Differential graded modules in a topos

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

The normalized complex of a simplicial ring $R$ is a graded commutative differential graded ring $NR$, concentrated in nonpositive degrees, and the normalized complex of an $R$-module is a differential graded $NR$-module. This lesson develops the derived categories of differential graded modules over a differential graded ring of a topos, with the features that hold when the ring is concentrated in nonpositive degrees: truncations, bounded derived categories that are full subcategories, truncated rings, resolutions that respect bounds, and bounds for derived Hom, derived internal Hom, derived tensor products and extension of scalars. Extension and restriction of scalars are functorial in the ring up to quasi-isomorphism. The derived functors on all differential graded modules come from the AI-integrated Stacks project, which constructs them with good and K-injective resolutions. The last section proves that normalization carries the derived tensor products and extensions of scalars of simplicial modules to those of differential graded modules.

Prerequisites. Proposition 2.1 of Simplicial rings and modules in a topos; Construction 1.2, Lemma 1.3 and Proposition 3.2 of Derived tensor products of simplicial modules; Proposition 1.1 of Spectral sequences for derived tensor products; Lemma 2.6 of Multisimplicial objects and spectral diagrams; Theorems 1.1 and 1.3 of The cotangent complex. From the AI-integrated Stacks project, for sheaves of differential graded algebras on a ringed site: algebras, modules and homotopies [Stacks, Tag 0FRF, Tag 0FRI, Tag 0FS0], the triangulated homotopy category [Tag 0FS7], good modules [Section 0FSA, Tag 0FSB, Tag 0FSC, Tag 0FSD, Tag 0FSH], K-injective modules [Tag 0FSU, Tag 0FT0], derived functors defined by essentially constant ind- and pro-objects [Tag 05S9], the derived category [Tag 0FT4, Tag 0FT5, Tag 0FT8, Tag 0FTN], derived tensor products and extension of scalars [Tag 0FTF, Tag 0FTG, Tag 0FTR, Tag 0FTZ]; bounded above complexes of flat sheaves are K-flat [Tag 06YQ] and bounded below complexes of injective sheaves are K-injective [Tag 070J]; resolutions by double complexes [Tag 0133]; limits of Mittag-Leffler systems [Tag 0598].

Conventions

$T$ is the topos of sheaves on a small site. A differential graded ring $A$ of $T$ is a sheaf of differential graded algebras over the constant ring $\mathbb Z$ [Stacks, Tag 0FRF]: a complex of abelian sheaves with an associative unital multiplication $A\otimes_{\mathbb Z}A\to A$ that is a map of complexes. Modules are right modules [Stacks, Tag 0FRI]; a left module is a right module over the opposite ring. $A$ is graded commutative if $ab=(-1)^{ij}ba$ for $a\in A^i$ and $b\in A^j$. Then a right module $M$ is a left module by $ax=(-1)^{ij}xa$ for $x\in M^j$, and tensor products and Hom complexes over $A$ are again $A$-modules. $A$ is in nonpositive degrees if $A^i=0$ for $i>0$.

$C(A)$ is the abelian category of differential graded $A$-modules. A homotopy between maps $f,g\colon M\to N$ is an $A$-linear graded map $h$ of degree $-1$ with $f-g=dh+hd$ [Stacks, Tag 0FS0]. The homotopy category $K(A)$ is triangulated by the usual cones [Stacks, Tag 0FS7]. A quasi-isomorphism is a map inducing isomorphisms on all cohomology sheaves. The quasi-isomorphisms form a saturated multiplicative system of $K(A)$, compatible with the triangulation, and the derived category $D(A)$ is the localization of $K(A)$ at it [Stacks, Tags 0FT4, 0FT5]. It is also the localization of $C(A)$ at the quasi-isomorphisms. Indeed, a homotopy $h$ between $f,g\colon M\to N$ defines a map $H(a,b,c)=f(a)+h(b)+g(c)$ to $N$ from the cylinder $M\oplus M[1]\oplus M$, whose differential is $(a,b,c)\mapsto(da+b,-db,dc-b)$. The maps $f$ and $g$ are $H$ composed with the inclusions of the two outer summands, and these inclusions have the common left inverse $(a,b,c)\mapsto a+c$, which is a homotopy equivalence. So $f$ and $g$ become equal in every localization of $C(A)$ at the quasi-isomorphisms. When $A$ is concentrated in degree $0$, $C(A)$ is the category of complexes of $A^0$-modules and $D(A)$ is its usual derived category.

For $p\leq q$ in $\mathbb Z$, $C^{\leq q}(A)$, $C^{\geq p}(A)$ and $C^{[p,q]}(A)$ are the full subcategories of the modules $M$ with $M^i=0$ for $i>q$, for $i<p$, and for $i\notin[p,q]$. $C^-(A)$, $C^+(A)$ and $C^b(A)$ are their unions over $q$, over $p$, and over the pairs $(p,q)$. Their localizations at the quasi-isomorphisms are written $D^{\leq q}(A)$, $D^{\geq p}(A)$, $D^{[p,q]}(A)$, $D^-(A)$, $D^+(A)$ and $D^b(A)$; Proposition 1.2 identifies them with full subcategories of $D(A)$.

For modules $E$ and $F$, $\operatorname{Hom}^\bullet_A(E,F)$ is the complex of abelian groups whose degree $k$ part consists of the $A$-linear graded maps $f\colon E\to F$ of degree $k$, with differential $df=d_F\circ f-(-1)^kf\circ d_E$. Thus $H^k\operatorname{Hom}^\bullet_A(E,F)=\operatorname{Hom}_{K(A)}(E,F[k])$. $\mathcal Hom^\bullet_A(E,F)$ is the complex of abelian sheaves $U\mapsto\operatorname{Hom}^\bullet_{A|_U}(E|_U,F|_U)$. For a complex $M$ of abelian sheaves, the induced and coinduced modules are $\operatorname{ind}(M)=M\otimes_{\mathbb Z}A$ and $\operatorname{coind}(M)=\mathcal Hom^\bullet_{\mathbb Z}(A,M)$, with $A$ acting by $(m\otimes b)a=m\otimes ba$ and $(\varphi a)(b)=\varphi(ab)$. Restriction along $m\mapsto m\otimes1$, evaluation at $1$ and associativity of the tensor product give isomorphisms, natural in modules $E$, $F$ and left modules $N$, $\begin{gathered} \operatorname{Hom}^\bullet_A(\operatorname{ind}M,F)=\operatorname{Hom}^\bullet_{\mathbb Z}(M,F),\\ \operatorname{Hom}^\bullet_A(E,\operatorname{coind}M)=\operatorname{Hom}^\bullet_{\mathbb Z}(E,M),\\ \operatorname{ind}(M)\otimes_AN=M\otimes_{\mathbb Z}N. \end{gathered}$

1. Rings in nonpositive degrees

In this section $A$ is in nonpositive degrees. For a module $M$, $Z^iM$ and $B^iM$ are the sheaves of cycles and of boundaries in degree $i$.

