Graded simplicial rings and the graded transitivity triangle

Written by Claude Opus 5.5 (Anthropic) on 23 October 2026. Self-checked by the writing AI. Public domain (CC0).

A graded simplicial ring is a simplicial object of the category of graded rings of a topos. The cotangent complex of a map of graded simplicial rings is formed, as for simplicial rings, from the graded standard resolutions of its terms, by taking the diagonal. This lesson shows that the result has the properties of its two special cases, maps of graded rings and maps of simplicial rings. Forgetting the grading gives the cotangent complex of the underlying simplicial rings, so it is invariant under quasi-isomorphisms. The graded cotangent complex corepresents graded derivations in the homotopy category of graded simplicial algebras. Maps $A\to B\to C$ give a graded transitivity triangle, whose part of degree $0$ is the transitivity triangle of the parts of degree $0$. For a map of rings $A\to B$ and a simplicial $B$-module $M$, the part of degree one of the transitivity triangle of $A\to B\to B\oplus M$ is the triangle of principal parts of $M$, so the Atiyah class of a simplicial module has a graded description.

Prerequisites. Simplicial rings and modules in a topos (Lemma 1.2, Section 3, Lemma 4.1, Theorem 4.2); Triangles and Ext groups of simplicial modules (Lemma 3.2); Derived tensor products of simplicial modules (Lemma 1.1, Lemma 2.1, Propositions 2.2 and 3.2); Graded modules and the graded cotangent complex (Definition 1.1, Lemma 1.2, Construction 2.1, Proposition 2.2, Lemma 2.4, Lemma 3.1, Definition 3.2, Theorems 3.3 and 3.4); The Atiyah class of a module (Lemmas 1.2 and 1.3, Construction 2.1, Definition 2.2, Theorem 2.4 and its proof); Cotangent complexes of simplicial rings (Proposition 1.3, Construction 2.1, Definition 2.3, Theorem 3.1, Corollary 3.4, Propositions 4.1 and 4.2).

Conventions

$T$ is a topos. Graded rings, graded modules and graded algebras, the functors $\Phi$ (forgetting the decomposition), $k^n$ (the component of degree $n$) and $U_{\mathrm{gr}}$, the categories $T^{\mathrm{gr}\geq0}$ and $A\text{-}\mathrm{Alggr}^+$, free graded algebras $A[X]$, graded differentials and graded standard resolutions $P_A(B)^{\mathrm{gr}}$ are those of the lesson on graded modules; graded rings sit in degrees $\geq0$. For simplicial rings, quasi-isomorphisms, $D_\bullet$, homotopies, fibrations and the cotangent complex $L^\Delta$ are those of the lessons on simplicial rings and on cotangent complexes of simplicial rings. Bisimplicial objects have an external index $n$ and an internal index $m$; their columns are obtained by fixing $n$, and the diagonal of a bisimplicial object $Z$ is $\Delta Z$, with $(\Delta Z)_n=Z_{n,n}$. A map of bisimplicial abelian sheaves that is a quasi-isomorphism on every column induces a quasi-isomorphism of diagonals (Lemma 1.1 of the lesson on derived tensor products).

1. Graded simplicial rings

Definition 1.1. A graded simplicial ring is a simplicial object of the category of graded rings of $T$. A graded module over a graded simplicial ring $A$ is a simplicial $A$-module $M$ with a decomposition $M=\bigoplus_nM^n$ into simplicial abelian subsheaves such that every $M_q$ is a graded $A_q$-module; graded algebras are defined in the same way. A map of graded modules is a quasi-isomorphism if the underlying map of simplicial modules is one, that is, if every component $k^n$ of it is one; a map of graded algebras, or of graded algebras over a graded algebra $B$, is a quasi-isomorphism if the underlying map of modules is one. Write $D_\bullet(A)_{\mathrm{gr}}$, $D(A\text{-}\mathrm{Alggr})$ and $D(A\text{-}\mathrm{Alggr}/B)$ for the localizations of the categories of graded $A$-modules, graded $A$-algebras and graded $A$-algebras over $B$ at the quasi-isomorphisms. A map of graded modules or algebras is a fibration if the underlying map of simplicial modules is one (Definition 3.1 of the lesson on simplicial rings).

