Graded modules and the graded cotangent complex

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

A graded ring \(A=\bigoplus_{n\geq0}A^n\) of a topos has graded modules, graded algebras and a graded cotangent complex \(L^{\mathrm{gr}}_{B/A}\): the cotangent complex computed with polynomial algebras on graded sheaves of generators, so that every term carries a grading. This lesson proves:

For a \(B\)-module \(M\), the algebra of dual numbers \(B\oplus M\) is graded with \(M\) in degree one. The lesson The Atiyah class of a module builds the Atiyah class of \(M\) from the degree-one part of the transitivity triangle of \(A\to B\to B\oplus M\).

One point needs care. The functor "degree \(n\) component" from graded \(A\)-modules to \(A^0\)-modules has a right adjoint, but that adjoint is a graded coinduction, not "place an \(A^0\)-module in degree \(n\)"; the latter is right adjoint to a different functor, which first applies \(A^0\otimes_A-\) (Lemma 1.3 and Example 1.4).

Prerequisites. The cotangent complex (Section 1 with its definition of polynomial resolutions, Lemma 2.1, Theorem 2.2, Lemma 4.A, Theorem 4.1 and its proof, and Section 10 with Lemma 10.2 and Theorem 10.3); Construction 1.2 of Derived tensor products of simplicial modules; Lemma 3.2 and Proposition 3.10 of Triangles and Ext groups of simplicial modules. From the AI-integrated Stacks project: K-injective resolutions in Grothendieck abelian categories [Stacks, Tag 079P]; derived adjunctions [Tag 0DVC, Tag 09T5, Tag 08BJ]; left derived functors through an adapted class [Tag 05T7, Tag 05TA]; bounded above flat complexes are K-flat [Tag 06YQ]; flatness of \(j_!\) [Tag 03EV]; sheafification [Tag 03EI, Tag 0GMW, Tag 08TP]; the standard simplicial object of an adjunction [Tag 08NC, Tag 08ND, Tag 08P2]; the cotangent complex of sheaves of rings [Tag 08SR, Tag 08SS, Tag 08SZ, Tag 08US]; products [Tag 08SY]; right adjoints of exact functors preserve injectives [Tag 015Z]; naive cotangent complexes of presentations [Tag 08TW, Tag 08TY]; torsors and \(H^1\) [Tag 03AJ]; sections of \(R\mathcal{H}om\) [Tag 08JA]; triangles of short exact sequences [Tag 0152].

Conventions

\(T\) is a topos, and we fix a small site \(\mathcal C\) with \(T=\operatorname{Sh}(\mathcal C)\). Rings are commutative with \(1\); modules are sheaves of modules. For a ring \(\Lambda\) of \(T\), \(D(\Lambda)\) is the derived category of \(\Lambda\)-modules. We write \(a\) for sheafification of presheaves on \(\mathcal C\); it is exact and commutes with finite limits, colimits, tensor products and modules of differentials [Tags 03EI, 0GMW, 08TP].

Simplicial modules are identified with complexes in cohomological degrees \(\leq0\) by normalization, \(N_n\) placed in degree \(-n\). A short exact sequence of complexes, or of simplicial modules, \(0\to E\to F\to G\to0\) gives the distinguished triangle \(E\to F\to G\xrightarrow{\delta}E[1]\) of [Tag 0152], or of Lemma 3.2 and Proposition 3.10 of the lesson on triangles; on cohomology \(\delta\) induces the connecting homomorphisms of the long exact sequence.

For a map of rings \(A\to B\) of \(T\), \(L_{B/A}=\Omega_{P/A}\otimes_PB\), with \(P=P_A(B)\) the standard resolution [Tags 08SR, 08SS], as in Section 10 of The cotangent complex.

1. Graded rings and modules

Definition 1.1. A graded sheaf is a family \(X=(X^n)_{n\in\mathbb Z}\) of sheaves of sets; graded sheaves form the topos \(T^{\mathrm{gr}}=T^{\mathbb Z}\). Write \(T^{\mathrm{gr}\geq0}\) and \(T^{\mathrm{gr}+}\) for the full subcategories of the \(X\) with \(X^n=\varnothing\) for \(n<0\), respectively for \(n\leq0\).

A graded ring is a ring \(A\) with a decomposition \(A=\bigoplus_{n\geq0}A^n\) into abelian subsheaves such that \(A^iA^j\subseteq A^{i+j}\) and \(1\in A^0\). (All graded rings in this lesson sit in degrees \(\geq0\).) Then \(A^0\) is a subring, \(A^+=\bigoplus_{n>0}A^n\) is an ideal, and \(A\to A/A^+=A^0\) is a ring map. A graded \(A\)-module is an \(A\)-module \(M\) with a decomposition \(M=\bigoplus_{n\in\mathbb Z}M^n\) such that \(A^iM^j\subseteq M^{i+j}\). A graded \(A\)-algebra is a graded ring \(B\) with a ring map \(A\to B\) carrying \(A^n\) into \(B^n\). Morphisms carry degree \(n\) into degree \(n\). We obtain the categories \(A\text{-}\mathrm{Modgr}\) and \(A\text{-}\mathrm{Alggr}\). If \(A\) is a ring, regarded as a graded ring concentrated in degree \(0\), \(A\text{-}\mathrm{Alggr}^+\) is the full subcategory of the graded \(A\)-algebras \(B\) with \(A\to B^0\) an isomorphism.

We write \(\Phi\) for the functors forgetting the decomposition, \(k^n(M)=M^n\) (an \(A^0\)-module), \(M(n)\) for the shifted module with \(M(n)^p=M^{n+p}\), \(U_{\mathrm{gr}}(M)=(M^n)_{n}\in T^{\mathrm{gr}}\) for the family of underlying sheaves of sets, and, for \(B\in A\text{-}\mathrm{Alggr}^+\), \(U_+(B)=(B^n)_{n\geq1}\in T^{\mathrm{gr}+}\). The tensor product \(E\otimes_AF\) of graded modules is graded: \(E\otimes_{\mathbb Z}F=\bigoplus_{i,j}E^i\otimes F^j\) is graded by \(i+j\), and the relations \(ae\otimes f-e\otimes af\) are generated by those with \(a,e,f\) homogeneous, so the quotient keeps the grading, with \((E\otimes_AF)^n\) the sum of the images of the \(E^i\otimes_{A^0}F^j\), \(i+j=n\). In the same way \(B\otimes_A-\), symmetric algebras \(S_A(M)\) (graded by total degree) and kernels and cokernels of graded maps are graded.

Lemma 1.2 (free objects). Let \(A\) be a graded ring.

  1. For \(X\in T^{\mathrm{gr}}\), the graded module \(A^{(X)}=\bigoplus_nA(-n)\otimes_{\mathbb Z}\mathbb Z^{(X^n)}\) satisfies \(\operatorname{Hom}_{A\text{-}\mathrm{Modgr}}(A^{(X)},M)=\) \(\operatorname{Hom}_{T^{\mathrm{gr}}}(X,U_{\mathrm{gr}}M)\), and \(\Phi A^{(X)}=(\Phi A)^{(\coprod_nX^n)}\).
  2. For \(X\in T^{\mathrm{gr}\geq0}\), the polynomial algebra \(A[X]=S_A(A^{(X)})\) is a graded \(A\)-algebra in degrees \(\geq0\), \(\operatorname{Hom}_{A\text{-}\mathrm{Alggr}}(A[X],B)=\) \(\operatorname{Hom}_{T^{\mathrm{gr}}}(X,U_{\mathrm{gr}}B)\), \(\Phi(A[X])=(\Phi A)[\coprod_nX^n]\), and \(k^0(A[X])=A^0[X^0]\).
  3. If \(A\) is a ring and \(X\in T^{\mathrm{gr}+}\), then \(A[X]\in A\text{-}\mathrm{Alggr}^+\), \(k^1(A[X])=A^{(X^1)}\), and \(\operatorname{Hom}_{A\text{-}\mathrm{Alggr}^+}(A[X],B)=\) \(\operatorname{Hom}_{T^{\mathrm{gr}+}}(X,U_+B)\).