Lemma 1.1 (truncations). For $n\in\mathbb Z$ and a module $M$ put $\begin{gathered} \tau^{\leq n}M=(\cdots\to M^{n-1}\\ \to Z^nM\to0\to\cdots),\\ \tau^{\geq n}M=(\cdots\to0\to M^n/B^nM\\ \to M^{n+1}\to\cdots). \end{gathered}$ Then $\tau^{\leq n}M$ is a submodule and $\tau^{\geq n}M$ a quotient module of $M$. The functors $\tau^{\leq n}$ and $\tau^{\geq n}$ preserve homotopies and quasi-isomorphisms. The natural maps $\tau^{\leq n}M\to M\to\tau^{\geq n}M$ induce isomorphisms on $H^i$ for $i\leq n$, respectively $i\geq n$, and $H^i\tau^{\leq n}M=0$ for $i>n$, $H^i\tau^{\geq n}M=0$ for $i<n$. On $C^{\leq n}(A)$ the functor $\tau^{\leq n}$ is the identity and the map $\tau^{\leq n}M\to M$ is the identity; the same holds for $\tau^{\geq n}$ on $C^{\geq n}(A)$.

Proof. Let $a\in A^i$ with $i\leq0$. For $x\in Z^nM$, $xa$ has degree $n+i$. If $i<0$ this is below $n$; if $i=0$, then $d(xa)=(dx)a+(-1)^nx\,da=0$ because $da\in A^1=0$. So $\tau^{\leq n}M$ is a submodule. The kernel of $M\to\tau^{\geq n}M$ is $\cdots\to M^{n-1}\to B^nM\to0$. For $y\in M^{n-1}$, $(dy)a=d(ya)-(-1)^{n-1}y\,da$ lies in $B^nM$ if $i=0$ and has degree below $n$ if $i<0$. So the kernel is a submodule. Maps of modules preserve cycles and boundaries, so both constructions are functors. A homotopy $h$ between two maps $M\to N$ has degree $-1$; it maps $\tau^{\leq n}M$ into $\tau^{\leq n}N$, and the kernel of $M\to\tau^{\geq n}M$ into the kernel of $N\to\tau^{\geq n}N$. So it restricts to a homotopy between the truncated maps and induces one between the quotient maps. The statements about cohomology and the last sentence follow from the definitions. $\square$

Proposition 1.2.

If $H^iX=0$ for $i>n$ and $H^iY=0$ for $i\leq n$, then $\operatorname{Hom}_{D(A)}(X,Y)=0$.
The canonical functors from $D^-(A)$, $D^+(A)$, $D^b(A)$, $D^{\leq q}(A)$, $D^{\geq p}(A)$ and $D^{[p,q]}(A)$ to $D(A)$ are fully faithful. Their essential images consist of the objects whose cohomology is bounded above, bounded below, bounded, zero in degrees above $q$, zero in degrees below $p$, and zero outside $[p,q]$. Hence the canonical functors among these categories, such as $D^b(A)\to D^-(A)$ or $D^{[p,q]}(A)\to D^{\leq q}(A)$, are fully faithful as well.

Proof. By Lemma 1.1 the truncations induce functors on $D(A)$, the localization of $C(A)$. A natural transformation between functors of modules that preserve quasi-isomorphisms remains natural on the localization, because the morphisms of a localization are composites of maps of modules and inverses of quasi-isomorphisms.

(1) Let $f\colon X\to Y$, and let $\beta_M\colon M\to\tau^{\geq n+1}M$ be the natural map. Naturality gives $\beta_Yf=\tau^{\geq n+1}(f)\beta_X$. The module $\tau^{\geq n+1}X$ is acyclic, hence zero in $D(A)$, and $\beta_Y$ is a quasi-isomorphism. So $f=0$.

(2) We use the following principle. Let $\mathcal C'\subseteq\mathcal C$ be full subcategories of $C(A)$, with localizations $D(\mathcal C')$ and $D(\mathcal C)$ at the quasi-isomorphisms and the induced functor $\iota\colon D(\mathcal C')\to D(\mathcal C)$. Suppose given a functor $\tau\colon\mathcal C\to\mathcal C'$ that is the identity on $\mathcal C'$, and a chain of natural transformations $\mathrm{id}=F_0,F_1,\ldots,F_m=\tau$, each between $F_{j-1}$ and $F_j$ in one of the two directions, where the $F_j\colon\mathcal C\to\mathcal C$ preserve quasi-isomorphisms, fix the objects of $\mathcal C'$, and the transformations are identities at the objects of $\mathcal C'$. Then $\iota$ is bijective on $\operatorname{Hom}(X,Y)$ for $X,Y\in\mathcal C'$. Indeed, $\tau$ induces $T\colon D(\mathcal C)\to D(\mathcal C')$, and $T\iota$ is the identity because it is induced by the identity of $\mathcal C'$; so $\iota$ is injective on morphisms. For $\varphi\colon X\to Y$ in $D(\mathcal C)$, naturality of the transformation between $F_{j-1}$ and $F_j$, whose components at $X$ and $Y$ are identities, gives $F_{j-1}(\varphi)=F_j(\varphi)$. So $\varphi=\tau(\varphi)=\iota T(\varphi)$.

Apply the principle with $\tau=\tau^{\leq q}$ and the transformation $\tau^{\leq q}\to\mathrm{id}$; with $\tau=\tau^{\geq p}$ and $\mathrm{id}\to\tau^{\geq p}$; and with $\tau=\tau^{\geq p}\tau^{\leq q}$ and $\mathrm{id}\leftarrow\tau^{\leq q}\to\tau^{\geq p}\tau^{\leq q}$. Taking $\mathcal C=C(A)$, this shows that $D^{\leq q}(A)$, $D^{\geq p}(A)$ and $D^{[p,q]}(A)$ are fully faithful in $D(A)$. Taking $\mathcal C=C^-(A)$, $C^+(A)$ and $C^b(A)$, which the truncations preserve, it shows that they are fully faithful in $D^-(A)$, $D^+(A)$ and $D^b(A)$. Two objects $X,Y$ of $C^-(A)$ lie in a common $C^{\leq q}(A)$, and $D^{\leq q}(A)\to D(A)$ factors through $D^-(A)$. Both maps out of $\operatorname{Hom}_{D^{\leq q}(A)}(X,Y)$ are bijective, hence so is $\operatorname{Hom}_{D^-(A)}(X,Y)\to\operatorname{Hom}_{D(A)}(X,Y)$. The same argument applies to $D^+(A)$ and $D^b(A)$. For the essential images: if $H^iX=0$ for $i>q$, then $\tau^{\leq q}X\to X$ is a quasi-isomorphism and $\tau^{\leq q}X\in C^{\leq q}(A)$; the other cases use $X\to\tau^{\geq p}X$ and $X\leftarrow\tau^{\leq q}X\to\tau^{\geq p}\tau^{\leq q}X$. $\square$

Proposition 1.3 (truncated rings). Let $n\geq0$. Then $I=A^{<-n}+dA^{-n-1}$ is a two-sided ideal of $A$ stable under $d$, and $\begin{gathered} t_n(A)=A/I=\\ (A^{-n}/dA^{-n-1}\to A^{-n+1}\to\cdots\to A^0) \end{gathered}$ is a differential graded ring in nonpositive degrees. The map $A\to t_n(A)$ induces isomorphisms on $H^i$ for $i\geq-n$. If $b-a\leq n$, then $I$ annihilates every module of $C^{[a,b]}(A)$. Hence restriction of scalars is an isomorphism of categories $C^{[a,b]}(t_n(A))\to C^{[a,b]}(A)$ that preserves homotopies and quasi-isomorphisms, and it induces an equivalence $D^{[a,b]}(t_n(A))\to D^{[a,b]}(A)$. By Proposition 1.2 this is an equivalence between the full subcategories of $D(t_n(A))$ and of $D(A)$ formed by the objects with cohomology in $[a,b]$.

