Equivariant extensions and deformations

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

Let \(G\) be a category scheme over a scheme \(S\), for instance a flat group scheme. A first-order thickening \(Y\to Y'\) of schemes with an action of \(G\) is an equivariant extension when its ideal behaves well along the nerve: the thickening of the nerves \(N(G,Y)\to N(G,Y')\) has an ideal that is cartesian along the faces fixing the first vertex. Equivariant extensions of \(Y\) over a \(G\)-scheme \(Z\) are exactly the extensions of the simplicial scheme \(N(G,Y)\) over \(N(G,Z)\) whose ideal has this property, and so they are classified by the first Ext group of the equivariant cotangent complex. In the same way, deforming a morphism of \(G\)-schemes equivariantly over an equivariant extension of its target is obstructed by a class in the second Ext group of the equivariant cotangent complex, and this class is the product of the equivariant Kodaira–Spencer class with the class of the extension. A spectral sequence over the nerve splits the obstruction into the obstruction to deforming the underlying morphism of schemes and further terms coming from the action. The lesson also computes the truncated equivariant cotangent complex of a closed immersion into a smooth \(G\)-scheme by the conormal sequence, and treats the deformation of a morphism between two given equivariant extensions.

Prerequisites. Equivariant cotangent complexes (all of it; its notation is used throughout); Deformations of diagrams of schemes (Definitions 2.1, 3.1, 4.1, Propositions 1.2, 1.3, 2.3, Theorems 2.4 and 4.3, Lemmas 3.2–3.4, Corollary 3.6, Construction 4.2); Quasi-coherent and perfect complexes on big fppf sites (Proposition 1.1, Corollary 2.2, Lemma 4.2); Equivariant sheaves on simplicial topoi (Propositions 2.2 and 2.3, Section 6); Mixed Hom complexes and Ext groups of equivariant modules (Theorem 3.3 and the map (3.1), Corollary 3.4); Cohomology of equivariant modules (Proposition 5.1); Décalage, nerves and the homotopy type of a category (Sections 2.3–2.5). From the programme lesson The cotangent complex: Theorems 1.1 and 2.2 and formula (2.5), Theorem 4.1 with its construction (4.3) for a polynomial middle ring, Proposition 5.1, Theorem 6.1 with formula (6.5), Theorem 7.2, Proposition 10.1 and Theorem 10.3 with its class (10.6). From the AI-integrated Stacks project: differentials of smooth morphisms and their base change [Stacks, Tag 02G1, Tag 01V0, Tag 01VB]; closed immersions and fibre products [Tag 01JU]; triangles [Tag 05QR, Tag 014A, cones Tag 014E, Tag 0152, Tag 06XS]; Tor independence on affine opens [Tag 0FXV]; complexes on affine schemes and maps out of complexes of projectives [Tag 06Z0, Tag 064B].

Conventions

The notation is that of Equivariant cotangent complexes: \(T\) is the big fppf topos of \(S\), \(G\) a category scheme over \(S\), \([G,X]\) the action category of a \(G\)-scheme \(X\), \(N(G,X)\) its nerve, \(v_0\colon N_n(G,X)\to X\) the first vertex, \(B[G,X]^{\mathrm o}\) the classifying category of the opposite category, \(\operatorname{ner}\) the nerve functor of Proposition 1.5 there, \(f_B^*\) the pullback of Lemma 3.1 there, and \(t_{[}L^G_{X/Y}\) the equivariant cotangent complex. For a module \(J\) of \(B[G,Y]^{\mathrm o}\) whose underlying module is quasi-coherent, \(\underline J\) is the termwise quasi-coherent module of \(\operatorname{Top}(N(G,Y)_{\mathrm{Zar}})\) with \(\varepsilon^*\underline J=\operatorname{ner}(J)\) (Proposition 3.2 (1) there); it is cartesian along every \(\alpha\) with \(\alpha(0)=0\). For a morphism \(f\colon X\to Y\) of \(G\)-schemes, \(\varphi=N(G,f)\), and the module of \(\operatorname{Top}(N(G,X)_{\mathrm{Zar}})\) belonging to \(f_B^*J\) is \(\varphi^*\underline J\), because \(\operatorname{ner}(f_B^*J)=\varphi_{\mathrm{big}}^*\varepsilon^*\underline J=\varepsilon^*\varphi^*\underline J\) (Lemma 3.1 there and Construction 4.1 of the lesson on big fppf sites). For \(E\) one of the projective systems of that lesson, \(\operatorname{Ext}^i(E,J)\) is \(\operatorname{Hom}(t_{[n}E,J[i])\) for \(n\leq-i\), independent of \(n\). Extensions, deformations and their ideals are those of Deformations of diagrams of schemes; "cartesian along \(\alpha\)" is as in the lesson on equivariant cotangent complexes.