Proof. (1) \(A(-n)\otimes\mathbb Z^{(X^n)}\) is the free \(A\)-module on \(X^n\) with the generators placed in degree \(n\); a graded map out of it is the same as a map of sheaves \(X^n\to M^n\). The sum over \(n\) gives the formula, and \(\bigoplus_n\mathbb Z^{(X^n)}=\mathbb Z^{(\coprod X^n)}\). (2) The symmetric algebra of a graded module is graded by total degree, as noted above, and it is in degrees \(\geq0\) when \(A\) and \(X\) are. An algebra map out of \(S_A(A^{(X)})\) is a module map out of \(A^{(X)}\); it is graded exactly when the module map is, which gives the adjunction by (1). Forgetting the grading commutes with \(S\) and with (1). A monomial \(a\,x_1\cdots x_k\) with \(a\in A^i\) and \(x_j\in X^{n_j}\) has degree \(i+\sum n_j\); as all these are \(\geq0\), it has degree \(0\) exactly when \(a\in A^0\) and all \(x_j\in X^0\). (3) Now \(A=A^0\) and the variables have degree \(\geq1\), so the monomials of degree \(0\) are the constants and those of degree \(1\) are the \(a\,x\) with \(x\in X^1\). A graded algebra map \(A[X]\to B\) is a map \(X\to U_{\mathrm{gr}}B\), and \(X^0=\varnothing\). \(\square\)

Lemma 1.3 (the degree-\(n\) functor). Let \(A\) be a graded ring, \(n\in\mathbb Z\), \(F\) an \(A^0\)-module and \(E\) a graded \(A\)-module.

  1. Put \(\ell_n(F)=A(-n)\otimes_{A^0}F\), with \(\ell_n(F)^p=A^{p-n}\otimes_{A^0}F\). Then \(\operatorname{Hom}_{A\text{-}\mathrm{Modgr}}(\ell_nF,E)\cong\) \(\operatorname{Hom}_{A^0}(F,E^n)\).
  2. Put \(r_n(F)^p=\mathcal Hom_{A^0}(A^{n-p},F)\) (internal Hom; zero for \(p>n\)), with \(a\in A^i\) acting by \((a\varphi)(b)=\varphi(ab)\). Then \(r_n(F)\) is a graded \(A\)-module and \(\operatorname{Hom}_{A\text{-}\mathrm{Modgr}}(E,r_nF)\cong\) \(\operatorname{Hom}_{A^0}(E^n,F)\), by \(\psi\mapsto(e\mapsto\psi(e)(1))\).
  3. Let \(F\langle n\rangle\) be \(F\) placed in degree \(n\), with \(A^+\) acting by zero. Then \(\operatorname{Hom}_{A\text{-}\mathrm{Modgr}}(E,F\langle n\rangle)\cong\) \(\operatorname{Hom}_{A^0}(k^n(A^0\otimes_AE),F)\), where \(k^n(A^0\otimes_AE)=E^n/\sum_{i>0}A^iE^{n-i}\).

So \(\ell_n\dashv k^n\dashv r_n\). If \(A=A^0\), then \(r_n(F)=F\langle n\rangle\).

Proof. (1) The degree-\(n\) part of \(\ell_nF\) is \(A^0\otimes_{A^0}F=F\), and it generates \(\ell_nF\) as an \(A\)-module; a graded \(A\)-linear map is determined by its restriction \(F\to E^n\), and every \(A^0\)-linear \(f\colon F\to E^n\) extends by \(a\otimes x\mapsto af(x)\).

(2) For \(a\in A^i\) and \(\varphi\in r_n(F)^p\), \(b\mapsto\varphi(ab)\) is defined on \(A^{n-p-i}\), so \(a\varphi\in r_n(F)^{p+i}\); associativity and the unit are clear. Given \(\psi\), the composite \(E^n\xrightarrow{\psi}r_n(F)^n=\mathcal Hom(A^0,F)\to F\), evaluation at \(1\), is \(A^0\)-linear. Conversely, \(f\colon E^n\to F\) gives \(\psi_f(e)(b)=f(be)\) for \(e\in E^p\) and \(b\in A^{n-p}\); then \(\psi_f(ae)(b)=\) \(f(bae)=\) \(\psi_f(e)(ba)=\) \((a\psi_f(e))(b)\), so \(\psi_f\) is \(A\)-linear. The two constructions are inverse: \(\psi_f(e)(1)=f(e)\) for \(e\in E^n\), and \(\psi(be)(1)=(b\psi(e))(1)=\psi(e)(b)\).

(3) An \(A^0\)-linear \(f\colon E^n\to F\) extends by zero in the other degrees to an \(A\)-linear map \(E\to F\langle n\rangle\) exactly when \(f(A^iE^{n-i})=0\) for \(i>0\), because \(A^i\) acts by zero on \(F\langle n\rangle\) and \(A^iE^{n-i}\subseteq E^n\). The degree-\(n\) part of \(A^0\otimes_AE=E/A^+E\) is the stated quotient. \(\square\)

Example 1.4. Let \(k\) be a ring of \(T\) and \(A=k[t]\) with \(t\) in degree \(1\), \(E=A\), \(n=1\), \(F=k\). Then \(k^1(A^0\otimes_AA)=k^1(k)=0\), so every graded map \(A\to k\langle1\rangle\) is zero, while \(\operatorname{Hom}_k(k^1A,k)=\operatorname{Hom}_k(kt,k)\cong k\). So \(F\mapsto F\langle n\rangle\) is not right adjoint to \(k^n\). The right adjoint \(r_1(k)\) is the graded dual of \(A\) shifted: \(r_1(k)^p=\operatorname{Hom}_k(k\,t^{1-p},k)\) for \(p\leq1\), with \(t\) acting by the shift.

Proposition 1.5. Let \(A\) be a graded ring.

  1. \(A\text{-}\mathrm{Modgr}\) is a Grothendieck abelian category. Kernels, cokernels and colimits are computed degree by degree, the functors \(k^n\) and \(\Phi\) are exact and commute with colimits, and a sequence of graded modules is exact if and only if its image under \(\Phi\) is, if and only if all its images under the \(k^n\) are.
  2. The modules \(A(-n)\otimes_{\mathbb Z}\mathbb Z_U\), for \(U\in\mathcal C\) and \(n\in\mathbb Z\), where \(\mathbb Z_U\) is the free abelian sheaf on the sheafified representable sheaf of \(U\), form a family of generators, and their images under \(\Phi\) are flat \(\Phi A\)-modules.
  3. Every graded module is a quotient of a graded module whose image under \(\Phi\) is flat (a flat graded module), and every complex of graded modules has a K-injective resolution. Write \(D(A)_{\mathrm{gr}}\) for the derived category of \(A\text{-}\mathrm{Modgr}\). A map of complexes of graded modules is a quasi-isomorphism if and only if its image under \(\Phi\) is, if and only if its images under all \(k^n\) are.

Proof. (1) The kernel and cokernel of a graded map, computed in \(A\)-modules, inherit decompositions because the map preserves them; colimits are computed degreewise. Filtered colimits are exact because they are in \(T\). (2) By Lemma 1.2 (1), \(\operatorname{Hom}(A(-n)\otimes\mathbb Z_U,M)=M^n(U)\), so a nonzero \(M\) receives a nonzero map from one of them. The image under \(\Phi\) is the extension by zero of \(A|_U\), which is flat [Tag 03EV]. (3) The first assertion follows from (2) by taking a sum of generators; the second from (1) and [Tag 079P]; the last from (1). \(\square\)

Proposition 1.6 (derived functors). Let \(A\) be a graded ring and \(n\in\mathbb Z\).