Proof. Let $c\in A^i$ with $i\leq0$. Products of $c$ with $A^{<-n}$ lie in $A^{<-n}$. For $y\in A^{-n-1}$, $c\,dy=(-1)^i(d(cy)-(dc)y)$. If $i<0$, both terms lie in $A^{<-n}$. If $i=0$, then $dc=0$ and $d(cy)\in dA^{-n-1}$. Products on the right are treated in the same way, and $dI\subseteq I$ because $d(A^{<-n})\subseteq A^{<-n}+dA^{-n-1}$. The quotient has the stated terms. Its cohomology in degree $-n$ is $Z^{-n}A/dA^{-n-1}=H^{-n}(A)$, and in higher degrees it agrees with that of $A$.

Let $M\in C^{[a,b]}(A)$ and $x\in M^j$. For $c\in A^{<-n}$, $xc$ has degree at most $b-n-1<a$, so $xc=0$. For $y\in A^{-n-1}$, $x\,dy=(-1)^j(d(xy)-(dx)y)$. Here $xy$ has degree $j-n-1<a$, so $xy=0$. If $j=b$ then $dx=0$; otherwise $(dx)y$ has degree $j-n<b-n\leq a$, so $(dx)y=0$. Hence $I$ acts by zero, and the action factors uniquely through $t_n(A)$. Since $A\to t_n(A)$ is surjective, $A$-linear maps and homotopies between $t_n(A)$-modules are $t_n(A)$-linear, and quasi-isomorphisms are defined on the underlying complexes. So the two categories and their quasi-isomorphisms correspond, and so do their localizations. $\square$

2. Resolutions and derived functors

A right module $P$ is K-flat if $P\otimes_AN$ is acyclic for every acyclic left module $N$; K-flat left modules are defined in the same way. Good modules in the sense of the Stacks project are K-flat [Stacks, Section 0FSA].

Lemma 2.1 (bounded resolutions). Let $A$ be in nonpositive degrees.

For every $q$ there is a functor $Q\colon C^{\leq q}(A)\to C^{\leq q}(A)$ with a natural quasi-isomorphism $Q(E)\to E$, such that every $Q(E)$ is good.
For every $F\in C^{\geq p}(A)$ there is a quasi-isomorphism $F\to J$ with $J\in C^{\geq p}(A)$ K-injective [Stacks, Tag 0FSU].

Proof. (1) For an object $U$ of the site and $k\in\mathbb Z$, let $S_{U,k}=(j_{U!}A_U)[-k]$, the module freely generated over $U$ by one element of degree $k$. Let $D_{U,k}$ be the cone of the identity of $S_{U,k+1}$, freely generated over $U$ by an element $x$ of degree $k$ and its differential $dx$. Both are good: $j_{U!}A_U$ is good [Stacks, Tag 0FSB], the conditions defining good modules do not change under shifts, and $D_{U,k}$ is an extension of $S_{U,k+1}[1]$ by $S_{U,k+1}$ split as graded modules [Stacks, Tag 0FSC]. Since $A$ is in nonpositive degrees, $S_{U,k}$ lies in degrees $\leq k$ and $D_{U,k}$ in degrees $\leq k+1$. Maps $S_{U,k}\to E$ correspond to the cycles of $E^k$ over $U$, and maps $D_{U,k}\to E$ to the sections of $E^k$ over $U$. For $E\in C^{\leq q}(A)$ let $Q_0(E)$ be the sum of a copy of $D_{U,k}$ for each section $s\in E^k(U)$ with $k<q$ and a copy of $S_{U,q}$ for each section $s\in E^q(U)$, mapped to $E$ through $s$; the sections of $E^q$ are cycles because $E^{q+1}=0$. Then $Q_0(E)$ is good [Stacks, Tag 0FSD], lies in $C^{\leq q}(A)$ and maps onto $E$ in every degree. A map $E\to E'$ carries the summand of $s$ to that of its image, so $Q_0$ is a functor. The kernel of $Q_0(E)\to E$ lies in $C^{\leq q}(A)$, and applying $Q_0$ to kernels repeatedly gives an exact sequence $\cdots\to Q_1(E)\to Q_0(E)\to E\to0$, functorial in $E$. Let $Q(E)$ be its total complex, with $Q_i(E)$ in horizontal degree $-i$; it is a module because the maps $Q_i(E)\to Q_{i-1}(E)$ are $A$-linear. In each degree it is a finite sum, because $Q_i(E)^{m+i}=0$ for $i>q-m$, and $Q(E)\in C^{\leq q}(A)$. The total complexes of $Q_j(E)\to\cdots\to Q_0(E)$ are good by induction on $j$, each being an extension of a shift of $Q_j(E)$ by the previous one, split as graded modules [Stacks, Tag 0FSC]. $Q(E)$ is their union, hence good [Stacks, Tag 0FSD]. The map $Q(E)\to E$ is a quasi-isomorphism by [Stacks, Tag 0133], applied in the opposite category, which exchanges resolutions and coresolutions.

(2) Choose injections $F^n\to I^n$ into injective abelian sheaves. Let $M$ be the complex with $M^n=I^n\oplus I^{n+1}$ for $n\geq p$, $M^n=0$ for $n<p$, and differential $(u,v)\mapsto(v,0)$. It is a complex of injective sheaves vanishing below $p$, and $x\mapsto(x,dx)$ is an injective map of complexes $F\to M$. By the isomorphism $\operatorname{Hom}_A(F,\operatorname{coind}M)=\operatorname{Hom}_{\mathbb Z}(F,M)$ of the Conventions, it corresponds to an $A$-linear map $F\to\operatorname{coind}(M)$, which is injective because its composite with evaluation at $1$ is. The module $\operatorname{coind}(M)$ lies in $C^{\geq p}(A)$: its degree $n$ term is the product of the sheaves $\mathcal Hom(A^k,M^{n+k})$ over $k\leq0$, and $M^{n+k}=0$ for $n+k<p$. It is K-injective: for an acyclic module $N$, $\operatorname{Hom}^\bullet_A(N,\operatorname{coind}M)=\operatorname{Hom}^\bullet_{\mathbb Z}(N,M)$ is acyclic, because a bounded below complex of injective sheaves is K-injective [Stacks, Tag 070J]. Repeating with cokernels gives an exact sequence $0\to F\to J^0\to J^1\to\cdots$ of K-injective modules in $C^{\geq p}(A)$. Let $J$ be its total complex. In each degree it is a finite sum, $J\in C^{\geq p}(A)$, and $F\to J$ is a quasi-isomorphism [Stacks, Tag 0133].