1. Closed immersions into smooth G-schemes

Let \(f\colon X\to Y\) be a morphism of diagrams of schemes indexed by \(I\), of type the identity, each \(f_i\) a closed immersion with ideal \(\mathcal I_i\subseteq\mathcal O_{Y_i}\). As in Definition 2.1 of the lesson on deformations of diagrams, the \(\mathcal I_i\) form an ideal \(\mathcal I\) of \(\mathcal O_Y\) on \(\operatorname{Top}(Y_{\mathrm{Zar}})\), and its conormal module \(\mathcal C_{X/Y}=f^*\mathcal I\), with the levels \(\mathcal I_i/\mathcal I_i^2\), is a termwise quasi-coherent \(\mathcal O_X\)-module. Let \(g\colon Y\to Z\) be a morphism of diagrams of type \(v\), each \(g_j\) smooth. Put \[ \begin{gathered}D_{X/Y/Z}=\bigl[\mathcal C_{X/Y}\xrightarrow{\ d\ }f^*\Omega_{Y/Z}\bigr],\\ d(\bar x)=1\otimes dx,\end{gathered} \] in degrees \(-1\) and \(0\). By [Tag 02G1], \(\Omega_{Y/Z}\) has finite locally free levels.

Lemma 1.1. In the transitivity triangle of the ring maps \(\mathbb Z\to\mathbb Z[t]\to\mathbb Z\), \(t\mapsto0\), the third map \(K\colon L_{\mathbb Z/\mathbb Z[t]}\to(\mathbb Z\,dt)[1]\) is an isomorphism, and on \(H^{-1}\) it sends the class \(\bar t\) of \(t\) in \((t)/(t^2)\) to \(dt\).

Proof. The middle term \(L_{\mathbb Z/\mathbb Z}\) is \(0\) by formula (2.5) of The cotangent complex, and the first term is \(\Omega_{\mathbb Z[t]/\mathbb Z}\otimes\mathbb Z=\mathbb Z\,dt\) in degree \(0\). A distinguished triangle with a zero vertex has its other map an isomorphism [Tag 05QR], so \(K\) is an isomorphism.

To compute \(K\) on \(H^{-1}\), use the construction of the triangle in Theorem 4.1 there for a polynomial middle ring. Let \(Q\to\mathbb Z\) be a polynomial resolution over \(\mathbb Z[t]\), for instance the standard one; every \(Q_n\) is a polynomial ring over \(\mathbb Z\). The triangle is the triangle [Tag 0152] of the short exact sequence of complexes associated with the sequence of simplicial modules \[ \begin{aligned}0&\to\mathbb Z\,dt\to\Omega_{Q/\mathbb Z}\otimes_Q\mathbb Z\\&\to\Omega_{Q/\mathbb Z[t]}\otimes_Q\mathbb Z\to0,\end{aligned} \] (4.3) there, with constant first term. By the convention of [Tag 0152], the third map of the triangle of a short exact sequence \(0\to E\xrightarrow{a}F\to G\to0\) of complexes is \(-p\circ q^{-1}\), with \(q\colon C(a)\to G\) and \(p\colon C(a)\to E[1]\) the maps from the cone \(C(a)^n=F^n\oplus E^{n+1}\), whose differential is \((y,x)\mapsto(dy+a(x),-dx)\) [Tag 014E]. It induces on cohomology the connecting map of the snake lemma: a cocycle of \(G\) of degree \(n\), lifted to \(y\) in \(F^n\), is the image under \(q\) of the cocycle \((y,-a^{-1}(dy))\) of \(C(a)\), which \(-p\) sends to \(a^{-1}(dy)\). These connecting maps may be computed on the unnormalized complexes of the simplicial modules, the normalized complex being a natural direct summand with acyclic complement (Theorem 1.1 there); the differential from degree \(-1\) to degree \(0\) is \(d_0-d_1\).