Proposition 1.2 (homotopy theory). Let $A$ be a graded simplicial ring and $B$ a graded $A$-algebra. Lemma 1.2, Section 3, Lemma 4.1 and Theorem 4.2 of the lesson on simplicial rings hold for graded $A$-modules, graded $A$-algebras and graded $A$-algebras over $B$, with homotopies, paths and cylinders formed as there. In particular the quasi-isomorphisms of graded $A$-modules up to homotopy admit calculi of left and right fractions, and the localization of the graded algebras over $B$ whose structure map is a fibration, up to homotopy over $B$, is equivalent to $D(A\text{-}\mathrm{Alggr}/B)$, the quasi-isomorphisms admitting a calculus of right fractions there. A short exact sequence of graded $A$-modules gives a distinguished triangle of $D_\bullet(A)_{\mathrm{gr}}$, as in Lemma 3.2 of the lesson on triangles, whose image under each $k^n$ is the triangle of the component of degree $n$.

Proof. The constructions of those statements are formed from $\Delta[k]\odot-$, from $\underline{\operatorname{Homs}}(K,-)$ for simplicial sets $K$ with finitely many nondegenerate simplices, from finite limits and colimits, and from normalization; the proofs use these constructions, the long exact sequences of homotopy, and conditions on the underlying modules. On graded modules each of these constructions is formed component by component. For $\underline{\operatorname{Homs}}(K,-)$ this holds because a map of simplicial sets from $K\times\Delta[q]$ to $\bigoplus_nF^n$ has, locally, only finitely many nonzero components, $K\times\Delta[q]$ having finitely many nondegenerate simplices; so $\underline{\operatorname{Homs}}(K,F)=\bigoplus_n\underline{\operatorname{Homs}}(K,F^n)$. For a graded algebra $F$, the pointwise product of $\underline{\operatorname{Homs}}(K,F)$ carries the components of degrees $i$ and $j$ to that of degree $i+j$, so path objects of graded algebras are graded algebras, and the maps of the constructions (evaluations, constant paths, contractions induced by maps of simplicial sets) are graded. Fibre products and pushouts of graded objects are graded, and fibrations, quasi-isomorphisms, monomorphisms and epimorphisms are conditions on the underlying simplicial modules, which hold for a graded map exactly when they hold for each component. So every construction of the proofs stays within graded objects, and every condition checked there is checked on the underlying modules. The proofs therefore apply as written. The last statement holds because the triangle of Lemma 3.2 of the lesson on triangles is formed from the cone of the inclusion, which is graded, componentwise. $\square$

2. The diagonal graded cotangent complex

Construction 2.1. Let $A\to B$ be a map of graded simplicial rings. For every $n$, $P_{A_n}(B_n)^{\mathrm{gr}}$ is a simplicial graded $A_n$-algebra augmented to $B_n$ (Construction 2.1 of the lesson on graded modules), functorial in commutative squares of graded rings. A map $\theta\colon[n']\to[n]$ of $\Delta$ gives a commutative square from $A_n\to B_n$ to $A_{n'}\to B_{n'}$, hence a map $P_{A_n}(B_n)^{\mathrm{gr}}\to P_{A_{n'}}(B_{n'})^{\mathrm{gr}}$. So $(n,m)\mapsto P_{A_n}(B_n)^{\mathrm{gr}}_m$ is a bisimplicial graded ring $P_A(B)^{\mathrm{gr}}$, an $A$-algebra augmented to $B$, $A$ and $B$ being constant in the internal direction. Its diagonal is the graded simplicial $A$-algebra $P^{\Delta}_A(B)^{\mathrm{gr}}$, augmented to $B$.

Lemma 2.2. $P^{\Delta}_A(B)^{\mathrm{gr}}$ is degreewise free graded over $A$: its term of degree $n$ is $A_n[X_n]$, with $X_n=U_{\mathrm{gr}}(P_{A_n}(B_n)^{\mathrm{gr}}_{n-1})\in T^{\mathrm{gr}\geq0}$ and $P_{A_n}(B_n)^{\mathrm{gr}}_{-1}=B_n$. The augmentation $P^{\Delta}_A(B)^{\mathrm{gr}}\to B$ is an epimorphism in each degree and a quasi-isomorphism.