The total complex $J_{\leq s}$ of $J^0\to\cdots\to J^s$ is K-injective, by induction on $s$: it is an extension of $J_{\leq s-1}$ by a shift of $J^s$, split as graded modules, which gives a distinguished triangle in $K(A)$, and $\operatorname{Hom}_{K(A)}(N,-)$ is homological. $J$ is the limit of the $J_{\leq s}$, reached at a finite stage in each degree, and the maps $J_{\leq s+1}\to J_{\leq s}$ split as graded modules. So for acyclic $N$, $\operatorname{Hom}^\bullet_A(N,J)$ is the limit of the acyclic complexes $K_s=\operatorname{Hom}^\bullet_A(N,J_{\leq s})$ of abelian groups, with surjective transition maps. Such a limit is acyclic. The kernel of $K_{s+1}\to K_s$ is acyclic, so a cycle of $K_s$ lifts to an element $k$ of $K_{s+1}$ whose differential is the differential of an element $l$ of that kernel, and $k-l$ is a cycle lifting it. So the cycles of the $K_s$ form an inverse system with surjective transition maps, the exact sequences $0\to Z^mK_s\to K^m_s\to Z^{m+1}K_s\to0$ stay exact in the limit [Stacks, Tag 0598], and every cycle of the limit is a boundary. $\square$

Lemma 2.2 (K-flat modules).

If $P$ is K-flat, $P\otimes_A-$ carries quasi-isomorphisms to quasi-isomorphisms.
If $P'\to P$ is a quasi-isomorphism of K-flat right modules, then $P'\otimes_AN\to P\otimes_AN$ is a quasi-isomorphism for every left module $N$.
If $P\to E$ and $Q\to N$ are quasi-isomorphisms from a K-flat right module and a K-flat left module, then the maps $P\otimes_AN\leftarrow P\otimes_AQ\to E\otimes_AQ$ are quasi-isomorphisms.

The statements with the two variables exchanged hold as well.

Proof. (1) $P\otimes_A-$ commutes with cones. (2) Choose a quasi-isomorphism $Q\to N$ with $Q$ good [Stacks, Tag 0FSH], applied to the opposite ring. In the commutative square formed by $P'\otimes_AQ\to P\otimes_AQ$ and $P'\otimes_AN\to P\otimes_AN$, the vertical maps are quasi-isomorphisms by (1). The upper map is one because its cone is the cone of $P'\to P$ tensored with $Q$, which is acyclic. So the lower map is a quasi-isomorphism. (3) The first map by (1); the cone of the second is the cone of $P\to E$ tensored with $Q$. $\square$

Proposition 2.3 (derived Hom). Let $A$ be a differential graded ring of $T$.

For modules $E$, $F$ and a quasi-isomorphism $F\to I$ into a K-injective module, put $\operatorname{RHom}_A(E,F)=\operatorname{Hom}^\bullet_A(E,I)$ and $R\mathcal Hom_A(E,F)=\mathcal Hom^\bullet_A(E,I)$. With the functorial choice of $I$ of [Stacks, Tag 0FT0], these are functors $D(A)^{\mathrm o}\times D(A)\to D(\mathrm{Ab})$ and $D(A)^{\mathrm o}\times D(A)\to D(\mathbb Z)$, into the derived categories of abelian groups and of abelian sheaves, exact in each variable. Every K-injective $I$ gives the same values up to canonical isomorphism. The ind-object of $D(\mathrm{Ab})$ formed by the complexes $\operatorname{Hom}^\bullet_A(E',F')$, over the quasi-isomorphisms $E'\to E$ and $F\to F'$, is essentially constant with value $\operatorname{RHom}_A(E,F)$; so $\operatorname{RHom}_A$ is the right derived functor of $\operatorname{Hom}^\bullet_A$ [Stacks, Tag 05S9]. If $A$ is graded commutative, $R\mathcal Hom_A$ takes values in $D(A)$.
$H^n\operatorname{RHom}_A(E,F)=$ $\operatorname{Hom}_{D(A)}(E,F[n])$, and $H^nR\mathcal Hom_A(E,F)$ is the sheaf associated with the presheaf $U\mapsto\operatorname{Hom}_{D(A|_U)}(E|_U,F|_U[n])$.
If $A$ is in nonpositive degrees, $\operatorname{RHom}_A$ and $R\mathcal Hom_A$ map $D^{\leq0}(A)^{\mathrm o}\times D^{\geq0}(A)$ into $D^{\geq0}$.
If $A$ is concentrated in degree $0$, they are the usual derived Hom functors of complexes of $A^0$-modules.

Proof. Let $I$ be K-injective. For an acyclic module $N$, $H^n\operatorname{Hom}^\bullet_A(N,I)=$ $\operatorname{Hom}_{K(A)}(N,I[n])=$ $0$. Since $\operatorname{Hom}^\bullet_A(-,I)$ carries cones to shifted cones, it carries quasi-isomorphisms to quasi-isomorphisms. A quasi-isomorphism $I\to I'$ between K-injective modules is a homotopy equivalence, because its cone is acyclic and K-injective, so the identity of the cone is zero in $K(A)$; this gives the independence of $I$. Hence $\operatorname{RHom}_A$ is a functor of two variables on $D(A)$, and $H^n\operatorname{RHom}_A(E,F)=$ $\operatorname{Hom}_{K(A)}(E,I[n])=$ $\operatorname{Hom}_{D(A)}(E,F[n])$ [Stacks, Tag 0FT8]. For a fixed $E'$, the ind-object of the $\operatorname{Hom}^\bullet_A(E',F')$ over $F\to F'$ is essentially constant with value $\operatorname{Hom}^\bullet_A(E',I)$ [Stacks, Tag 0FTN], naturally in $E'$; and the $\operatorname{Hom}^\bullet_A(E',I)$ form a constant ind-object over $E'\to E$ up to isomorphism, because every quasi-isomorphism $E''\to E'$ induces a quasi-isomorphism. An ind-object over a product of two index categories whose partial ind-objects are essentially constant, naturally, is essentially constant with the iterated value. In the first variable, $\operatorname{Hom}^\bullet_A(-,I)$ carries cone triangles to cone triangles. In the second, a distinguished triangle is isomorphic to the cone triangle of a map of K-injective modules, since cones of K-injective modules are K-injective, and $\operatorname{Hom}^\bullet_A(E,-)$ carries it to a cone triangle. So $\operatorname{RHom}_A$ is exact in each variable.

For an object $U$ of the site, $I|_U$ is K-injective over $A|_U$: extension by zero $j_{U!}$ is exact and left adjoint to restriction, compatibly with homotopies, so $\operatorname{Hom}_{K(A|_U)}(N,I|_U)=$ $\operatorname{Hom}_{K(A)}(j_{U!}N,I)=$ $0$ for acyclic $N$. So the sections of $\mathcal Hom^\bullet_A(E,I)$ over $U$ form $\operatorname{RHom}_{A|_U}(E|_U,F|_U)$, whose cohomology is $\operatorname{Hom}_{D(A|_U)}(E|_U,F|_U[n])$; the cohomology sheaves are the associated sheaves. The arguments given for $\operatorname{RHom}_A$ apply over every $U$. If $A$ is graded commutative, $\mathcal Hom^\bullet_A(E,I)$ is an $A$-module and all the maps above are $A$-linear.