The term of degree \(-1\) of \(\Omega_{Q/\mathbb Z[t]}\otimes_Q\mathbb Z\) is free over \(\mathbb Z\) on the \(dx\), \(x\) a variable of \(Q_1\) over \(\mathbb Z[t]\), so every class \(\xi\in H^{-1}(L_{\mathbb Z/\mathbb Z[t]})\) is represented by a cycle \(dy\otimes1\), with \(y\) a \(\mathbb Z\)-linear combination of variables. Put \(g=d_0y-d_1y\), which lies in the kernel \(I_0\) of the augmentation \(Q_0\to\mathbb Z\), the augmentation being a map of simplicial rings. By formula (6.5) of Theorem 6.1 there, the identification \(H^{-1}(L_{\mathbb Z/\mathbb Z[t]})=(t)/(t^2)\) sends \(\xi\) to the element whose image under the map of presentations \(\mathbb Z[t]\to Q_0\), \((t)/(t^2)\to I_0/I_0^2\), is the class of \(g\). The connecting map sends \(\xi\) to \((d_0-d_1)(dy\otimes1)=dg\otimes1\) in \(\Omega_{Q_0/\mathbb Z}\otimes_{Q_0}\mathbb Z\), an element of the image of \(\mathbb Z\,dt\). The map \(I_0/I_0^2\to\Omega_{Q_0/\mathbb Z}\otimes_{Q_0}\mathbb Z\), \(\bar g\mapsto dg\otimes1\), is well defined. For the class \(\xi\) corresponding to \(\bar t\), the classes of \(g\) and of \(t\) in \(I_0/I_0^2\) agree, so \(dg\otimes1=dt\otimes1\), and \(K\) sends \(\bar t\) to \(dt\). \(\square\)

As in the proof of Lemma 1.1, the third map of the triangle [Tag 0152] of a short exact sequence of complexes induces on cohomology the connecting maps of the snake lemma. For a map \(\delta\colon M\to N\) of modules, the triangle \(N\to[M\xrightarrow{\delta}N]\to M[1]\xrightarrow{\partial}N[1]\) of the short exact sequence \(0\to N\to[M\to N]\to M[1]\to0\), with \(M\) in degree \(-1\), therefore has \(H^{-1}(\partial)=\delta\): the connecting homomorphism lifts \(m\in M\) to the degree \(-1\) of the middle complex and applies its differential.

Proposition 1.2. In this situation:

1. \(L_{Y/Z}\cong\Omega_{Y/Z}[0]\) and \(Lf^*L_{Y/Z}=f^*\Omega_{Y/Z}[0]\); \(L_{X/Y}\in D^{\leq-1}\), with \(H^{-1}(L_{X/Y})=\mathcal C_{X/Y}\).
2. On \(H^{-1}\), the Kodaira–Spencer class \(K(X/Y/Z)\) is \(d\colon\mathcal C_{X/Y}\to f^*\Omega_{Y/Z}\).
3. There is an isomorphism \(\psi\colon\tau_{\geq-1}L_{X/Z}\to D_{X/Y/Z}\) in \(D(\mathcal O_X)\) compatible with the maps from \(f^*\Omega_{Y/Z}\), whose composite with \(D_{X/Y/Z}\to\mathcal C_{X/Y}[1]\) is the map \(\tau_{\geq-1}L_{X/Z}\to\tau_{\geq-1}L_{X/Y}=\mathcal C_{X/Y}[1]\). It induces the identity of \(\Omega_{X/Z}\) on \(H^0\), and the identity of \(\ker d\subseteq\mathcal C_{X/Y}\) on \(H^{-1}\). Two such isomorphisms differ by a map of the form \(\tau_{\geq-1}L_{X/Z}\to\) \(\Omega_{X/Z}\xrightarrow{\ \sigma\ }\) \(f^*\Omega_{Y/Z}\to\) \(D_{X/Y/Z}\).
4. If \(L_{X/Z}\) has no cohomology in degrees \(<-1\), for instance if every \(g_jf_j\) is a local complete intersection morphism, then \(L_{X/Z}\cong D_{X/Y/Z}\) in the same way.
5. Let \(\mathcal A=(gf)^{-1}\mathcal O_Z\), \(\mathcal A'=f^{-1}\mathcal O_Y\) and \(\mathcal B=\mathcal O_X\), rings of \(\operatorname{Top}(X_{\mathrm{Zar}})\). For a surjection \(\mathcal R\to\mathcal B\) of \(\mathcal A\)-algebras with kernel \(\mathcal K\) put \(\operatorname{NL}_{\mathcal A}(\mathcal R)=\bigl[\mathcal K/\mathcal K^2\xrightarrow{d}\Omega_{\mathcal R/\mathcal A}\otimes_{\mathcal R}\mathcal B\bigr]\), in degrees \(-1\) and \(0\). Let \(\mathcal A[\mathcal B]\) and \(\mathcal A'[\mathcal B]\) be the polynomial algebras on the underlying sheaf of sets of \(\mathcal B\). Then \(\operatorname{NL}_{\mathcal A}(\mathcal A[\mathcal B])\) is the naive complex of \(\mathcal B\) over \(\mathcal A\), and \(\operatorname{NL}_{\mathcal A}(\mathcal A')=D_{X/Y/Z}\). The maps \(\mathcal A[\mathcal B]\to\mathcal A'[\mathcal B]\) and \(\mathcal A'\to\mathcal A'[\mathcal B]\) of \(\mathcal A\)-algebras over \(\mathcal B\) induce quasi-isomorphisms \[ \begin{aligned}&\operatorname{NL}_{\mathcal A}(\mathcal A[\mathcal B])\xrightarrow{\kappa}\operatorname{NL}_{\mathcal A}(\mathcal A'[\mathcal B])\\&\quad\xleftarrow{\iota}D_{X/Y/Z},\end{aligned} \] and \(\psi_{\mathrm{can}}=\iota^{-1}\kappa\), composed with the isomorphism \(\tau_{\geq-1}L_{X/Z}\cong\operatorname{NL}_{\mathcal A}(\mathcal A[\mathcal B])\) of Theorem 6.1 of The cotangent complex, is an isomorphism as in (3). It involves no choices and is natural for commutative diagrams of data \(X\to Y\to Z\) as above.

Proof. (1) By Proposition 1.2 of the lesson on deformations of diagrams, the levels of \(L_{Y/Z}\) are the \(L_{Y_j/Z_{v(j)}}\), which have cohomology \(\Omega\) in degree \(0\) only (Propositions 5.1 and 10.1 of The cotangent complex). So the canonical map \(L_{Y/Z}\to H^0(L_{Y/Z})[0]=\Omega_{Y/Z}[0]\) is an isomorphism, and since \(\Omega_{Y/Z}\) has flat levels, \(Lf^*\) of it is \(f^*\Omega_{Y/Z}\). For the surjections \(f_i^{-1}\mathcal O_{Y_i}\to\mathcal O_{X_i}\), the sheaf form of Theorem 6.1 there gives \(H^0=0\), \(H^{-1}=\mathcal I_i/\mathcal I_i^2\), and no cohomology in positive degrees; these identifications are natural, hence compatible with the structure maps.

(2) Write \(\delta=H^{-1}(K(X/Y/Z))\). It is a map of sheaves, so it suffices to compute it on local sections. Let \(U\) be an open subset of some \(Y_i\) and \(x\in\mathcal I_i(U)\). The ring maps \(\mathbb Z\to\mathbb Z[t]\to\mathbb Z\), as constant sheaves on \(U\cap X_i\), map to \((gf)^{-1}\mathcal O_Z\to f^{-1}\mathcal O_Y\to\mathcal O_X\) at the level \(i\) by \(t\mapsto x\); since \(x\) vanishes on \(X_i\) this is a map of triples. The transitivity triangle of the constant triple is the inverse image of that of Lemma 1.1, and the triangle is natural in maps of triples (Theorem 4.1 of The cotangent complex, in its sheaf form). On \(H^{-1}\) the map of triples sends \(\bar t\) to \(\bar x\), and on \(H^0\) of the first terms it sends \(dt\) to \(dx\). So \(\delta(\bar x)=dx\). The classes \(\bar x\) generate \(\mathcal C_{X/Y}\), so \(\delta=d\).