Proof. The first statement is Proposition 2.2 (1) of the lesson on graded modules in each degree. The augmentation is the diagonal of the map of bisimplicial graded rings $P_A(B)^{\mathrm{gr}}\to B$, whose column $n$ is the augmentation of $P_{A_n}(B_n)^{\mathrm{gr}}$, surjective in each degree and a quasi-isomorphism (Proposition 2.2 (2) there). Apply the Conventions to each component $k^d$. $\square$

Definition 2.3. The diagonal graded cotangent complex of $A\to B$ is the graded simplicial $B$-module $\begin{aligned}L^{\Delta\mathrm{gr}}_{B/A}&=\Omega_{P^\Delta/A}\otimes_{P^\Delta}B,\\ P^\Delta&=P^{\Delta}_A(B)^{\mathrm{gr}},\end{aligned}$ the diagonal of the bisimplicial graded $B$-module with the columns $L^{\mathrm{gr}}_{B_n/A_n}$ (Definition 3.2 of the lesson on graded modules). Its term of degree $n$ is the free graded $B_n$-module $B_n^{(X_n)}$, so it is degreewise free, hence flat. It is functorial in commutative squares of graded simplicial rings, through the maps of Construction 2.1. If $A$ and $B$ are constant, it is $L^{\mathrm{gr}}_{B/A}$.

Theorem 2.4. Let $A\to B$ be a map of graded simplicial rings.

The maps $\theta_{B_n/A_n}$ of Lemma 2.4 (1) of the lesson on graded modules induce a quasi-isomorphism $\Phi L^{\Delta\mathrm{gr}}_{B/A}\to L^\Delta_{\Phi B/\Phi A}$, natural in $A\to B$.
For a commutative square of graded simplicial rings from $A\to B$ to $A'\to B'$ whose vertical maps are quasi-isomorphisms, the maps $L^{\Delta\mathrm{gr}}_{B/A}\to L^{\Delta\mathrm{gr}}_{B'/A'}$ and $L^{\Delta\mathrm{gr}}_{B/A}\otimes_BB'\to L^{\Delta\mathrm{gr}}_{B'/A'}$ are quasi-isomorphisms.
$k^0(L^{\Delta\mathrm{gr}}_{B/A})=L^\Delta_{B^0/A^0}$, compatibly with the augmentations.
Suppose that $A$ is concentrated in degree $0$ and that $B^0=A$, that is, every $B_n$ lies in $A_n\text{-}\mathrm{Alggr}^+$. Then $k^0(L^{\Delta\mathrm{gr}}_{B/A})$ is acyclic, and there is an isomorphism $k^1(L^{\Delta\mathrm{gr}}_{B/A})\cong B^1$ in $D_\bullet(A)$, natural in $B$.

Proof. (1) The maps $\theta_{B_n/A_n}\colon\Phi P_{A_n}(B_n)^{\mathrm{gr}}\to P_{\Phi A_n}(\Phi B_n)$ are natural in the maps of graded rings $A_n\to B_n$, so they form a map of bisimplicial $\Phi A$-algebras from $\Phi P_A(B)^{\mathrm{gr}}$ to the bisimplicial standard resolution of $\Phi A\to\Phi B$ (Construction 2.1 of the lesson on cotangent complexes of simplicial rings), compatible with the augmentations. The induced map of bisimplicial modules has the columns $\Phi L^{\mathrm{gr}}_{B_n/A_n}\to L_{\Phi B_n/\Phi A_n}$, which are quasi-isomorphisms by Theorem 3.3 (1) of the lesson on graded modules. Its diagonal is the map of the statement, a quasi-isomorphism by the Conventions. Naturality follows from that of the $\theta$.

(2) A map of graded modules is a quasi-isomorphism if its image under $\Phi$ is one. By the naturality in (1), $\Phi$ carries the first map to the map $L^\Delta_{\Phi B/\Phi A}\to L^\Delta_{\Phi B'/\Phi A'}$ up to the quasi-isomorphisms of (1), and that map is a quasi-isomorphism by Theorem 3.1 (2) of the lesson on cotangent complexes of simplicial rings. For the second map, $\Phi(L^{\Delta\mathrm{gr}}_{B/A}\otimes_BB')=\Phi L^{\Delta\mathrm{gr}}_{B/A}\otimes_{\Phi B}\Phi B'$, and the map $\Phi L^{\Delta\mathrm{gr}}_{B/A}\otimes_{\Phi B}\Phi B'\to L^\Delta_{\Phi B/\Phi A}\otimes_{\Phi B}\Phi B'$ induced by (1) is a quasi-isomorphism, both modules before the tensor product having flat terms (Proposition 2.2 of the lesson on derived tensor products). Theorem 3.1 (2) there gives the rest by two out of three.