  1. \(r_n\) has a right derived functor \(Rr_n\colon D(A^0)\to D(A)_{\mathrm{gr}}\), right adjoint to \(k^n\colon D(A)_{\mathrm{gr}}\to D(A^0)\).
  2. For \(E\in D^-(A)_{\mathrm{gr}}\) and \(F\in D(A^0)\), \(\operatorname{Hom}_{D(A)_{\mathrm{gr}}}(E,F\langle n\rangle)\cong\) \(\operatorname{Hom}_{D(A^0)}(k^n(A^0\otimes^{\mathbf L}_AE),F)\), where \(A^0\otimes^{\mathbf L}_A-\) is computed with bounded above complexes of flat graded modules.
  3. For \(E\in D^-(A)_{\mathrm{gr}}\) and \(F\in D(A)_{\mathrm{gr}}\), the derived tensor product \(E\otimes^{\mathbf L}_AF\) is defined by \(P\otimes_AF\) (direct sum totalization) for a quasi-isomorphism \(P\to E\) from a bounded above complex of flat graded modules, and \(\Phi(E\otimes^{\mathbf L}_AF)=\Phi E\otimes^{\mathbf L}_{\Phi A}\Phi F\). The same holds for \(B\otimes^{\mathbf L}_A-\), \(B\) a graded \(A\)-algebra.

Proof. (1) \(k^n\) is exact and left adjoint to \(r_n\) (Lemma 1.3), so \(r_n\) carries K-injective complexes to K-injective complexes [Tag 08BJ]; K-injective resolutions exist in \(A^0\)-modules [Tag 079P], so \(Rr_n\) exists, and the adjunction is [Tag 09T5].

(3) By Proposition 1.5 (3) every bounded above complex is the target of a quasi-isomorphism from a bounded above complex of flat graded modules [Tag 05T7]. If \(P\) is such a complex and is acyclic, then \(\Phi(P\otimes_AF)=\Phi P\otimes_{\Phi A}\Phi F\) is acyclic, because \(\Phi P\) is K-flat [Tag 06YQ]. Hence \(P\otimes_AF\) is acyclic, and the derived functor exists and is computed by such resolutions [Tag 05TA]; the formula for \(\Phi\) holds on these representatives. The argument for \(B\otimes_A-\) is the same.

(2) The functor \(G=k^n(A^0\otimes_A-)\) is left adjoint to the exact functor \(F\mapsto F\langle n\rangle\) (Lemma 1.3 (3)). By the proof of (3), \(LG\) is defined on \(D^-(A)_{\mathrm{gr}}\) and computed by bounded above flat resolutions, and \(F\langle n\rangle\) is its own right derived functor. The isomorphism is [Tag 0DVC]. \(\square\)

2. Standard resolutions

Construction 2.1. Let \(A\) be a graded ring and \(B\) a graded \(A\)-algebra. The adjunction \(A[-]\dashv U_{\mathrm{gr}}\) between \(A\text{-}\mathrm{Alggr}\) and \(T^{\mathrm{gr}\geq0}\) (Lemma 1.2 (2)) gives, by the standard construction [Tag 08NC], an augmented simplicial graded \(A\)-algebra \[ \begin{gathered}P_A(B)^{\mathrm{gr}}\to B,\\ P_A(B)^{\mathrm{gr}}_n =G^{n+1}(B),\quad G\\ =A[U_{\mathrm{gr}}(-)],\end{gathered} \] with faces \(d_i=G^i\varepsilon G^{n-i}\) and degeneracies \(s_i=G^i\delta G^{n-i}\), where the counit \(\varepsilon\) evaluates a variable at the element labelling it and the comultiplication \(\delta\) replaces a label \(x\) by the variable \([x]\). Explicitly, on the variables \([x]\) of \(P_n=A[U_{\mathrm{gr}}P_{n-1}]\): \(d_0[x]=x\), \(d_i[x]=[d_{i-1}x]\) for \(1\leq i\leq n\), \(s_0[x]=[[x]]\) and \(s_i[x]=[s_{i-1}x]\) for \(i\geq1\). If \(A\) is a ring and \(B\in A\text{-}\mathrm{Alggr}^+\), the adjunction \(A[-]\dashv U_+\) of Lemma 1.2 (3) gives in the same way \(P^+_A(B)\to B\), a simplicial object of \(A\text{-}\mathrm{Alggr}^+\).

Proposition 2.2.

  1. \(P_A(B)^{\mathrm{gr}}_n=A[X_n]\) with \(X_n=U_{\mathrm{gr}}(P_A(B)^{\mathrm{gr}}_{n-1})\in T^{\mathrm{gr}\geq0}\) (and \(X_0=U_{\mathrm{gr}}B\)); \(P^+_A(B)_n=A[Y_n]\) with \(Y_n\in T^{\mathrm{gr}+}\).
  2. The augmentations \(P_A(B)^{\mathrm{gr}}\to B\) and \(P^+_A(B)\to B\) are surjective in each simplicial degree and are quasi-isomorphisms of simplicial graded \(A\)-modules.
  3. \(k^0(P_A(B)^{\mathrm{gr}})=P_{A^0}(B^0)\), the standard resolution of \(B^0\) over \(A^0\), compatibly with the augmentations.
  4. If \(A\) is a ring and \(B\in A\text{-}\mathrm{Alggr}^+\), then \(k^0(P^+_A(B))=A\) (constant) and \(k^1(P^+_A(B))=F_A(B^1)\), the standard free resolution of the \(A\)-module \(B^1\) (Construction 1.2 of the lesson on derived tensor products), compatibly with the augmentations.

Proof. (1) holds by construction. (2) The augmentation \(P_0=A[U_{\mathrm{gr}}B]\to B\) is surjective (the variable \([b]\) maps to \(b\)), and \(P_n\to B\) is its composite with a vertex map, which has a section given by degeneracies. By [Tag 08ND], applying \(U_{\mathrm{gr}}\) to \(P_A(B)^{\mathrm{gr}}\to B\) gives a homotopy equivalence of simplicial objects of \(T^{\mathrm{gr}\geq0}\). Taking sections over \(U\in\mathcal C\) of the degree-\(k\) components preserves the homotopy identities, so \(k^k(P_A(B)^{\mathrm{gr}})(U)\to B^k(U)\) is a map of simplicial abelian groups which is a homotopy equivalence of simplicial sets, hence a quasi-isomorphism [Tag 08P2]. Homology sheaves are the sheafifications of these sectionwise homology groups, so the augmentation is a quasi-isomorphism. The same argument applies to \(P^+\). (3) By Lemma 1.2 (2), \(k^0(A[X])=A^0[X^0]\), and \(k^0\) of \(U_{\mathrm{gr}}\) is the underlying sheaf of \(k^0\). So \(k^0\circ G\) is the comonad \(A^0[U(-)]\) applied to \(k^0\), with the same counit and comultiplication on variables, and \(k^0(P_n)=(A^0[U(-)])^{n+1}(B^0)\) with the same faces and degeneracies. (4) By Lemma 1.2 (3), \(k^0(A[Y])=A\) and \(k^1(A[Y])=A^{(Y^1)}\), and \(k^1\) of \(U_+\) is the underlying sheaf of \(k^1\). So in degree one the comonad \(A[U_+(-)]\) is the comonad \(A^{(U(-))}\) of free \(A\)-modules, with its counit and comultiplication, whose standard simplicial object is \(F_A\). \(\square\)

Lemma 2.3 (sectionwise description). The constructions above make sense for presheaves on \(\mathcal C\) (polynomial algebras, symmetric algebras and tensor products formed objectwise). For a map of presheaves of graded rings \(A\to B\):

  1. \(P_A(B)^{\mathrm{gr},\mathrm{psh}}(U)=P_{A(U)}(B(U))^{\mathrm{gr}}\), the graded standard resolution in the topos of sets, for every \(U\in\mathcal C\);
  2. \(a\big(P_A(B)^{\mathrm{gr},\mathrm{psh}}\big)=P_{aA}(aB)^{\mathrm{gr}}\) as augmented simplicial graded algebras, and likewise for \(P^+\), for modules of differentials and for tensor products.