(3) Write \(A=f^*\Omega_{Y/Z}[0]\), \(B=L_{X/Z}\) and \(C=L_{X/Y}\), with the triangle \(A\to B\to C\xrightarrow{K}A[1]\). Since \(A[1]\in D^{\geq-1}\), \(K\) factors uniquely as \(C\to\tau_{\geq-1}C=\mathcal C_{X/Y}[1]\xrightarrow{K'}A[1]\) [Tag 06XS], and \(H^{-1}(K')=d\) by (2). Choose a distinguished triangle \(A\to B'\to\tau_{\geq-1}C\xrightarrow{K'}A[1]\) and a morphism of triangles from \(A\to B\to C\) to it which is the identity on \(A\) and the truncation on \(C\). On cohomology it is an isomorphism in degrees \(\geq-1\) by the five lemma (in degree \(-1\) the map \(H^{-2}(C)\to H^{-2}(\tau_{\geq-1}C)=0\) is surjective and \(H^0(A)\) is unchanged), and \(B'\in D^{[-1,0]}\); so \(B\to B'\) induces \(\tau_{\geq-1}B\cong B'\). The triangle of the short exact sequence \(0\to A\to D\to\mathcal C_{X/Y}[1]\to0\), where \(D=D_{X/Y/Z}\), is \(A\to D\to\mathcal C_{X/Y}[1]\xrightarrow{\partial}A[1]\) with \(H^{-1}(\partial)=d\). Maps \(\mathcal C_{X/Y}[1]\to A[1]\) are determined by their effect on \(H^{-1}\) [Tag 06XS], so \(K'=\partial\). The identities of \(A\) and of \(\mathcal C_{X/Y}[1]\) therefore extend to a morphism of triangles \(\psi\colon\tau_{\geq-1}B\to D\), an isomorphism by [Tag 014A]. The effect on cohomology follows from the compatibilities: \(H^0\) of both is the cokernel of \(d\) on \(f^*\Omega_{Y/Z}\), and \(H^{-1}\) of both maps injectively to \(\mathcal C_{X/Y}\) with image \(\ker d\). If \(\psi'\) is another such isomorphism, \(\psi-\psi'\) vanishes on \(A\) and after composition with \(D\to\mathcal C_{X/Y}[1]\); so it factors as \(A\to D\) after a map \(\tau_{\geq-1}B\to A\), which factors through \(H^0(\tau_{\geq-1}B)=\Omega_{X/Z}\) because \(\tau_{\geq-1}B\in D^{\leq0}\) and \(A\) sits in degree \(0\) [Tag 06XS].

(4) Then \(L_{X/Z}=\tau_{\geq-1}L_{X/Z}\). For a local complete intersection, \(L_{X/Z}\) has tor amplitude in \([-1,0]\) by Theorem 7.2 of The cotangent complex on affine opens, hence no cohomology outside \([-1,0]\).

(5) The kernel of \(\mathcal A'\to\mathcal B\) is \(f^{-1}\mathcal I\), with \(f^{-1}\mathcal I/(f^{-1}\mathcal I)^2=\mathcal C_{X/Y}\), and \(\Omega_{\mathcal A'/\mathcal A}\otimes_{\mathcal A'}\mathcal B=f^*\Omega_{Y/Z}\), as differentials commute with inverse images; so \(\operatorname{NL}_{\mathcal A}(\mathcal A')=D_{X/Y/Z}\). Quasi-isomorphisms are detected on the stalks at the points of the levels, and all the constructions commute with stalks. Let \(R\to R'\to S\) be the stalks of \(\mathcal A\to\mathcal A'\to\mathcal B\) at a point; \(R'\) is a localization of a smooth \(R\)-algebra, and \(R'\to S\) is surjective with kernel \(J\). Let \(J'\) be the kernel of \(R'[S]\to S\). A choice of a preimage in \(R'\) of every element of \(S\) gives a map \(\rho\colon R'[S]\to R'\) of \(R'\)-algebras over \(S\) with \(\rho\iota=1\). The maps \(\iota\rho\) and \(1\) of \(R'[S]\) agree on \(R'\) and over \(S\), so \(\iota\rho-1\) induces an \(R'\)-derivation \(R'[S]\to J'/J'^2\), hence a map \(h\colon\Omega_{R'[S]/R}\otimes S\to J'/J'^2\), and \(\iota\rho-1=dh+hd\) on \(\operatorname{NL}_R(R'[S])\). So \(\iota\) is a homotopy equivalence with inverse \(\rho\). The composite \(\rho\kappa\) induces the identity of \(\Omega_{S/R}\) on \(H^0\), both complexes having \(H^0=\Omega_{S/R}\) by the conormal sequence. Followed by the map \(\operatorname{NL}_R(R')\to J/J^2[1]\) that forgets the differentials, \(\rho\kappa\) induces on \(H^{-1}\) the functoriality map \(H^{-1}(L_{S/R})\to H^{-1}(L_{S/R'})=J/J^2\), because Theorem 6.1 there is compatible with ring maps. In the transitivity triangle of \(R\to R'\to S\) (Theorem 4.1 there), \(L_{R'/R}=\Omega_{R'/R}[0]\) with \(\Omega_{R'/R}\) free (Proposition 5.1 there), and by (2) the connecting map on \(H^{-1}\) is \(d\). So the functoriality map is injective with image \(\ker d=H^{-1}(\operatorname{NL}_R(R'))\). Hence \(\rho\kappa\) and \(\kappa\) are quasi-isomorphisms.

