(c))\) for the objects \((q,c)\). Equivariance with respect to the zero section of \(C_q(A(1)^\times)(o_q,())=A\langle0\rangle=A\), together with \((o_q,())\otimes c=c\), gives \(\ell_c=0\cdot\ell_c\). The action of a section \(a\) of \(A\langle0\rangle\) on \(M\langle n\rangle(r)=M^P\) multiplies every coordinate by \(a\) (Definition 3.2 of the lesson on graded rings), so \(0\cdot\ell_c=0\). For the derived statement take \(G=0\) in Theorem 4.3. The spectrum of \(0\) is the final \(S_1\)-graded object, so the pseudo-category of \(G=0\) is \(\mathcal X_A\), \(p\) is the identity and \(X_0=\mathbb Z(C_\bullet(e_{S_1}))=\mathbb Z\), the constant module, whose maps into a module are its global sections. Hence \(R\Gamma(BA,C_\bullet(M(1)))=\) \(\operatorname{RHom}_Q(\mathbb Z,C_\bullet(M(1)))\cong\) \(\operatorname{RHom}_A(0,M)=\) \(0\), and Theorem 4.3 gives the same for the total topos.

**5.3.** (a) \(\mathbb F_p\otimes p\mathbb Z\) is free of rank one on \(e=1\otimes p\), and \(de=1\otimes p=p(1\otimes1)=0\). The product of two elements of degree \(-1\) vanishes by Lemma 1.1 of the lesson on the stable ring.

(b) Both have zero differential, with \(\mathbb F_p\) in degrees \(-1\) and \(0\), so they are isomorphic as complexes of vector spaces. For a differential graded \(\mathbb F_p[e]\)-module \(N\), the cohomology \(H^*(N)\) is a graded module over \(H^*(\mathbb F_p[e])=\mathbb F_p[e]\), and quasi-isomorphisms of modules induce isomorphisms of these graded modules; hence so do all isomorphisms of \(D(\mathbb F_p[e])\), which are composites of quasi-isomorphisms and their inverses. On \(H^*(E)\), \(e\) maps \(H^0\) isomorphically onto \(H^{-1}\); on \(H^*(E')\) it is \(0\). So \(E\not\cong E'\).

(c) Present both factors. The differential graded ring \(P_A\otimes(p\mathbb Z\to\mathbb Z)\) maps to \(A\otimes(p\mathbb Z\to\mathbb Z)=\mathbb F_p[e]\) through \(w\), and to \(P_A\otimes\mathbb F_p=R_{\mathcal O}\) through \((p\mathbb Z\to\mathbb Z)\to\mathbb F_p\). Both maps are homomorphisms under \(P_A\), and quasi-isomorphisms, because the complexes \(P_A\) and \(p\mathbb Z\to\mathbb Z\) are bounded with flat terms, hence K-flat. By Proposition 2.5 (3) of the lesson on differential graded modules, they induce equivalences \(D(\mathbb F_p[e])\simeq D(R_{\mathcal O})\) compatible with restriction to \(P_A\). Let \(M\) and \(M'\) correspond to \(E\) and \(E'\). Since \(w_!w^*\cong\mathrm{id}\), \(M|_A\) and \(M'|_A\) are the underlying complexes of \(E\) and \(E'\), restricted along \(\mathbb F_p\to\mathbb F_p[e]\), which are isomorphic by (b). By Theorem 3.3 (3), \(\Psi(M)|_Q\cong C_\bullet((M|_A)(1))\cong\Psi(M')|_Q\). But \(M\not\cong M'\), both lie in \(D^{[-1,0]}(R_{\mathcal O})\), and \(\Psi\) is fully faithful there (Proposition 3.4), so it reflects isomorphisms: \(\Psi(M)\not\cong\Psi(M')\).

## Sources

The expression of the Ext groups of modules over a ring of a topos as the cohomology of a diagram of products of copies of the ring and the modules, through the nerve of the action of the spectrum of the ring on the spectrum of a module, is due to Illusie, in his work on the cotangent complex, as are the comparison of modules over \(A\otimes^{\mathbf L}_{\mathbb Z}\mathcal O\) with modules over \(Q_A\otimes\mathcal O\) and the relative form for a homomorphism of modules. Mac Lane's theorem, which supplies the main input, goes back to Mac Lane's work on the homology of rings.