(3) Column by column this is Theorem 3.3 (4) of the lesson on graded modules, which is natural in the maps of graded rings, and the diagonal of the bisimplicial module with the columns $L_{B_n^0/A_n^0}$ is $L^\Delta_{B^0/A^0}$ (Definition 2.3 of the lesson on cotangent complexes of simplicial rings).

(4) Let $P^+_{A_n}(B_n)$ be the resolutions of Construction 2.1 of the lesson on graded modules. They form a bisimplicial graded $A$-algebra $P^+_A(B)$, and the maps $\theta^+_{B_n}$ of Lemma 2.4 (2) there form a map $P^+_A(B)\to P_A(B)^{\mathrm{gr}}$. By Theorem 3.3 (5) there, the induced maps of columns $\Omega_{P^+_{A_n}(B_n)/A_n}\otimes B_n\to L^{\mathrm{gr}}_{B_n/A_n}$ are quasi-isomorphisms, the sources have $k^0=0$, and their $k^1$ are the standard free resolutions $F_{A_n}(B^1_n)$, with the augmentations to $B_n^1$ quasi-isomorphisms (Proposition 2.2 (4) there). Taking diagonals, $k^0(L^{\Delta\mathrm{gr}}_{B/A})$ is quasi-isomorphic to $0$, and $k^1(L^{\Delta\mathrm{gr}}_{B/A})$ is related to $B^1$ by the two quasi-isomorphisms of simplicial $A$-modules $\begin{gathered}k^1L^{\Delta\mathrm{gr}}_{B/A}\longleftarrow\Delta F_A(B^1)\longrightarrow B^1,\end{gathered}$ natural in $B$. $\square$

Corollary 2.5 (rings in degree $0$). Let $A\to B$ be a map of graded simplicial rings concentrated in degree $0$. Then $k^0(L^{\Delta\mathrm{gr}}_{B/A})=L^\Delta_{B/A}$, and its inclusion into $L^{\Delta\mathrm{gr}}_{B/A}$ is a quasi-isomorphism of graded $B$-modules. So the components of $L^{\Delta\mathrm{gr}}_{B/A}$ of positive degree are acyclic. They need not vanish: the graded standard resolutions have variables of every positive degree, labelled by the zero elements of the components $B_n^d=0$, whose differentials give nonzero terms of positive degree.

Proof. By Theorem 2.4 (3), $k^0(L^{\Delta\mathrm{gr}}_{B/A})=L^\Delta_{B^0/A^0}=L^\Delta_{B/A}$. On the parts of degree $0$ the maps $\theta$ of Lemma 2.4 (1) of the lesson on graded modules are the identity, since they send the variable of an element of degree $0$ to the variable of the same element. So the composite of the inclusion $k^0(L^{\Delta\mathrm{gr}}_{B/A})\to\Phi L^{\Delta\mathrm{gr}}_{B/A}$ with the quasi-isomorphism of Theorem 2.4 (1) is the identity of $L^\Delta_{B/A}$, and the inclusion is a quasi-isomorphism by two out of three. Its cokernel, the sum of the components of positive degree, is therefore acyclic. $\square$

3. Graded derivations

Fix a graded simplicial ring $A$ and a graded $A$-algebra $B$. For a graded $B$-module $M$, let $B\oplus M$ be the graded trivial square-zero extension, with $k^n(B\oplus M)=B^n\oplus M^n$, a graded $A$-algebra over $B$ whose structure map is a fibration. A quasi-isomorphism $M\to M'$ induces a quasi-isomorphism $B\oplus M\to B\oplus M'$, so $M\mapsto B\oplus M$ induces a functor $D_\bullet(B)_{\mathrm{gr}}\to D(A\text{-}\mathrm{Alggr}/B)$.

Proposition 3.1. Let $\Phi^{\mathrm{gr}}(C)=\Omega_{C/A}\otimes_CB$ for graded $A$-algebras $C$ over $B$.