For the compatibilities of (3), use the isomorphisms of Theorem 6.1. The map \(f^*\Omega_{Y/Z}\to\tau_{\geq-1}L_{X/Z}\) becomes the map \(\operatorname{NL}_{\mathcal A}(\mathcal A[\mathcal A'])\otimes_{\mathcal A'}\mathcal B\to\operatorname{NL}_{\mathcal A}(\mathcal A[\mathcal B])\), \([a]\mapsto[\bar a]\), \(\bar a\) being the image of \(a\) in \(\mathcal B\), preceded by the inverse of the projection onto \(H^0=f^*\Omega_{Y/Z}\); the projection is a quasi-isomorphism by (1), \(\Omega_{\mathcal A'/\mathcal A}\) being flat. Followed by \(\kappa\), this map differs from the inclusion \(f^*\Omega_{Y/Z}\to D_{X/Y/Z}\) followed by \(\iota\) by \(dh'+h'd\), where \(h'(d[a])\) is the class of \([\bar a]-a\) in \(\mathcal J'/\mathcal J'^2\), \(\mathcal J'\) the kernel of \(\mathcal A'[\mathcal B]\to\mathcal B\). Indeed, in degree \(0\), \(d[\bar a]-da=d([\bar a]-a)\); in degree \(-1\), a polynomial \(j\) in the variables \([a]\) that vanishes in \(\mathcal A'\) goes to \(j([\bar a])=j([\bar a])-j(a)\equiv\) \(\sum_a\frac{\partial j}{\partial[a]}\,([\bar a]-a)\) modulo \(\mathcal J'^2\), which is \(h'(dj)\). The map \(\tau_{\geq-1}L_{X/Z}\to\tau_{\geq-1}L_{X/Y}=\mathcal C_{X/Y}[1]\) becomes \(\kappa\) followed by the map \(\operatorname{NL}_{\mathcal A}(\mathcal A'[\mathcal B])\to\operatorname{NL}_{\mathcal A'}(\mathcal A'[\mathcal B])\) that passes to differentials over \(\mathcal A'\), and the inverse of the homotopy equivalence \(\operatorname{NL}_{\mathcal A'}(\mathcal A')=\mathcal C_{X/Y}[1]\to\operatorname{NL}_{\mathcal A'}(\mathcal A'[\mathcal B])\). That map composed with \(\iota\) is the projection \(D_{X/Y/Z}\to\mathcal C_{X/Y}[1]\) followed by this homotopy equivalence. So \(\psi_{\mathrm{can}}\) is compatible with the maps from \(f^*\Omega_{Y/Z}\) and to \(\mathcal C_{X/Y}[1]\), and the rest of (3) follows as there. Every map is induced by a functorial construction, which gives the naturality. \(\square\)

The isomorphism of (3) is not unique in general (Exercise 6.3); it is characterized by the stated compatibilities only up to the maps described there. The isomorphism \(\psi_{\mathrm{can}}\) of (5) is one of them, singled out by its construction.

Proposition 1.3. Let \(X\xrightarrow{f}Y\xrightarrow{g}Z\) be morphisms of \(G\)-schemes, \(f\) a closed immersion with ideal \(\mathcal I\), \(g\) smooth, with \(f\) and \(G\), and \(g\) and \(G\), Tor independent.