Proof. (1) Every construction on presheaves is objectwise. (2) For a presheaf of graded sets \(X\) and a sheaf of graded rings \(R\), \(\operatorname{Hom}(a(A[X]^{\mathrm{psh}}),R)=\) \(\operatorname{Hom}(A[X]^{\mathrm{psh}},R)=\) \(\operatorname{Hom}(X,U_{\mathrm{gr}}R)=\) \(\operatorname{Hom}(aX,U_{\mathrm{gr}}R)\), so \(a(A[X]^{\mathrm{psh}})=(aA)[aX]\); and \(a(U_{\mathrm{gr}}B)=U_{\mathrm{gr}}(aB)\), sheafification commuting with finite limits. Hence \(a\circ G^{\mathrm{psh}}\cong G\circ a\), compatibly with counits and comultiplications (both are defined on variables), and induction on \(n\) gives the assertion. Differentials and tensor products commute with \(a\) [Tags 08TP, 0GMW]. \(\square\)

Lemma 2.4 (comparison maps).

  1. Let \(A\to B\) be a map of graded rings. There is a unique map of augmented simplicial \(\Phi A\)-algebras \(\theta_{B/A}\colon\Phi P_A(B)^{\mathrm{gr}}\to P_{\Phi A}(\Phi B)\) given on variables by \(\theta_n[x]=[\theta_{n-1}(x)]\), with \(\theta_{-1}=\mathrm{id}_B\); in degree \(0\) it sends the variable of a homogeneous element \(b\) to the variable of \(b\). It is natural in \(A\to B\), and \(\theta\) for sheaves is the sheafification of \(\theta\) for the presheaves of Lemma 2.3.
  2. If \(A\) is a ring and \(B\in A\text{-}\mathrm{Alggr}^+\), the same formula defines a map of augmented simplicial graded \(A\)-algebras \(\theta^+_B\colon P^+_A(B)\to P_A(B)^{\mathrm{gr}}\).

Proof. (1) By Lemma 1.2 (2), \(\Phi P_n\) is the polynomial algebra over \(\Phi A\) on the sheaf of homogeneous elements of \(P_{n-1}\), so the formula defines algebra maps \(\theta_n\) once \(\theta_{n-1}\) is defined. With the explicit faces and degeneracies of Construction 2.1, on a variable \([x]\): \(d_0\theta_n[x]=\theta_{n-1}(x)=\theta_{n-1}(d_0[x])\); for \(i\geq1\), \(d_i\theta_n[x]=[d_{i-1}\theta_{n-1}x]\) \(=[\theta_{n-2}d_{i-1}x]=\) \(\theta_{n-1}(d_i[x])\) by induction; \(s_0\theta_n[x]=[[\theta_{n-1}x]\) \(]=[\theta_n[x]]=\theta_{n+1}(s_0[x])\); and similarly for \(s_i\), \(i\geq1\). Algebra maps agreeing on variables agree, so \(\theta\) is simplicial, and \(\varepsilon\theta_0[b]=b\). Naturality and compatibility with sheafification hold because the formula is the same everywhere. (2) The same verification, the variables of \(P^+_n\) being among those of \(P_n\) by induction. \(\square\)

3. Graded differentials and the graded cotangent complex

Lemma 3.1 (graded differentials). Let \(A\to B\to C\) be maps of graded rings, \(I=\ker(B\otimes_AB\to B)\), and \(\Omega_{B/A}=I/I^2\) with \(\mathrm db=1\otimes b-b\otimes1\).

  1. \(\Omega_{B/A}\) is a graded \(B\)-module, \(\mathrm d\) preserves degrees, \(\Phi\Omega_{B/A}=\Omega_{\Phi B/\Phi A}\), and for a graded \(B\)-module \(M\), \(\operatorname{Hom}_{B\text{-}\mathrm{Modgr}}(\Omega_{B/A},M)\) is the group \(\operatorname{Der}_A(B,M)_{\mathrm{gr}}\) of \(A\)-derivations carrying \(B^n\) into \(M^n\); this is also the set of graded \(A\)-algebra sections of the dual numbers \(B\oplus M\to B\), graded by \(k^n(B\oplus M)=B^n\oplus M^n\).
  2. The sequence of graded \(C\)-modules \(\Omega_{B/A}\otimes_BC\to\Omega_{C/A}\to\Omega_{C/B}\to0\) is exact, and it is split exact on the left if \(C=B[X]\) for some \(X\in T^{\mathrm{gr}\geq0}\).
  3. \(k^0(\Omega_{B/A})=\Omega_{B^0/A^0}\).
  4. If \(A\) is a ring and \(B\in A\text{-}\mathrm{Alggr}^+\), then \(k^0(\Omega_{B/A})=0\) and \(b\mapsto\mathrm db\) is an isomorphism \(B^1\to k^1(\Omega_{B/A})\), natural in \(B\).

Proof. (1) \(I\) is the kernel of a graded map, hence graded, and so is \(I^2\). For rings of sections the remaining assertions are the usual properties of \(I/I^2\); they hold for sheaves because differentials, kernels and quotients sheafify [Tag 08TP]. A \(B\)-linear map out of \(\Omega_{B/A}\) is graded exactly when its composite with \(\mathrm d\) is, because the \(\mathrm db\) with \(b\) homogeneous generate \(\Omega_{B/A}\). A derivation \(D\) corresponds to the section \(b\mapsto(b,Db)\), which is graded exactly when \(D\) is.

(2) The maps are graded; exactness can be checked after \(\Phi\), where the sequence is the sheafification of the corresponding exact sequences for rings of sections. For \(C=B[X]\), \(\Omega_{C/A}=(\Omega_{B/A}\otimes_BC)\oplus C^{(X)}\), the second summand spanned by the \(\mathrm dx\).

(3) All rings are in degrees \(\geq0\), so \(k^0(B\otimes_AB)=B^0\otimes_{A^0}B^0\): the relations \(ab\otimes b'-b\otimes ab'\) with \(a\) of positive degree lie in positive degrees. Hence \(k^0(I)=\ker(B^0\otimes_{A^0}B^0\to B^0)\), and \(k^0(I^2)=k^0(I)^2\), a product of homogeneous elements of which one has positive degree having positive degree.

(4) By (3), \(k^0\Omega_{B/A}=\Omega_{A/A}=0\). As \(B^0=A\), \(k^1(B\otimes_AB)=\) \((B^1\otimes_AA)\oplus(A\otimes_AB^1)=\) \(B^1\oplus B^1\), and the multiplication is \((x,y)\mapsto x+y\) there. So \(k^1(I)=\{(-x,x)\}\), the anti-diagonal copy of \(B^1\), which is the set of the \(\mathrm dx=1\otimes x-x\otimes1\). Since \(k^0(I)=0\), \(k^1(I^2)=k^0(I)k^1(I)=0\). Hence \(x\mapsto\mathrm dx\) is an isomorphism \(B^1\to k^1(I/I^2)\). \(\square\)

Definition 3.2. Let \(A\to B\) be a map of graded rings and \(P=P_A(B)^{\mathrm{gr}}\). The graded cotangent complex is the simplicial graded \(B\)-module \[ L^{\mathrm{gr}}_{B/A}=\Omega_{P/A}\otimes_PB, \] regarded by normalization as a complex of graded \(B\)-modules in degrees \(\leq0\), and as an object of \(D(B)_{\mathrm{gr}}\). Its terms \(\Omega_{P_n/A}\otimes_{P_n}B=B^{(X_n)}\) are free graded \(B\)-modules (Lemmas 1.2 (1) and 3.1 (2)), and the maps \(P_0\to B\) give an augmentation \(L^{\mathrm{gr}}_{B/A}\to\Omega_{B/A}\). It is functorial in commutative squares of graded rings, through the maps of standard resolutions.