The functor $C\mapsto L^{\Delta\mathrm{gr}}_{C/A}\otimes_CB$ carries quasi-isomorphisms over $B$ to quasi-isomorphisms and homotopic maps over $B$ to homotopic maps, and so does $\Phi^{\mathrm{gr}}$ on homotopic maps.
For every graded $A$-algebra $C$ over $B$ whose structure map is a fibration, the pro-object $C'\mapsto\Phi^{\mathrm{gr}}(C')$ of $D_\bullet(B)_{\mathrm{gr}}$, indexed by the quasi-isomorphisms $C'\to C$ between such algebras up to homotopy over $B$, is essentially constant with value $L^{\Delta\mathrm{gr}}_{C/A}\otimes_CB$.

Proof. The proof of Proposition 4.1 of the lesson on cotangent complexes of simplicial rings applies, with the following graded inputs. In (1), Theorem 2.4 (2) replaces Theorem 3.1 (2) there, and the terms of $L^{\Delta\mathrm{gr}}_{C/A}$ are free. In (2), the cofinal subcategory consists of the quasi-isomorphisms $C'\to C$ with $C'$ degreewise free graded over $A$; for every quasi-isomorphism $C''\to C$, the composite $P^\Delta_A(C'')^{\mathrm{gr}}\to C''\to C$ lies in it, since $P^\Delta_A(C'')^{\mathrm{gr}}\to C''$ is an epimorphism and a quasi-isomorphism (Lemma 2.2), hence a fibration (Proposition 1.2). The calculus of right fractions of Proposition 1.2 makes the index category cofiltered. For a quasi-isomorphism $C''\to C'$ between degreewise free graded $A$-algebras, the map $\Omega_{C''/A}\otimes_{C''}C'\to\Omega_{C'/A}$ is a quasi-isomorphism: its image under $\Phi$ is the corresponding map for the degreewise free algebras $\Phi C''\to\Phi C'$ over $\Phi A$ (Lemma 1.2 (2) and Lemma 3.1 (1) of the lesson on graded modules), a quasi-isomorphism by Corollary 3.4 of the lesson on cotangent complexes of simplicial rings. Tensoring with $B$ over $C'$, the transition maps of the pro-object are quasi-isomorphisms, and its value at $P^\Delta_A(C)^{\mathrm{gr}}$ is $L^{\Delta\mathrm{gr}}_{C/A}\otimes_CB$. $\square$

Theorem 3.2 (graded derivations). For every graded $B$-module $M$ there is an isomorphism, natural in $M\in D_\bullet(B)_{\mathrm{gr}}$, $\begin{aligned}&\operatorname{Hom}_{D_\bullet(B)_{\mathrm{gr}}}(L^{\Delta\mathrm{gr}}_{B/A},M)\\&\quad\cong\operatorname{Hom}_{D(A\text{-}\mathrm{Alggr}/B)}(B,B\oplus M).\end{aligned}$ So $L^{\Delta\mathrm{gr}}_{B/A}$ corepresents graded derivations of $B$ over $A$ in the homotopy category of graded algebras over $B$, and this property characterizes it in $D_\bullet(B)_{\mathrm{gr}}$.

Proof. The proof of Proposition 4.2 of the lesson on cotangent complexes of simplicial rings applies. For strict maps, Lemma 3.1 (1) of the lesson on graded modules gives, degree by degree and compatibly with the simplicial maps, $\begin{aligned}&\operatorname{Hom}_{B\text{-}\mathrm{Modgr}}(\Phi^{\mathrm{gr}}(C),F)\\&\quad\cong\operatorname{Der}_A(C,F)_{\mathrm{gr}}\\&\quad\cong\operatorname{Hom}_{A\text{-}\mathrm{Alggr}/B}(C,B\oplus F)\end{aligned}$ for a graded $B$-module $F$. The path object $\operatorname{Path}(F)$ is a graded $B$-module (Proposition 1.2), and the path object of $B\oplus F$ over $B$ is $B\oplus\operatorname{Path}(F)$, so homotopies correspond. The localization step uses the calculi of fractions of Proposition 1.2 and Proposition 3.1 (2) for $C=B$. Every step is natural in $M$. An object corepresenting a functor on $D_\bullet(B)_{\mathrm{gr}}$ is unique up to unique isomorphism, by the Yoneda lemma. $\square$