1. \(N(G,f)\) is levelwise a closed immersion, \(N(G,g)\) is levelwise smooth, and the terms of \(D=D_{N(G,X)/N(G,Y)/N(G,Z)}\) are cartesian along every \(\alpha\) with \(\alpha(0)=0\); their levels are \(v_0^*\mathcal C_{X/Y}\) and \(v_0^*f^*\Omega_{Y/Z}\).
2. There is a complex \(D^G_{X/Y/Z}\) of modules of \(B[G,X]^{\mathrm o}\), unique up to unique isomorphism, with \(\operatorname{ner}(D^G_{X/Y/Z})=\varepsilon^*D\); its underlying complex is \(\varepsilon^*D_{X/Y/Z}\).
3. \(t_{[-1}L^G_{X/Z}\cong D^G_{X/Y/Z}\) in \(D^b(B[G,X]^{\mathrm o})\), by an isomorphism whose nerve is \(\varepsilon^*\) of an isomorphism as in Proposition 1.2 (3) for \(N(G,X)\to N(G,Y)\to N(G,Z)\). There is a unique one whose nerve is \(\varepsilon^*\psi_{\mathrm{can}}\), with \(\psi_{\mathrm{can}}\) of Proposition 1.2 (5); it is natural in the data.
4. If \(gf\) is a local complete intersection morphism, \(L^G_{X/Z}\cong D^G_{X/Y/Z}\) (Proposition 2.5 (2) of the lesson on equivariant cotangent complexes).

Proof. (1) By Lemma 1.3 of the lesson on equivariant cotangent complexes, \(N_n(G,X)=N_n(G,Y)\times_YX\), so \(N_n(G,f)\) is a closed immersion with ideal \(\mathcal I_n\) generated by the pullback of \(\mathcal I\) [Tag 01JU]; and \(N_n(G,Y)=N_n(G,Z)\times_ZY\), so \(N_n(G,g)\) is smooth [Tag 01VB] and \(\Omega_{N_n(G,Y)/N_n(G,Z)}=v_0^*\Omega_{Y/Z}\) [Tag 01V0]. On an affine open \(\operatorname{Spec}A\) of \(Y\) with \(\mathcal I=\tilde I\) there, and an affine open \(\operatorname{Spec}A'\) of \(N_n(G,Y)\) over it, \(\operatorname{Tor}^A_1(A/I,A')=0\) by the Tor independence of \(f\) and \(G\) [Tag 0FXV]. Tensoring \(0\to I\to A\to A/I\to0\) with \(A'\) gives \(I\otimes_AA'\cong IA'\), hence \[ \begin{aligned}&IA'/(IA')^2 =I\otimes_AA'/IA'\\&\quad =I/I^2\otimes_{A/I}A'/IA',\end{aligned} \] so \(\mathcal I_n/\mathcal I_n^2=v_0^*\mathcal C_{X/Y}\). The structure maps of \(D\) along \(\alpha\) with \(\alpha(0)=0\) are compatible with these isomorphisms, since \(v_0\circ N(\alpha)=v_0\); so they are isomorphisms.

(2) The terms of \(\varepsilon^*D\) have quasi-coherent levels and are cartesian along every \(\alpha\) with \(\alpha(0)=0\), so each is the nerve of a module of \(B[G,X]^{\mathrm o}\) (Proposition 2.3 (1) of the lesson on equivariant sheaves, for modules, through Proposition 1.5 of the lesson on equivariant cotangent complexes), and the differential lifts uniquely because \(\operatorname{ner}\) is fully faithful on modules (Proposition 2.2 (2) there). The level \(0\) of a nerve is the underlying module.