Theorem 3.3. Let \(A\to B\) be a map of graded rings.

  1. The map \(\Phi L^{\mathrm{gr}}_{B/A}\to L_{\Phi B/\Phi A}\) induced by \(\theta_{B/A}\) (Lemma 2.4) is a quasi-isomorphism, natural in \(A\to B\).
  2. The augmentation induces an isomorphism of graded modules \(H^0(L^{\mathrm{gr}}_{B/A})\cong\Omega_{B/A}\).
  3. If \(B=A[X]\) with \(X\in T^{\mathrm{gr}\geq0}\), the augmentation \(L^{\mathrm{gr}}_{B/A}\to\Omega_{B/A}\) is a quasi-isomorphism.
  4. \(k^0(L^{\mathrm{gr}}_{B/A})=L_{B^0/A^0}\) as simplicial \(B^0\)-modules, compatibly with the augmentations.
  5. If \(A\) is a ring and \(B\in A\text{-}\mathrm{Alggr}^+\), then \(\theta^+_B\) induces a quasi-isomorphism \(\Omega_{P^+_A(B)/A}\otimes B\to L^{\mathrm{gr}}_{B/A}\). Consequently \(L^{\mathrm{gr}}_{B/A}\) is represented by a complex of free graded \(B\)-modules generated in degrees \(\geq1\), \(k^0(L^{\mathrm{gr}}_{B/A})\) is acyclic, and \(k^1\) of the augmentation, followed by Lemma 3.1 (4), is an isomorphism \(k^1(L^{\mathrm{gr}}_{B/A})\cong B^1\) in \(D(A)\), natural in \(B\).

Proof. (1) By Lemmas 2.3 and 2.4 (1), source and target are the sheafifications of the sectionwise constructions, and the map is the sheafification of the maps induced by \(\theta\) over each \(U\in\mathcal C\). Over \(U\), \(\Phi P_{A(U)}(B(U))^{\mathrm{gr}}\) and \(P_{\Phi A(U)}(\Phi B(U))\) are polynomial resolutions of \(B(U)\) over \(A(U)\) in the sense of Section 1 of The cotangent complex: their terms are polynomial algebras (Lemma 1.2 (2)), and their augmentations are surjective and quasi-isomorphisms (Proposition 2.2 (2) in the topos of sets), hence trivial Kan fibrations. By Lemma 2.1 and Theorem 2.2 there, the map of resolutions \(\theta\) induces a quasi-isomorphism of the complexes \(\Omega\otimes B(U)\). Sheafification is exact.

(2) Under \(\Phi\) the augmentation becomes the augmentation of \(L_{\Phi B/\Phi A}\) composed with the map of (1), because \(\theta_0\) is compatible with the maps to \(B\) and with differentials; the former induces an isomorphism on \(H^0\) [Tag 08US].

(3) By (1) it suffices to show that \(L_{R[E]/R}\to\Omega_{R[E]/R}\) is a quasi-isomorphism for a ring \(R\) and a sheaf of sets \(E\). By Lemma 2.3 this is the sheafification of the same map for the polynomial rings \(R(U)[E^{\mathrm{psh}}(U)]\), where it holds by (2.5) of The cotangent complex.

(4) By Proposition 2.2 (3), \(k^0P=P_{A^0}(B^0)\); by Lemma 3.1 (3), \(k^0\Omega_{P/A}=\Omega_{k^0P/A^0}\); and \(k^0(Y\otimes_PB)=k^0Y\otimes_{k^0P}B^0\) for graded modules in degrees \(\geq0\), by the argument of Lemma 3.1 (3).

(5) The quasi-isomorphism: as in (1), after \(\Phi\) the map \(\theta^+_B\) is, over each \(U\), a map of polynomial resolutions of \(B(U)\) over \(A(U)\) (Proposition 2.2 (1), (2) and Lemma 1.2 (3)), and quasi-isomorphisms are detected after \(\Phi\) (Proposition 1.5 (3)). The terms \(\Omega_{A[Y_n]/A}\otimes B=B^{(Y_n)}\) with \(Y_n\in T^{\mathrm{gr}+}\) are free and generated in degrees \(\geq1\), so their \(k^0\) is zero and their \(k^1\) is \(A^{(Y^1_n)}=k^1(P^+_n)\). By Proposition 2.2 (4) this is \(F_A(B^1)\), and its augmentation to \(B^1\), which is a quasi-isomorphism, is \(k^1\) of the augmentation to \(\Omega_{B/A}\) followed by Lemma 3.1 (4): the variable \(y\in Y_0^1=B^1\) goes to \(\mathrm dy\), and \(\mathrm dy\) to \(y\). The augmentations are compatible with \(\theta^+_B\), which gives the naturality. \(\square\)

Theorem 3.4 (graded transitivity). Let \(A\to B\to C\) be maps of graded rings. There is a distinguished triangle of \(D(C)_{\mathrm{gr}}\) \[ \begin{aligned}&L^{\mathrm{gr}}_{B/A}\otimes^{\mathbf L}_BC \to L^{\mathrm{gr}}_{C/A} \to L^{\mathrm{gr}}_{C/B}\\&\quad \to(L^{\mathrm{gr}}_{B/A}\otimes^{\mathbf L}_BC)[1],\end{aligned} \] whose first two maps are the functoriality maps, natural in the diagram \(A\to B\to C\). Its image under \(\Phi\), identified by Theorem 3.3 (1), is the transitivity triangle of \(\Phi A\to\Phi B\to\Phi C\), and its image under \(k^0\), identified by Theorem 3.3 (4), is the transitivity triangle of \(A^0\to B^0\to C^0\).

Proof. The construction is that of the proof of Theorem 4.1 of The cotangent complex, carried out with graded standard resolutions. First let \(B\to C\) be a monomorphism. Put \(P=P_A(B)^{\mathrm{gr}}\), \(Q=P_A(C)^{\mathrm{gr}}\), \(R=P_B(C)^{\mathrm{gr}}\), \(S=P_B(B)^{\mathrm{gr}}\), \(\overline Q=Q\otimes_PB\) and \(\overline R=R\otimes_SB\), with the maps \(P\to Q\to R\) and \(P\to S\to R\) given by functoriality. Consider the sequence of simplicial graded \(C\)-modules \[ \begin{aligned}&0 \to\Omega_{P/A}\otimes_PC\\&\quad \to\Omega_{Q/A}\otimes_QC\\&\quad \to\Omega_{\overline Q/B}\otimes_{\overline Q}C \to0\end{aligned} \tag{3.1} \] and the maps \(\Omega_{R/B}\otimes_RC\to\) \(\Omega_{\overline R/B}\otimes_{\overline R}C\leftarrow\) \(\Omega_{\overline Q/B}\otimes_{\overline Q}C\). By Lemma 2.3 all these are sheafifications of the corresponding constructions for the rings \(A(U)\to B(U)\to C(U)\), and \(B(U)\to C(U)\) is injective. For these rings the proof of Theorem 4.1 of The cotangent complex applies to graded standard resolutions without change: \(Q_n\) is a polynomial algebra over \(P_n\), the label set of \(P_n\) being a subset of that of \(Q_n\), and likewise \(R_n\) over \(S_n\); so (3.1) is split exact in each degree, \(\overline Q\) and \(\overline R\) are polynomial resolutions of \(C(U)\) over \(B(U)\) by Lemma 4.A there, and the two comparison maps are quasi-isomorphisms by Lemma 2.1 there. Sheafification being exact, (3.1) is exact and the comparison maps are quasi-isomorphisms of simplicial graded \(C\)-modules (Proposition 1.5 (3)). The first term of (3.1) is \(L^{\mathrm{gr}}_{B/A}\otimes_BC\), which computes \(L^{\mathrm{gr}}_{B/A}\otimes^{\mathbf L}_BC\) because the terms of \(L^{\mathrm{gr}}_{B/A}\) are free; the second is \(L^{\mathrm{gr}}_{C/A}\); the third is identified with \(L^{\mathrm{gr}}_{C/B}\) by the comparison maps. The triangle is the triangle of (3.1).