(3) If $E\in D^{\leq0}(A)$, $F\in D^{\geq0}(A)$ and $n<0$, then $F[n]$ has no cohomology in degrees $\leq0$, so $\operatorname{Hom}_{D(A)}(E,F[n])=0$ by Proposition 1.2 (1). The same holds over every $U$, because restriction preserves the bounds on cohomology, so the associated sheaves vanish as well. (4) For $A$ concentrated in degree $0$, modules, homotopies, quasi-isomorphisms and K-injective modules are those of complexes of $A^0$-modules. $\square$

Proposition 2.4 (derived tensor product). Let $A$ be a differential graded ring of $T$. For a right module $E$ and a left module $F$, let $E\otimes^{\mathbf L}_AF$ be $P\otimes_AF$ for a quasi-isomorphism $P\to E$ from a K-flat module, or equivalently $E\otimes_AQ$ for a quasi-isomorphism $Q\to F$ from a K-flat left module. This defines a functor $D(A)\times D(A^{\mathrm{op}})\to D(\mathbb Z)$, exact in each variable, with values in $D(A)$ if $A$ is graded commutative. The pro-object formed by the complexes $E'\otimes_AF'$, over the quasi-isomorphisms $E'\to E$ and $F'\to F$, is essentially constant with value $E\otimes^{\mathbf L}_AF$, so this is the left derived functor of $\otimes_A$ [Stacks, Tag 05S9]. If $A$ is in nonpositive degrees, it maps $D^{\leq0}\times D^{\leq0}$ into $D^{\leq0}$.

Proof. For a fixed left module $F$, $E\mapsto E\otimes^{\mathbf L}_AF$ is the derived tensor product of the Stacks project, with $F$ regarded as a bimodule [Stacks, Tags 0FTF, 0FTG]; it is computed by good resolutions, and by Lemma 2.2 every K-flat resolution computes it, in either variable. A map $F\to F'$ induces a natural transformation, and by Lemma 2.2 (1) a quasi-isomorphism induces an isomorphism. So the construction is a functor of two variables on the derived categories. For a fixed $F'$, the pro-object of the $E'\otimes_AF'$ over $E'\to E$ is essentially constant with value $E\otimes^{\mathbf L}_AF'$ [Stacks, Tag 0FTF], naturally in $F'$, and these values form an essentially constant pro-object over $F'\to F$, because a quasi-isomorphism $F''\to F'$ induces an isomorphism; as in Proposition 2.3 this gives the statement about the pro-object over both variables. It is exact in each variable, because $P\otimes_A-$ and $-\otimes_AQ$ carry cone triangles to cone triangles. If $A$ is graded commutative, regard $F$ as a bimodule over $A$ on both sides. For the bound, $E\cong\tau^{\leq0}E$ and $F\cong\tau^{\leq0}F$ by Proposition 1.2. With $P=Q(\tau^{\leq0}E)$ from Lemma 2.1 (1), the module $P\otimes_A\tau^{\leq0}F$ vanishes in positive degrees. $\square$

Proposition 2.5 (extension of scalars). Let $\varphi\colon A\to B$ be a map of differential graded rings of $T$, and let $E\mapsto E\otimes^{\mathbf L}_AB$ be the derived extension of scalars $D(A)\to D(B)$ [Stacks, Tag 0FTG]. It is computed by $P\otimes_AB$ for every quasi-isomorphism $P\to E$ from a K-flat module, by Lemma 2.2 (2).

Extension of scalars is left adjoint to restriction of scalars $D(B)\to D(A)$. For $E\in D(A)$ and $F\in D(B)$ there is an isomorphism $\operatorname{RHom}_B(E\otimes^{\mathbf L}_AB,F)\cong$ $\operatorname{RHom}_A(E,F)$, natural in $E$ and $F$, which induces the adjunction bijections on $H^n$.
If $A$ and $B$ are in nonpositive degrees, extension of scalars maps $D^{\leq q}(A)$ into $D^{\leq q}(B)$ for every $q$.
If $\varphi$ is a quasi-isomorphism, extension and restriction of scalars are inverse equivalences, and for $E,F\in D(B)$ restriction of scalars induces an isomorphism $\operatorname{RHom}_B(E,F)\to\operatorname{RHom}_A(E,F)$.

Proof. (1) The adjunction is [Stacks, Tag 0FTR] for the identity morphism of the topos ringed by $\mathbb Z$. There the derived pushforward is restriction of scalars, which preserves quasi-isomorphisms, and the adjunction is induced by the adjunction between $-\otimes_AB$ and restriction on modules; on a good module $P$ its unit is $P\to P\otimes_AB$, $x\mapsto x\otimes1$. Let $P\to E$ be a good resolution, $F\to I$ a K-injective resolution over $B$, and $I\to J$ one over $A$. The map of complexes $\begin{gathered} \operatorname{Hom}^\bullet_B(P\otimes_AB,I)=\operatorname{Hom}^\bullet_A(P,I)\\ \to\operatorname{Hom}^\bullet_A(P,J) \end{gathered}$ induces on $H^n$ the composite of $\operatorname{Hom}_{K(B)}(P\otimes_AB,I[n])=$ $\operatorname{Hom}_{D(B)}(E\otimes^{\mathbf L}_AB,F[n])$ with the adjunction bijection, by the description of the unit, and the cohomology of its target is $\operatorname{Hom}_{D(A)}(E,F[n])$ (Proposition 2.3). So it is a quasi-isomorphism, and it is natural in $E$ and $F$.

(2) By Proposition 1.2, $E\cong\tau^{\leq q}E$. With $P=Q(\tau^{\leq q}E)$ from Lemma 2.1 (1), $P\otimes_AB$ lies in $C^{\leq q}(B)$.

(3) Extension of scalars is an equivalence [Stacks, Tag 0FTZ], so its right adjoint is a quasi-inverse, and the counit $E\otimes^{\mathbf L}_AB\to E$ is an isomorphism for $E\in D(B)$. The restriction map $\operatorname{RHom}_B(E,F)\to\operatorname{RHom}_A(E,F)$ is the composite of the isomorphism induced by the counit with the isomorphism of (1): on the complexes of (1), with $P\to E$ a good resolution over $A$, both send $\psi\in\operatorname{Hom}^\bullet_B(E,I)$ to the composite of $P\to E$, $\psi$ and $I\to J$. $\square$

Proposition 2.6 (functoriality in the ring). Let $\mathbf{Andg}(T)$ be the category of differential graded rings of $T$ in nonpositive degrees, and $D(\mathbf{Andg}(T))$ its localization at the quasi-isomorphisms. To every morphism $u\colon A\to B$ of $D(\mathbf{Andg}(T))$ are attached functors $u_!\colon D(A)\to D(B)$ and $u^*\colon D(B)\to D(A)$, each defined up to isomorphism, such that $u_!$ is left adjoint to $u^*$; $(vu)_!\cong v_!u_!$ and $(vu)^*\cong u^*v^*$; for a map $\varphi$ of rings, $\varphi_!$ and $\varphi^*$ are extension and restriction of scalars; and $u_!$, $u^*$ preserve $D^{\leq q}$ and $D^-$. In particular $\operatorname{Hom}_{D(B)}(u_!E,F)\cong\operatorname{Hom}_{D(A)}(E,u^*F)$ for $E\in D^-(A)$ and $F\in D^-(B)$.