4. The graded transitivity triangle

Theorem 4.1. Let $A\to B\to C$ be maps of graded simplicial rings. There is an exact sequence of graded simplicial $C$-modules $\begin{aligned}&0\to L^{\Delta\mathrm{gr}}_{B/A}\otimes_BC\\&\quad\to L^{\Delta\mathrm{gr}}_{C/A}\to L'\to0,\end{aligned}\tag{4.1}$ whose first two maps are the functoriality maps, together with quasi-isomorphisms of graded $C$-modules relating $L'$ to $L^{\Delta\mathrm{gr}}_{C/B}$, all natural in $A\to B\to C$. It gives a distinguished triangle of $D_\bullet(C)_{\mathrm{gr}}$ $\begin{aligned}&L^{\Delta\mathrm{gr}}_{B/A}\otimes^\ell_BC\to L^{\Delta\mathrm{gr}}_{C/A}\\&\quad\to L^{\Delta\mathrm{gr}}_{C/B}\to\sigma(L^{\Delta\mathrm{gr}}_{B/A}\otimes^\ell_BC),\end{aligned}$ natural in the diagram. Its image under $k^0$, identified by Theorem 2.4 (3), is the triangle of the same construction for $A^0\to B^0\to C^0$, and its image under $\Phi$, identified by Theorem 2.4 (1), is the triangle of the same construction for $\Phi A\to\Phi B\to\Phi C$. For graded simplicial rings concentrated in degree $0$ it is a transitivity triangle for maps of simplicial rings.

Proof. First let every $B_n\to C_n$ be a monomorphism. For each $n$, the proof of Theorem 3.4 of the lesson on graded modules gives, for $A_n\to B_n\to C_n$, the exact sequence (3.1) there, $\begin{aligned}&0\to\Omega_{P/A_n}\otimes_PC_n\to\Omega_{Q/A_n}\otimes_QC_n\\&\quad\to\Omega_{\overline Q/B_n}\otimes_{\overline Q}C_n\to0,\end{aligned}$ with $P=P_{A_n}(B_n)^{\mathrm{gr}}$, $Q=P_{A_n}(C_n)^{\mathrm{gr}}$, $\overline Q=Q\otimes_PB_n$, and the quasi-isomorphisms $\begin{aligned}&\Omega_{R/B_n}\otimes_RC_n\to\Omega_{\overline R/B_n}\otimes_{\overline R}C_n\\&\quad\leftarrow\Omega_{\overline Q/B_n}\otimes_{\overline Q}C_n,\end{aligned}$ with $R=P_{B_n}(C_n)^{\mathrm{gr}}$, $S=P_{B_n}(B_n)^{\mathrm{gr}}$ and $\overline R=R\otimes_SB_n$. All these are formed from standard resolutions, tensor products and differentials, functorially in the triple of graded rings. So they are the columns of bisimplicial graded $C$-modules and maps, and their diagonals give an exact sequence of graded simplicial $C$-modules and two quasi-isomorphisms (Conventions). The first term of the diagonal sequence is $L^{\Delta\mathrm{gr}}_{B/A}\otimes_BC$, the second $L^{\Delta\mathrm{gr}}_{C/A}$, and the diagonal of the columns $\Omega_{R/B_n}\otimes_RC_n=L^{\mathrm{gr}}_{C_n/B_n}$ is $L^{\Delta\mathrm{gr}}_{C/B}$. This gives (4.1) with $L'$ the diagonal of the third terms. In general, $B\to B\times C$, $b\mapsto(b,\bar b)$, is a monomorphism of graded simplicial rings in each degree, and $B\times C\to C$ is flat in each degree, $C_n$ being the direct factor cut out by the idempotent $(0,1)$ of degree $0$. Apply the first case to $A\to B\to B\times C$ and tensor with $C$. The maps induced by the projections identify the terms after $\otimes_{B\times C}C$ with those of the statement, being quasi-isomorphisms on every column by the proof of Theorem 3.4 there.