In general, \(B\to B\times C\), \(b\mapsto(b,\bar b)\), is a monomorphism of graded rings (the product graded degreewise), and the projection \(B\times C\to C\) is flat, \(C\) 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 projections induce maps \(L^{\mathrm{gr}}_{(B\times C)/A}\to L^{\mathrm{gr}}_{B/A}\oplus L^{\mathrm{gr}}_{C/A}\) and \(L^{\mathrm{gr}}_{(B\times C)/B}\to L^{\mathrm{gr}}_{B/B}\oplus L^{\mathrm{gr}}_{C/B}\), which are quasi-isomorphisms because their images under \(\Phi\) are, by Theorem 3.3 (1) and the product formula for cotangent complexes of sheaves of rings [Tag 08SY]. This identifies the terms after \(\otimes_{B\times C}C\) with those of the statement.

The maps \(\theta\) of Lemma 2.4 for \(P,Q,R,S\) are compatible with the maps between these resolutions, so \(\Phi\) carries (3.1) to the sequence that defines the ungraded triangle, and the comparison of Theorem 3.3 (1) is an isomorphism of triangles. For \(k^0\), Proposition 2.2 (3) and Lemma 3.1 (3) identify \(k^0\) of (3.1) and of the comparison maps with the corresponding constructions for \(A^0\to B^0\to C^0\). \(\square\)

4. Graded extensions

Proposition 4.1 (surjections). Let \(P\to B\) be a surjective map of graded rings with kernel \(K\). Then \(H^0(L^{\mathrm{gr}}_{B/P})=0\), and there is an isomorphism of graded \(B\)-modules \(H^{-1}(L^{\mathrm{gr}}_{B/P})\cong K/K^2\), natural in the surjection; hence \(\tau_{\geq-1}L^{\mathrm{gr}}_{B/P}\cong(K/K^2)[1]\).

Proof. \(H^0=\Omega_{B/P}=0\) by Theorem 3.3 (2). Let \(Q=P_P(B)^{\mathrm{gr}}\) and \(J=\ker(Q_0\to B)\), \(Q_0=P[U_{\mathrm{gr}}B]\). The rule \(\mathrm df\otimes b\mapsto(d_0f-d_1f)b\) defines a graded map \(L^{\mathrm{gr}}_{B/P}\to\) \(\operatorname{NL}(Q_0\to B)=[J/J^2\to\Omega_{Q_0/P}\otimes_{Q_0}B]\), the naive cotangent complex of the presentation in degrees \(-1,0\): the formula of [Tag 08US], which preserves degrees. The inclusion \(P\subseteq Q_0\) is a map of presentations of \(B\) over \(P\), from the presentation \(P\to B\) with no variables, and induces a graded map \(\operatorname{NL}(P\to B)=[K/K^2\to0]\) \(\to\operatorname{NL}(Q_0\to B)\). Both maps induce isomorphisms on \(H^{-1}\); this may be checked after \(\Phi\). For the second, \(\Phi\) of both complexes are naive cotangent complexes of presentations \(\Phi P[E]\to\Phi B\) by polynomial algebras on sheaves of sets [Tag 08TW], and a map of such presentations induced by a map of label sheaves \(f\colon E_1\to E_2\) over \(\Phi B\) is a quasi-isomorphism. Indeed, by the proof of [Tag 08TY] the inclusions of \(E_1\) and of \(E_2\) into \(E_1\amalg E_2\) induce quasi-isomorphisms; the map \(E_1\amalg E_2\to E_2\) equal to \(f\) on \(E_1\) and to the identity on \(E_2\) is a retraction of the second inclusion, so it also induces a quasi-isomorphism; and \(f\) is its composite with the first inclusion. For the first, \(\theta_{B/P}\) maps it to the corresponding map for the ungraded standard resolution, \(\theta_0\) carrying \(J\) into the kernel of \(\Phi P[U\Phi B]\to\Phi B\) and commuting with \(d_0,d_1\); the ungraded map is a quasi-isomorphism on \(\tau_{\geq-1}\) [Tag 08US], \(\theta\) is a quasi-isomorphism (Theorem 3.3 (1)), and the map of presentations induced by \(\theta_0\), which comes from the map of label sheaves \(\coprod_nB^n\to\Phi B\), is a quasi-isomorphism by the argument just given. Naturality is clear from the formulas. \(\square\)

Proposition 4.2 (first truncation). Let \(A\to B\) be a map of graded rings and \(P=A[X]\to B\) a surjective map of graded \(A\)-algebras, \(X\in T^{\mathrm{gr}\geq0}\), with kernel \(K\). There is an isomorphism in \(D(B)_{\mathrm{gr}}\) \[ \tau_{\geq-1}L^{\mathrm{gr}}_{B/A}\cong\big[K/K^2\xrightarrow{\ \mathrm d\ }\Omega_{P/A}\otimes_PB\big] \] (degrees \(-1,0\)), inducing on \(H^0\) the identification with \(\Omega_{B/A}\), and natural in the presentation.

Proof. The graded map \(L^{\mathrm{gr}}_{B/A}\to\operatorname{NL}(P_0\to B)\), \(P_0=A[U_{\mathrm{gr}}B]\) the degree-\(0\) term of \(P_A(B)^{\mathrm{gr}}\), given by the formula of [Tag 08US], induces isomorphisms on \(H^0\) and \(H^{-1}\): after \(\Phi\) this follows, as in the proof of Proposition 4.1, from [Tags 08US, 08TY] and Theorem 3.3 (1). Since its target is in degrees \([-1,0]\), it factors through an isomorphism \(\tau_{\geq-1}L^{\mathrm{gr}}_{B/A}\cong\operatorname{NL}(P_0\to B)\). The rule \(x\mapsto[\pi(x)]\), \(\pi\colon P\to B\), defines a graded algebra map \(P\to P_0\) over \(B\) (Lemma 1.2 (2)), hence a map of presentations, induced by the map of label sheaves \(\coprod_nX^n\to\coprod_nB^n\), and a graded map \(\operatorname{NL}(P\to B)\to\operatorname{NL}(P_0\to B)\), which is a quasi-isomorphism because its image under \(\Phi\) is (proof of Proposition 4.1). On \(H^0\) all these maps are compatible with the identifications with \(\Omega_{B/A}\) (Lemma 3.1 (2)). A map of presentations \(P\to P'\) of \(B\) over \(A\) commutes with the maps \(P\to P_0\) and \(P'\to P_0\), whence the naturality. \(\square\)

Lemma 4.3 (coefficients with acyclic components). Let \(B\) be a graded ring and \(I\) an injective \(B^0\)-module. Put \(r(I)=\prod_nr_n(I)\) (product of graded modules, degree by degree).