Proof. Let $\mathcal K$ be the category whose objects are categories and whose morphisms are isomorphism classes of functors. Restriction of scalars is strictly functorial, and $(\psi\varphi)_!\cong\psi_!\varphi_!$ because both are left adjoint to $\varphi^*\psi^*$. So $A\mapsto D(A)$ gives functors from $\mathbf{Andg}(T)$ to $\mathcal K$, through $\varphi\mapsto[\varphi_!]$, and from its opposite, through $\varphi\mapsto[\varphi^*]$. By Proposition 2.5 (3) they carry quasi-isomorphisms to isomorphisms of $\mathcal K$, so they factor uniquely through the localization. This defines $[u_!]$ and $[u^*]$ and the composition isomorphisms. Every morphism of the localization is a composite of morphisms $\varphi$ and inverses $w^{-1}$ of quasi-isomorphisms $w$. For $\varphi$, $\varphi_!$ is left adjoint to $\varphi^*$ by Proposition 2.5 (1). For $w^{-1}$, $(w^{-1})_!\cong w^*$ and $(w^{-1})^*\cong w_!$, and $w^*$ is left adjoint to $w_!$ because they are inverse equivalences. Composites of adjunctions are adjunctions, and replacing functors by isomorphic ones preserves adjunctions. The functors $\varphi_!$ preserve $D^{\leq q}$ by Proposition 2.5 (2), and the $\varphi^*$ preserve all bounds on cohomology, so all $u_!$ and $u^*$ preserve $D^{\leq q}$ and $D^-$. $\square$

Proposition 2.7 (modules over an algebra). Let $A$ be graded commutative, and $\varphi\colon A\to B$ a map of differential graded rings with $\varphi(a)b=(-1)^{ij}b\varphi(a)$ for $a\in A^i$ and $b\in B^j$. For a $B$-module $E$ and an $A$-module $F$, $E\otimes_AF$ is a $B$-module and $\mathcal Hom^\bullet_A(E,F)$ a left $B$-module, through the action of $B$ on $E$. There are functors $\begin{gathered} \otimes^{\mathbf L}_A\colon D(B)\times D(A)\to D(B),\\ R\mathcal Hom_A\colon D(B)^{\mathrm o}\times D(A)\to D(B^{\mathrm{op}}), \end{gathered}$ computed by $E\otimes_AQ$ for a quasi-isomorphism $Q\to F$ from a K-flat $A$-module, and by $\mathcal Hom^\bullet_A(E,I)$ for a quasi-isomorphism $F\to I$ into a K-injective $A$-module. Composed with restriction of scalars to $A$, they are the functors of Propositions 2.4 and 2.3 applied to the restriction of $E$. If $A$ and $B$ are in nonpositive degrees, $\otimes^{\mathbf L}_A$ maps $D^{\leq0}(B)\times D^{\leq0}(A)$ into $D^{\leq0}(B)$, and $R\mathcal Hom_A$ maps $D^{\leq0}(B)^{\mathrm o}\times D^{\geq0}(A)$ into $D^{\geq0}(B^{\mathrm{op}})$.

Proof. The condition on $\varphi$ makes $E$ an $(A,B)$-bimodule, with $A$ acting on the left by $ae=(-1)^{ij}e\varphi(a)$. So $F\mapsto F\otimes^{\mathbf L}_AE$ is the derived tensor product of the Stacks project [Stacks, Tag 0FTF], with values in $D(B)$, and $F\otimes_AE\cong E\otimes_AF$ by the sign rule. As in Proposition 2.4, it is functorial in $E$ and carries quasi-isomorphisms of $E$ to isomorphisms, and it is computed by $E\otimes_AQ$ by Lemma 2.2. For $R\mathcal Hom_A$, Proposition 2.3 shows that $\mathcal Hom^\bullet_A(E,I)$ carries quasi-isomorphisms in $E$ to quasi-isomorphisms and does not depend on $I$; the action of $B$ commutes with all the maps used. The underlying complexes are those computing the functors of Propositions 2.4 and 2.3 for the restriction of $E$, and the bounds follow from those propositions, because they concern the underlying complexes. $\square$

3. Normalized complexes of simplicial modules

Let $R$ be a simplicial ring of $T$, commutative as in Simplicial rings and modules in a topos, and place $N_nR$ in degree $-n$. By Proposition 2.1 of that lesson, $A=NR$ is a graded commutative differential graded ring in nonpositive degrees, and for an $R$-module $E$ the shuffle map followed by the action makes $NE$ a left $A$-module, hence an $A$-module by the sign rule. Normalization is exact and preserves quasi-isomorphisms, so it induces a functor $D_\bullet(R)\to D^{\leq0}(A)$.

For a bisimplicial abelian sheaf $M$, the shuffle map $\operatorname{sh}_M\colon\operatorname{Tot}NM\to N\operatorname{diag}M$ of Lemma 2.6 of Multisimplicial objects and spectral diagrams is given on $x\in M_{p,q}$ by $\operatorname{sh}_M(x)=\sum_{(\mu,\nu)}(-1)^{\epsilon(\mu,\nu)}s^{(1)}_\nu s^{(2)}_\mu x.$ The sum runs over the partitions of $\{0,\ldots,p+q-1\}$ into increasing lists $\mu$ of length $p$ and $\nu$ of length $q$, $\epsilon(\mu,\nu)$ is the sign of the shuffle, $s^{(1)}$ and $s^{(2)}$ are the degeneracies in the first and the second index, and $s_\nu=s_{\nu_q}\cdots s_{\nu_1}$. For $M_{p,q}=V_p\otimes W_q$ it is the shuffle map $NV\otimes NW\to N(V\otimes W)$ of Theorem 1.3 of The cotangent complex. We write $\nabla$ for the shuffle maps of pairs of simplicial objects. The total complex uses the sign convention of the lesson on multisimplicial objects, so a bisimplicial $R$-module $Y$ whose $R$-module structure is in the first index makes $\operatorname{Tot}NY$ a left $A$-module, through the shuffle maps in the first index followed by the action; these shuffle maps commute with the faces and degeneracies in the second index.

Lemma 3.1 (shuffles and diagonals). Let $M$ be a bisimplicial abelian sheaf, $L$ a simplicial abelian sheaf, and $L'$ the bisimplicial sheaf $(p,q)\mapsto L_p$, constant in the second index. For $y\in L_k$ and $x\in M_{p,q}$, let $\nabla_1(y\otimes x)\in(L'\otimes M)_{k+p,q}$ be the shuffle map of $L$ and the simplicial sheaf $M_{\bullet,q}$. Then $\operatorname{sh}_{L'\otimes M}(\nabla_1(y\otimes x))=\nabla(y\otimes\operatorname{sh}_M(x))$ in $(L\otimes\operatorname{diag}M)_{k+p+q}=$ $\operatorname{diag}(L'\otimes M)_{k+p+q}$.