The terms of $L^{\Delta\mathrm{gr}}_{B/A}$ are free, so $L^{\Delta\mathrm{gr}}_{B/A}\otimes_BC$ represents $L^{\Delta\mathrm{gr}}_{B/A}\otimes^\ell_BC$ (Proposition 3.2 of the lesson on derived tensor products, on each component). The short exact sequence (4.1) gives the triangle by Proposition 1.2. The statements on $k^0$ and $\Phi$ hold column by column by the last paragraph of the proof of Theorem 3.4 there, and pass to the diagonals. For rings concentrated in degree $0$, Theorem 2.4 (3) identifies the parts of degree $0$ with the cotangent complexes $L^\Delta$ of the simplicial rings. $\square$

5. The Atiyah class of a simplicial module

Let $A\to B$ be a map of rings of $T$, $P=P_A(B)$, and let $\operatorname{at}_{B/A}(M)\colon M\to\sigma(L_{B/A}\otimes^\ell_BM)$ be the Atiyah class of a simplicial $B$-module $M$, defined by the principal parts of $M$ (Definition 2.2 of the lesson on the Atiyah class), the third map of the triangle (2.2) there.

Theorem 5.1 (the graded description). Let $M$ be a simplicial $B$-module and $C=B\oplus M$, a graded simplicial $B$-algebra with $C^0=B$ and $C^1=M$. The part of degree one of the graded transitivity triangle of $A\to B\to C$ (Theorem 4.1), with its first term identified with $L_{B/A}\otimes^\ell_BM$ through Corollary 2.5 and its third term $k^1(L^{\Delta\mathrm{gr}}_{C/B})\cong M$ (Theorem 2.4 (4)), is isomorphic to the triangle (2.2) of the lesson on the Atiyah class, by an isomorphism that is the identity on the first and third terms. In particular $\operatorname{at}_{B/A}(M)$ is the third map of the part of degree one of the graded transitivity triangle. The same holds for every graded simplicial $B$-algebra $C$ with $C^0=B$ and $C^1=M$.

Proof. The proof of Theorem 2.4 of the lesson on the Atiyah class applies degree by degree. Let $R=P^{\Delta}_A(C)^{\mathrm{gr}}$. The rings $A$ and $B$ are constant and $k^0(C_n)=B$ for every $n$, so $k^0P_A(C_n)^{\mathrm{gr}}=P$ for every $n$ (Proposition 2.2 (3) of the lesson on graded modules), and $k^0R=P$. Its part of degree one $F=k^1R$ is a simplicial $P$-module with free terms $F_n=P_n^{(X_n^1)}$, and the part of degree one of the augmentation, $\varepsilon\colon F\to M$, is a quasi-isomorphism, being the diagonal of the parts of degree one of the augmentations of the columns (Proposition 2.2 (2) there). Moreover every term $P_A(C_n)^{\mathrm{gr}}_m$ of the bisimplicial resolution is a graded polynomial algebra over $P_m$ on generators of positive degree. The map $B\to C$ is injective in each degree. The argument of the proof of Theorem 2.4 of the lesson on the Atiyah class, column by column, shows that the graded transitivity triangle of $A\to B\to C$ is isomorphic to the triangle of the diagonal of the exact sequences formed from $P$ and the columns $P_A(C_n)^{\mathrm{gr}}$, by an isomorphism that is the identification of Corollary 2.5 on the first terms and is compatible with the augmentations to $\Omega_{C/B}$ on the third terms; the comparison maps of the columns are natural, so they pass to the diagonals. In simplicial degree $n$ this diagonal is the sequence of column $n$ in degree $n$. So the computation of the part of degree one in the proof of Theorem 2.4 there, which uses Lemma 1.3 of that lesson in each degree, gives the exact sequence of simplicial $P$-modules $\begin{aligned}&0\to\Omega_{P/A}\otimes_PM\to Y\\&\quad\to F\otimes_PB\to0,\end{aligned}$ with $Y$ the pushout of $P^1_{P/A}(F)\otimes_PB$ along $\Omega_{P/A}\otimes_PF\otimes_PB\to\Omega_{P/A}\otimes_PM$, and with the third term identified with $M$ by $x\otimes b\mapsto b\,\varepsilon(x)$. The maps of short exact sequences from the sequence of principal parts of $F$ to the sequence (2.1) of that lesson for $M$ and to this sequence are formed degree by degree, as there, and they are quasi-isomorphisms by the same five lemma argument: $1\otimes\varepsilon$ is a quasi-isomorphism because $\Omega_{P/A}$ has flat terms, and $F\to F\otimes_PB$ is one because $F$ has free terms and $P\to B$ is a quasi-isomorphism (Lemma 2.1 of the lesson on derived tensor products). This gives the isomorphism of triangles, the identity on the first and third terms. For another graded simplicial $B$-algebra $C'$ with $C'^0=B$ and $C'^1=M$, the quotient map $C'\to C'/C'^{\geq2}=B\oplus M$ induces a morphism of graded transitivity triangles whose part of degree one is the identity on the first and third terms, by the naturality in Theorems 4.1 and 2.4 (4). So the third maps agree. $\square$