  1. \(\operatorname{Hom}_{B\text{-}\mathrm{Modgr}}(K,r(I))\cong\) \(\operatorname{Hom}_{B^0}(\textstyle\bigoplus_nK^n,I)\), so \(r(I)\) is injective in \(B\text{-}\mathrm{Modgr}\).
  2. \(k^m(r(I))=\mathcal Hom_{B^0}(B,I)\) for every \(m\), and \(H^i(T/V,\mathcal Hom_{B^0}(B,I)|_V)=0\) for all \(i>0\) and \(V\in T\).
  3. Every graded \(B\)-module \(K\) embeds into \(r(I)\) for an injective \(B^0\)-module \(I\).

Proof. (1) By Lemma 1.3 (2), \(\operatorname{Hom}(K,\prod_nr_nI)=\) \(\prod_n\operatorname{Hom}_{B^0}(K^n,I)=\) \(\operatorname{Hom}_{B^0}(\bigoplus_nK^n,I)\). The functor \(K\mapsto\bigoplus_nK^n\), to \(B^0\)-modules, is exact, so its right adjoint preserves injective objects [Tag 015Z]. (2) \(k^m(\prod_nr_nI)=\) \(\prod_{n\geq m}\mathcal Hom_{B^0}(B^{n-m},I)=\) \(\mathcal Hom_{B^0}(\bigoplus_{j\geq0}B^j,I)\). The restriction of \(I\) to \(T/V\) is injective, its restriction functor having the exact left adjoint "extension by zero"; so \(\operatorname{Ext}^i(B|_V,I|_V)=0\) for \(i>0\), and \(H^i(T/V,\mathcal Hom(B,I)|_V)=\) \(\operatorname{Ext}^i(B|_V,I|_V)\) because \(\mathcal Hom(B,I)=R\mathcal Hom(B,I)\) [Tag 08JA]. (3) Choose an injective \(B^0\)-module \(I\supseteq\bigoplus_nK^n\). The map \(K\to r(I)\) corresponding by (1) to this inclusion sends \(x\in K^m\) to \(b\mapsto bx\), which is injective (take \(b=1\)). \(\square\)

Lemma 4.4 (lifting generators). Let \(A\to B\) be a map of graded rings, \(E\to B\) a surjective map of graded \(A\)-algebras whose kernel \(K\) is square zero and has components with \(H^1(T/V,K^n|_V)=0\) for all \(n\) and \(V\in T\) (for instance \(K=r(I)\), Lemma 4.3). Every graded \(A\)-algebra map \(A[X]\to B\), \(X\in T^{\mathrm{gr}\geq0}\), lifts to a graded \(A\)-algebra map \(A[X]\to E\).

Proof. By Lemma 1.2 (2) we must lift the maps \(X^n\to B^n\) to \(E^n\). On the slice topos \(T/X^n\), the sheaf of lifts is a torsor under the restriction of \(K^n\): two lifts differ by a unique section of \(K^n\), and lifts exist locally because \(E^n\to B^n\) is an epimorphism. Torsors under an abelian sheaf are classified by \(H^1\) [Tag 03AJ], so this torsor is trivial and has a global section, which is the required lift. \(\square\)

Theorem 4.5 (graded extensions). Let \(A\to B\) be a map of graded rings and \(K\) a graded \(B\)-module. Let \(\operatorname{Exal}_A(B,K)_{\mathrm{gr}}\) be the group (under Baer sum) of isomorphism classes of graded square-zero extensions of \(B\) by \(K\): surjective maps of graded \(A\)-algebras \(E\to B\) with square-zero kernel identified with \(K\) as graded \(B\)-module. There are natural isomorphisms \[ \begin{gathered}\operatorname{Exal}_A(B,K)_{\mathrm{gr}}\\ \cong\operatorname{Ext}^1_B(L^{\mathrm{gr}}_{B/A},K)_{\mathrm{gr}},\\ \operatorname{Aut}(E/B,K)\cong\operatorname{Der}_A(B,K)_{\mathrm{gr}},\end{gathered} \] the Ext group being taken in \(D(B)_{\mathrm{gr}}\), and the first sends \(E\) to the composite \(\kappa_K(E)\colon\) \(L^{\mathrm{gr}}_{B/A}\to\) \(L^{\mathrm{gr}}_{B/E}\to\) \(\tau_{\geq-1}L^{\mathrm{gr}}_{B/E}\cong\) \(K[1]\) (Proposition 4.1).

Proof. The proof of Theorem 10.3 of The cotangent complex applies to graded objects with the following changes. Graded maps replace maps throughout; the transitivity triangle is Theorem 3.4 and the identification \(\tau_{\geq-1}L^{\mathrm{gr}}_{B/E}=K[1]\) is Proposition 4.1, which defines \(\kappa_K\), natural and additive by the same arguments; the presentation \(\mathcal P=A[U_{\mathrm{gr}}B]\to B\) is graded, with \(\tau_{\geq-1}L^{\mathrm{gr}}_{B/A}\cong[\mathcal N\to\mathcal F]\), \(\mathcal N=\mathcal K/\mathcal K^2\), \(\mathcal F=\Omega_{\mathcal P/A}\otimes B\) (Proposition 4.2); and injective coefficients \(I\) are replaced by the graded modules \(r(I)\) of Lemma 4.3, which are injective in \(B\text{-}\mathrm{Modgr}\), into which every graded module embeds, and over which polynomial generators lift (Lemma 4.4, replacing Lemma 10.2 there). With these, for \(K=r(I)\) a lift \(\mathcal P\to E\) exists, extensions are the pushouts of \(\mathcal P/\mathcal K^2\) along graded maps \(\mathcal N\to r(I)\) modulo maps \(\mathcal F\to r(I)\), and the exactness of \(\operatorname{Hom}_{\mathrm{gr}}(-,r(I))\) identifies this cokernel with \(\operatorname{Hom}_{\mathrm{gr}}(H^{-1}L^{\mathrm{gr}},r(I))=\operatorname{Ext}^1_{\mathrm{gr}}(L^{\mathrm{gr}},r(I))\); the five-term sequence (10.7) there, with graded derivations, and the same diagram chase give the isomorphism for every \(K\). Automorphisms of a graded extension are \(e\mapsto e+D(\bar e)\) with \(D\) a graded derivation, and \(\operatorname{Der}_A(B,K)_{\mathrm{gr}}=\) \(\operatorname{Hom}_{\mathrm{gr}}(\Omega_{B/A},K)=\) \(\operatorname{Ext}^0_B(L^{\mathrm{gr}}_{B/A},K)_{\mathrm{gr}}\) by Lemma 3.1 (1) and Theorem 3.3 (2). \(\square\)

Remark 4.6. The same formalism applies to graded simplicial rings, with the diagonal of bisimplicial standard resolutions, and it gives a graded transitivity triangle for maps of graded simplicial rings and a description of the graded cotangent complex of a graded simplicial algebra as corepresenting graded derivations in the homotopy category of graded simplicial algebras. These statements rest on the corresponding ungraded ones for simplicial rings, which are not proved in this course yet; they are recorded as course obligations. This lesson and the lesson on the Atiyah class use only constant graded rings.

5. Exercises

Exercise 5.1. Let \(k\) be a ring of \(T\) and \(A=k[t]\), \(t\) in degree \(1\). Compute \(\operatorname{Ext}^1_A(k\langle0\rangle,k\langle1\rangle)_{\mathrm{gr}}\) with Proposition 1.6 (2), and exhibit the corresponding extension of graded modules.

Exercise 5.2. Let \(A\) be a ring of \(T\) and \(B=A[x]/(x^2)\) with \(x\) in degree \(1\), so \(B\in A\text{-}\mathrm{Alggr}^+\). Using Proposition 4.2 with \(P=A[x]\), compute \(\tau_{\geq-1}L^{\mathrm{gr}}_{B/A}\) degree by degree. Check Theorem 3.3 (5) in degrees \(0\) and \(1\), and compute the cohomology of the degree-\(2\) part.