Proof. For a subset $J=\{j_1<\cdots<j_m\}$ of $\{0,\ldots,n-1\}$ write $s_J=s_{j_m}\cdots s_{j_1}$, which repeats the entries at the positions in $J$. If $J$ has $m$ elements and $J'\subseteq\{0,\ldots,n-m-1\}$, then $s_Js_{J'}=s_{J\cup\iota(J')}$, where $\iota$ is the increasing bijection from $\{0,\ldots,n-m-1\}$ onto the complement of $J$. Let $n=k+p+q$. By this rule both sides are sums over the decompositions of $\{0,\ldots,n-1\}$ into sets $C$, $A$, $B$ of sizes $k$, $p$, $q$, the positions of the directions of $y$ and of the first and second directions of $x$. On the left, $\nabla_1$ places $C$ and $A$ among the first $k+p$ directions and $\operatorname{sh}$ places $B$; on the right, $\operatorname{sh}_M$ places $A$ and $B$ among $p+q$ directions and $\nabla$ places $C$. In both cases the term of $(C,A,B)$ is $\pm(s_{A\cup B}\,y)\otimes(s^{(1)}_{C\cup B}s^{(2)}_{C\cup A}\,x),$ because the degeneracies of $L'$ in the second index are identities. Its sign is the product of the signs of the two shuffles, which in both cases is the sign of the permutation listing the elements of $C$, of $A$ and of $B$, in this order and each in increasing order. $\square$

Lemma 3.2 (induced modules). Let $V$ be a simplicial abelian sheaf with flat terms and $F=R\otimes_{\mathbb Z}V$. The shuffle map $NR\otimes_{\mathbb Z}NV\to NF$ is an $A$-linear quasi-isomorphism from a K-flat $A$-module. For every $R$-module $E$ and every $A$-linear quasi-isomorphism $G\to NF$ from a K-flat module $G$, the composite $\begin{aligned}&NE\otimes_AG \to NE\otimes_ANF\\&\quad \to N(E\otimes_RF),\end{aligned}$ whose second map is the shuffle map, is a quasi-isomorphism.

Proof. The shuffle map is $A$-linear by associativity of shuffle maps, and a quasi-isomorphism by the Eilenberg–Zilber theorem (Theorem 1.3 of The cotangent complex, whose homotopies are natural and pass to sheaves). The terms of $NV$ are direct summands of those of $V$ (Theorem 1.1 there), so $NV$ is a bounded above complex of flat sheaves and is K-flat [Stacks, Tag 06YQ]. For a right module $N'$, $N'\otimes_A(NR\otimes_{\mathbb Z}NV)=N'\otimes_{\mathbb Z}NV$, so $G_0=NR\otimes_{\mathbb Z}NV$ is K-flat. Under the identifications $NE\otimes_AG_0=NE\otimes_{\mathbb Z}NV$ and $E\otimes_RF=E\otimes_{\mathbb Z}V$, the composite for $G_0$ is the shuffle map $NE\otimes_{\mathbb Z}NV\to N(E\otimes_{\mathbb Z}V)$, by associativity of shuffle maps; it is a quasi-isomorphism by the Eilenberg–Zilber theorem.

Let $c\colon NE\otimes_AG\to N(E\otimes_RF)$ be the composite for $G$, $c_0$ the one for $G_0$, and $h\colon NE\otimes_ANF\to N(E\otimes_RF)$ the shuffle map. Choose a quasi-isomorphism $P\to NE$ from a K-flat module, and let $m\colon P\otimes_ANF\to NE\otimes_ANF$. In $D(\mathbb Z)$, $cv=hmt$ and $c_0v_0=hmt_0$, where $v\colon P\otimes_AG\to NE\otimes_AG$, $t\colon P\otimes_AG\to P\otimes_ANF$ and $v_0$, $t_0$ are the corresponding maps for $G_0$; these four maps are quasi-isomorphisms by Lemma 2.2. Since $c_0$ is a quasi-isomorphism, $hm$ is an isomorphism in $D(\mathbb Z)$, and so is $c=hmtv^{-1}$. $\square$

Theorem 3.3. Let $E$ and $F$ be $R$-modules, and $P(F)\to F$ the flat replacement of Derived tensor products of simplicial modules. The composite $\begin{aligned} \Phi\colon NE\otimes^{\mathbf L}_ANF&\cong NE\otimes^{\mathbf L}_ANP(F)\\ &\to NE\otimes_ANP(F)\\ &\to N(E\otimes_RP(F))\\ &\cong N(E\otimes^\ell_RF) \end{aligned}$ of the isomorphism induced by $P(F)\to F$, the canonical map from the derived to the ordinary tensor product, the shuffle map (Proposition 1.1 of Spectral sequences for derived tensor products) and the isomorphism of Proposition 3.2 (1) of the lesson on derived tensor products, is an isomorphism in $D(A)$, natural in $E$ and $F$. For a map $R\to R'$ of simplicial rings, the map $NR'\otimes^{\mathbf L}_ANF\to N(R'\otimes^\ell_RF)$ obtained in the same way is an isomorphism in $D(NR')$.

Proof. Let $Y=F_R(F)$ be the standard free resolution (Construction 1.2 of the lesson on derived tensor products): a bisimplicial $R$-module, with the $R$-module structure in the first index, whose diagonal is $P(F)$. For each $q$, $Y_{\bullet,q}=R^{(X_q)}=R\otimes_{\mathbb Z}\mathbb Z^{(X_q)}$ for a simplicial sheaf of sets $X_q$, so Lemma 3.2 applies to it. The maps in the second index are $R$-linear, so $q\mapsto NY_{\bullet,q}$ is a simplicial object of $C^{\leq0}(A)$. Let $Q$ be the functor of Lemma 2.1 (1) on $C^{\leq0}(A)$. Let $G$ be the total complex of the double complex obtained by normalizing the simplicial object $q\mapsto Q(NY_{\bullet,q})$, with $q$ in horizontal degree $-q$. Normalization in the second index is the image of a natural idempotent made of faces and degeneracies (Theorem 1.1 of The cotangent complex), so it commutes with additive functors, and the same construction applied to $q\mapsto NY_{\bullet,q}$ gives $\operatorname{Tot}NY$.

$G$ is K-flat. For an acyclic right module $N'$, $N'\otimes_AG$ is the total complex of a double complex whose columns are direct summands of the acyclic complexes $N'\otimes_AQ(NY_{\bullet,q})$. It is the union of the total complexes of its first $s$ columns, which are acyclic, and filtered colimits of acyclic complexes of sheaves are acyclic. The same argument, applied to the cones of the column maps, shows that $G\to\operatorname{Tot}NY$ is a quasi-isomorphism. The map $\operatorname{sh}_Y\colon\operatorname{Tot}NY\to N\operatorname{diag}Y=NP(F)$ is a quasi-isomorphism by Lemma 2.6 of the lesson on multisimplicial objects. It is $A$-linear: by naturality of $\operatorname{sh}$ for the action $R'\otimes Y\to Y$, where $R'$ is $R$ made constant in the second index, this follows from Lemma 3.1 with $L=R$. So the composite $g\colon G\to\operatorname{Tot}NY\to NP(F)$ is an $A$-linear quasi-isomorphism from a K-flat module, and by Lemma 2.2, $\Phi$ is represented by the composite of $1\otimes g\colon NE\otimes_AG\to NE\otimes_ANP(F)$ with the shuffle map to $N(E\otimes_RP(F))$.

Let $cE\otimes_RY$ be the bisimplicial module $(p,q)\mapsto E_p\otimes_{R_p}Y_{p,q}$, whose diagonal is $E\otimes_RP(F)$. The module $NE\otimes_AG$ is the total complex of the double complex with columns the normalized parts of the $NE\otimes_AQ(NY_{\bullet,q})$. The maps $\begin{gathered} NE\otimes_AQ(NY_{\bullet,q})\to NE\otimes_ANY_{\bullet,q}\\ \to N(E\otimes_RY_{\bullet,q}), \end{gathered}$ the second being the shuffle map, have quasi-isomorphisms as composites by Lemma 3.2, and they are compatible with the maps in the second index. By the argument used for $G$, the composites induce a quasi-isomorphism $NE\otimes_AG\to\operatorname{Tot}N(cE\otimes_RY)$. Composed with the quasi-isomorphism $\operatorname{sh}_{cE\otimes_RY}$ to $N(E\otimes_RP(F))$, it agrees with the map representing $\Phi$, by Lemma 3.1 with $L=E$ and $M=Y$ followed by the projections from tensor products over $\mathbb Z$ to tensor products over $R$. So $\Phi$ is an isomorphism. All the maps are natural in $E$ and $F$ and $A$-linear.