6. Exercises

Exercise 6.1. Let $A\to B\to C$ be maps of graded simplicial rings, with $B\to C$ a quasi-isomorphism. Show that $L^{\Delta\mathrm{gr}}_{C/B}$ is acyclic and that $L^{\Delta\mathrm{gr}}_{B/A}\otimes_BC\to L^{\Delta\mathrm{gr}}_{C/A}$ is a quasi-isomorphism.

Exercise 6.2. Let $A$ be a simplicial ring, regarded as a graded simplicial ring concentrated in degree $0$, and $Y$ a simplicial object of $T^{\mathrm{gr}+}$. Let $B$ be the graded simplicial $A$-algebra with $B_n=A_n[Y_n]$. Compute $k^1(L^{\Delta\mathrm{gr}}_{B/A})$ in $D_\bullet(A)$.

Exercise 6.3. Let $B$ be a ring of $T$ and $M$ a simplicial $B$-module. Show with Theorem 5.1 that $\operatorname{at}_{B/B}(M)=0$.

7. Complete solutions

6.1. Apply Theorem 2.4 (2) to the square with top row $B\to B$, bottom row $B\to C$, and the vertical maps $1_B$ and $B\to C$. It shows that $L^{\Delta\mathrm{gr}}_{B/B}\to L^{\Delta\mathrm{gr}}_{C/B}$ is a quasi-isomorphism. And $L^{\Delta\mathrm{gr}}_{B/B}$ is acyclic: by Theorem 2.4 (1) its image under $\Phi$ is quasi-isomorphic to $L^\Delta_{\Phi B/\Phi B}$, whose columns $L_{\Phi B_n/\Phi B_n}$ are acyclic, $\Phi B_n$ being the polynomial algebra over itself on no variables (Proposition 1.3 of the lesson on cotangent complexes of simplicial rings), so its diagonal is acyclic. The triangle of Theorem 4.1 then has an acyclic third term, so its first map is an isomorphism of $D_\bullet(C)_{\mathrm{gr}}$, and the first map of (4.1) is a quasi-isomorphism.

6.2. Every $B_n$ lies in $A_n\text{-}\mathrm{Alggr}^+$, so Theorem 2.4 (4) gives $k^1(L^{\Delta\mathrm{gr}}_{B/A})\cong B^1$ in $D_\bullet(A)$. By Lemma 1.2 (3) of the lesson on graded modules, $k^1(A_n[Y_n])=A_n^{(Y_n^1)}$. So $k^1(L^{\Delta\mathrm{gr}}_{B/A})$ is isomorphic to the simplicial $A$-module $A^{(Y^1)}$, free on the simplicial sheaf of sets $Y^1$ in each degree.

6.3. By Theorem 5.1 with $A=B$, $\operatorname{at}_{B/B}(M)$ is the third map of the part of degree one of the graded transitivity triangle of $B\to B\to B\oplus M$, whose first term is $L_{B/B}\otimes^\ell_BM$. The cotangent complex $L_{B/B}$ is acyclic, $B$ being the polynomial algebra over itself on no variables (Proposition 1.3 of the lesson on cotangent complexes of simplicial rings). So the third map is a morphism to $\sigma$ of an object isomorphic to $0$, and it vanishes.

Sources

Graded simplicial rings, the diagonal graded cotangent complex, its universal property for graded derivations, the graded transitivity triangle and the graded description of the Atiyah class of a simplicial module are due to Illusie, in his work on the cotangent complex, where the extension from graded rings and from simplicial rings to graded simplicial rings is stated to follow by the same arguments. The proofs given here reduce each statement to its two special cases, column by column and through the diagonal of bisimplicial objects.