Exercise 5.3. Let \(T\) be the topos of sets, \(A\) a ring, \(B=A[x]/(x^2)\) as in Exercise 5.2 and \(K=A\langle2\rangle\). Compute \(\operatorname{Exal}_A(B,K)_{\mathrm{gr}}\) with Theorem 4.5 and describe the extensions.

Exercise 5.4. Let \(A\) be a graded ring, \(B=A[X]\) with \(X\in T^{\mathrm{gr}\geq0}\), and \(K\) a graded \(B\)-module. Show that \(\operatorname{Exal}_A(B,K)_{\mathrm{gr}}\cong\prod_nH^1(T/X^n,K^n)\), describe the class of an extension, and deduce that the group vanishes when \(T\) is the topos of sets.

6. Complete solutions

Solution 5.1. The sequence \(0\to A(-1)\xrightarrow{t}A\to k\langle0\rangle\to0\) is a flat resolution. Applying \(A^0\otimes_A-=k\otimes_A-\) gives \(k\langle1\rangle\xrightarrow{0}k\langle0\rangle\), because \(t\) acts by zero on \(k\). Its degree-\(1\) part is \(k\) in cohomological degree \(-1\); so \(k^1(A^0\otimes^{\mathbf L}_Ak\langle0\rangle)=k[1]\) in \(D(k)\). By Proposition 1.6 (2), \(\operatorname{Ext}^1_A(k\langle0\rangle,k\langle1\rangle)_{\mathrm{gr}}=\) \(\operatorname{Hom}_{D(A)_{\mathrm{gr}}}(k\langle0\rangle,k\langle1\rangle[1])\cong\) \(\operatorname{Hom}_{D(k)}(k[1],k[1])=\) \(\Gamma(T,k)\), the ring of global sections of \(k\).

The extension \(0\to k\langle1\rangle\to E\to k\langle0\rangle\to0\) with \(E=A/(t^2)\), the generator of \(k\langle1\rangle\) going to \(t\), does not split: a graded section would send \(1\) to \(1\), and then \(t\cdot1=t\neq0\) contradicts \(t\cdot1=0\) in \(k\langle0\rangle\). Its class is a basis of the \(\Gamma(T,k)\)-module above. To see this, apply the exact functor \(k^1(A^0\otimes^{\mathbf L}_A-)\) to the triangle of the extension. From \(0\to A(-2)\xrightarrow{t^2}A\to E\to0\) we get \(A^0\otimes^{\mathbf L}_AE=[k\langle2\rangle\xrightarrow{0}k\langle0\rangle]\), whose degree-\(1\) part is \(0\); from \(0\to A(-2)\xrightarrow{t}A(-1)\to k\langle1\rangle\to0\) we get \(k^1(A^0\otimes^{\mathbf L}_Ak\langle1\rangle)=k\), the counit of the adjunction being the identity of \(k\). So the map that Proposition 1.6 (2) attaches to the class of \(E\) is the connecting map of a triangle whose middle term is zero, an isomorphism \(k[1]\to k[1]\), that is, a unit of \(\Gamma(T,k)\). (By Lemma 1.3 (2), the Ext group into \(r_1(k)\) is instead \(\operatorname{Ext}^1_k(k^1k\langle0\rangle,k)=0\).)

Solution 5.2. Here \(K=(x^2)\subseteq P\), \(K/K^2=B\,\overline{x^2}\) is free on a generator of degree \(2\), \(\Omega_{P/A}\otimes_PB=B\,\mathrm dx\) is free on a generator of degree \(1\), and \(\mathrm d(x^2)=2x\,\mathrm dx\). So \(\tau_{\geq-1}L^{\mathrm{gr}}_{B/A}\cong[B\,\overline{x^2}\to B\,\mathrm dx]\), \(\overline{x^2}\mapsto2x\,\mathrm dx\). In degree \(0\) both terms vanish; in degree \(1\) the complex is \([0\to A\,\mathrm dx]\), which is \(A\cong B^1\), as in Theorem 3.3 (5). In degree \(2\) it is \([A\,\overline{x^2}\to A\,x\,\mathrm dx]\), multiplication by \(2\): \(H^{-1}\) is the \(2\)-torsion \(A[2]\) and \(H^0=A/2A\). In degree \(3\), \(B^1\overline{x^2}=A\,x\,\overline{x^2}\to B^2\mathrm dx=0\), so \(H^{-1}=A\) and \(H^0=0\); in degrees \(\geq4\) both terms vanish. These are the groups \(H^0\) and \(H^{-1}\) of \(L^{\mathrm{gr}}_{B/A}\) in each degree; the truncation says nothing about \(H^{-2}\) and below.

Solution 5.3. In the topos of sets free modules are projective, so \(\operatorname{Ext}^1\) of the two-term complex of free modules of Solution 5.2 into \(K\) is the cokernel of \(\operatorname{Hom}_{\mathrm{gr}}(B\,\mathrm dx,K)\to\operatorname{Hom}_{\mathrm{gr}}(B\,\overline{x^2},K)\). A graded map out of \(B\,\mathrm dx\) sends \(\mathrm dx\) into \(K^1=0\), and a graded map out of \(B\,\overline{x^2}\) is the choice of the image of \(\overline{x^2}\) in \(K^2=A\). So \(\operatorname{Exal}_A(B,K)_{\mathrm{gr}}\cong A\). The extension of class \(c\in A\) is \(E_c=A\oplus Ax\oplus Ay\), with \(x\) in degree \(1\), \(y\) in degree \(2\), \(x^2=cy\) and \(xy=y^2=0\); it is \(A[x]/(x^3)\) for \(c=1\). A graded isomorphism of extensions fixes \(x\), because \(K^1=0\), so the \(E_c\) are pairwise non-isomorphic, as the theorem says.

Solution 5.4. By Theorem 4.5 and Theorem 3.3 (3), \(\operatorname{Exal}_A(B,K)_{\mathrm{gr}}\cong\operatorname{Ext}^1_B(\Omega_{B/A},K)_{\mathrm{gr}}\), and \(\Omega_{B/A}=B^{(X)}\) (Lemma 3.1 (2)). By Lemma 1.2 (1), \(\operatorname{Hom}_{\mathrm{gr}}(B^{(X)},M)=\prod_n\Gamma(T/X^n,M^n)\) for graded \(B\)-modules \(M\). A graded injective module is a direct summand of a module \(r(I)\) (Lemma 4.3 (1), (3)), so its components have no higher cohomology on any slice (Lemma 4.3 (2)). Computing with an injective resolution of \(K\) in \(B\text{-}\mathrm{Modgr}\), we get \(\operatorname{Ext}^1_{\mathrm{gr}}(B^{(X)},K)=\prod_nH^1(T/X^n,K^n)\). Directly: an extension \(E\) is split by a graded algebra section \(B\to E\) exactly when the maps \(X^n\to B^n\) lift to \(E^n\) (Lemma 1.2 (2)); the lifts of \(X^n\to B^n\) form a torsor under \(K^n\) on \(T/X^n\) (proof of Lemma 4.4), and the family of the classes of these torsors is the class of \(E\). In the topos of sets every torsor under a group on a set has a section, so the group vanishes: every graded extension of a graded polynomial algebra over the topos of sets splits. In a general topos the torsors of lifts need not be trivial.

Sources

The graded cotangent complex, its degree-zero and degree-one parts, the graded transitivity triangle and the classification of graded extensions are due to Illusie, in his work on the cotangent complex, where they serve to express deformations of modules through deformations of graded algebras. The cotangent complex of sheaves of rings and the standard resolution follow the AI-integrated Stacks project, Chapter The Cotangent Complex, and the lesson The cotangent complex; the standard simplicial object of an adjunction is due to Godement.