For the second statement take $E=R'$. All the maps above are $NR'$-linear for the action on the first factor, by associativity of shuffle maps, and $NR'\otimes_AG$, isomorphic to $G\otimes_ANR'$ by the sign rule, represents the extension of scalars of $NF$ to $NR'$ by Proposition 2.5. $\square$

4. Exercises

4.1. Let $A$ be graded commutative and $E,F,G\in D(A)$. Show that there are isomorphisms, natural in $E$, $F$ and $G$, $\begin{gathered} R\mathcal Hom_A(E\otimes^{\mathbf L}_AF,G)\\ \cong R\mathcal Hom_A(E,R\mathcal Hom_A(F,G)),\\[4pt] \operatorname{RHom}_A(E\otimes^{\mathbf L}_AF,G)\\ \cong\operatorname{RHom}_A(E,R\mathcal Hom_A(F,G)),\\[4pt] \operatorname{Hom}_{D(A)}(E\otimes^{\mathbf L}_AF,G)\\ \cong\operatorname{Hom}_{D(A)}(E,R\mathcal Hom_A(F,G)). \end{gathered}$

4.2. Let $A$ be in nonpositive degrees and $a\in\mathbb Z$. Regard a module $V$ over the sheaf of rings $H^0(A)$ as an $A$-module through $A\to H^0(A)$. Show that $V\mapsto V[-a]$ is an equivalence from the category of $H^0(A)$-modules to the full subcategory of $D(A)$ of the objects with cohomology only in degree $a$, with quasi-inverse $H^a$.

4.3. Let $R$ be the simplicial ring of the punctual topos whose ring $R_n$ is the polynomial ring over $\mathbb Z$ in the $n$-simplices of $\Delta[1]$, and let $E=F=R$. Show that the Alexander–Whitney map $N(E\otimes_{\mathbb Z}F)\to NE\otimes_{\mathbb Z}NF$, followed by the projection to $NE\otimes_{NR}NF$, does not factor through $N(E\otimes_RF)$. So the comparison map of Theorem 3.3 cannot be defined with the Alexander–Whitney map in place of the shuffle map.

4.4. Let $f\colon R\to R'$ be a quasi-isomorphism of simplicial rings. Show that $NR\to NR'$ is a quasi-isomorphism of differential graded rings, that extension of scalars $D(NR)\to D(NR')$ is an equivalence, and that $N(R'\otimes^\ell_RE)\cong NR'\otimes^{\mathbf L}_{NR}NE$ in $D(NR')$ for every $R$-module $E$.

5. Complete solutions

4.1. Choose quasi-isomorphisms $P\to E$ and $Q\to F$ from K-flat modules and $G\to I$ into a K-injective module. The module $P\otimes_AQ$ is K-flat, since $(P\otimes_AQ)\otimes_AN=P\otimes_A(Q\otimes_AN)$, and it computes $E\otimes^{\mathbf L}_AF$ by Lemma 2.2. The module $\mathcal Hom^\bullet_A(Q,I)$ computes $R\mathcal Hom_A(F,G)$ (Proposition 2.3), and it is K-injective: for an acyclic module $N$, $\operatorname{Hom}^\bullet_A(N,\mathcal Hom^\bullet_A(Q,I))\cong$ $\operatorname{Hom}^\bullet_A(N\otimes_AQ,I)$ is acyclic, because $N\otimes_AQ$ is acyclic. The tensor–Hom isomorphism of complexes of sheaves $\begin{gathered} \mathcal Hom^\bullet_A(P\otimes_AQ,I)\\ \cong\mathcal Hom^\bullet_A(P,\mathcal Hom^\bullet_A(Q,I)), \end{gathered}$ $\varphi\mapsto(x\mapsto(y\mapsto\varphi(x\otimes y)))$ with the usual signs, therefore gives the first isomorphism. Its global sections give the second, and $H^0$ of the second gives the third, by Proposition 2.3 (2).

4.2. Since $A^1=0$, $t_0(A)=A^0/dA^{-1}=H^0(A)$. By Proposition 1.3 with $n=0$ and $b=a$, restriction of scalars gives an equivalence $D^{[a,a]}(H^0(A))\to D^{[a,a]}(A)$, and by Proposition 1.2 the target is the full subcategory of objects of $D(A)$ with cohomology only in degree $a$. A module of $C^{[a,a]}(H^0(A))$ is an $H^0(A)$-module placed in degree $a$, with zero differential. Homotopies between maps of such modules vanish for degree reasons, and quasi-isomorphisms between them are isomorphisms, so $D^{[a,a]}(H^0(A))$ is the category of $H^0(A)$-modules. Under these identifications, $V\mapsto V[-a]$ corresponds to the identity, and $H^a$ is a quasi-inverse.

4.3. Let $t\in R_1$ be the variable of the nondegenerate $1$-simplex, $t_0=d_1t$ and $t_1=d_0t$ the variables of the vertices, and $t_{00}=s_0t_0$, $t_{11}=s_0t_1$. The element $x=t^2\otimes1-t\otimes t$ of $(E\otimes_{\mathbb Z}F)_1$ maps to $0$ in $E\otimes_RF=R$. In degree $1$ the Alexander–Whitney map is $v\otimes w\mapsto d_1v\otimes w+v\otimes d_0w$. Since $1\in R_1$ is degenerate, it vanishes in $N_1R$, and the image of $x$ is $t^2\otimes1-t_0\otimes t-t\otimes t_1$. Multiplication $NR\otimes_{NR}NR\to NR$ is an isomorphism, and the product of $u\in N_0R$ with $w\in N_1R$ is $s_0(u)w$. So the image of $x$ in $NE\otimes_{NR}NF\cong NR$ is the class of $t^2-t_{00}t-t\,t_{11}$ in $N_1R=R_1/s_0(R_0)$. It is not zero, because $s_0(R_0)=\mathbb Z[t_{00},t_{11}]$ does not contain this polynomial.

4.4. Normalization preserves quasi-isomorphisms, and $Nf$ is a map of differential graded rings by naturality in Proposition 2.1 of the lesson on simplicial rings and modules. Extension of scalars along a quasi-isomorphism is an equivalence by Proposition 2.5 (3). The isomorphism $N(R'\otimes^\ell_RE)\cong NR'\otimes^{\mathbf L}_{NR}NE$ is the second statement of Theorem 3.3.

Sources

Differential graded modules over a differential graded ring of a topos, their bounded derived categories when the ring is concentrated in nonpositive degrees, the truncated rings, the bounds for derived Hom and derived tensor products, the functoriality of extension of scalars in the ring and the comparison with simplicial modules through shuffle maps are due to Illusie, in his work on the cotangent complex. The existence of derived functors for all differential graded modules over a sheaf of differential graded algebras, through good and K-injective resolutions, is taken from the Stacks project. The shuffle maps and the Eilenberg–Zilber theorem go back to Eilenberg, Mac Lane and Zilber.