(3) Proposition 1.2 (3) applies to \(N(G,X)\to N(G,Y)\to N(G,Z)\) by (1) and gives \(\tau_{\geq-1}L\cong D\), where \(L=L_{N(G,X)/N(G,Z)}\). By (0.2) of the lesson on equivariant cotangent complexes, \(t_{[-1}L\varepsilon^*L\cong t_{[-1}L\varepsilon^*\tau_{\geq-1}L\cong t_{[-1}L\varepsilon^*D\). The terms of \(D\) in degree \(0\) are flat, so in the long exact sequence of \(L\varepsilon^*\) of the triangle of \(D\), \(H^0(L\varepsilon^*D)=\operatorname{coker}(\varepsilon^*d)\) and \(H^{-1}(L\varepsilon^*D)=\ker(\varepsilon^*d)\). So the canonical map \(L\varepsilon^*D\to\varepsilon^*D\) induces isomorphisms on \(H^{-1}\) and \(H^0\), and \(t_{[-1}L\varepsilon^*D\cong\varepsilon^*D=\operatorname{ner}(D^G)\). On the other side \(t_{[-1}L\varepsilon^*L=\operatorname{ner}(t_{[-1}L^G_{X/Z})\) by Theorem 2.4 there, \(gf\) and \(G\) being Tor independent (Proposition 3.3 (1) there). Full faithfulness of \(\operatorname{ner}\) on \(D^b\) (Proposition 1.5 (3) there) gives the isomorphism.

(4) By Theorem 7.2 of The cotangent complex, \(L_{X/Z}\) has tor amplitude in \([-1,0]\), so \(L^G_{X/Z}=t_{[-1}L^G_{X/Z}\) by Proposition 2.5 (2) of the lesson on equivariant cotangent complexes, and (3) applies. \(\square\)

2. Equivariant extensions

Lemma 2.1 (nerves of action categories). Let \(G\) be an internal category of \(T\).

1. The functor \(L\mapsto N(G,L)\), from \(G\)-sheaves to simplicial objects of \(T\) over \(N(G)\), is fully faithful.
2. A simplicial object \(q\colon N\to N(G)\) over \(N(G)\) is isomorphic over \(N(G)\) to some \(N(G,L)\) if and only if every map \((q_n,v_0)\colon N_n\to N_n(G)\times_{G_0}N_0\) is an isomorphism. Then \(L=N_0\), over \(G_0\) by \(q_0\), with the action \(g\cdot x=d_0(\ell(g,x))\), where \(\ell\) is the inverse of \((q_1,v_0)\) and \(v_0=d_1\) on \(N_1\).

Proof. By the Yoneda lemma it suffices to argue with points with values in an object of \(T\), that is, with sets. (1) A morphism \(w\colon N(G,L)\to N(G,M)\) over \(N(G)\) commutes with the first vertex, so \(w_n(g;x)=(g;u(x))\) with \(u=w_0\) (Lemma 1.3 of the lesson on equivariant cotangent complexes). Compatibility with \(d_0\) on \(N_1\) gives \(u(g\cdot x)=g\cdot u(x)\). Conversely, for an equivariant \(u\) the maps \((g;x)\mapsto(g;u(x))\) commute with the formula of that Lemma 1.3.

(2) The condition holds for \(N(G,L)\) by that Lemma 1.3. Conversely, let it hold. Then \(q_0(g\cdot x)=\) \(q_0d_0\ell(g,x)=\) \(d_0q_1\ell(g,x)=\) \(d_0(g)\). For a point \(x\) of \(N_0\), \(s_0(x)\) satisfies \(q_1s_0(x)=s_0q_0(x)\) and \(d_1s_0(x)=x\), so \(\ell(1,x)=s_0(x)\) and \(1\cdot x=d_0s_0(x)=x\). For composable \(g\), \(h\) and \(x\) over the source of \(g\), let \(\sigma\in N_2\) be the point with \(q_2(\sigma)=(g,h)\) and first vertex \(x\). Its edges are \(d_2\sigma=\ell(g,x)\), \(d_0\sigma=\ell(h,g\cdot x)\) and \(d_1\sigma=\ell(hg,x)\), because each is determined by its image in \(N_1(G)\) and its source, the sources being \(d_1d_2\sigma=x\), \(d_1d_0\sigma=d_0d_2\sigma=g\cdot x\) and \(d_1d_1\sigma=d_1d_2\sigma=x\). Comparing targets, \(d_0d_1\sigma=d_0d_0\sigma\) gives \((hg)\cdot x=h\cdot(g\cdot x)\). So \(N_0\) is a \(G\)-sheaf. For a point \(z\) of \(N_n\) with \(q_n(z)=(g_1,\ldots,g_n)\) and first vertex \(x\), induction along the edges shows that its vertex \(k\) is \(g_k\cdots g_1\cdot x\); hence the maps \(z\mapsto(q_n(z);v_0(z))\), isomorphisms by hypothesis, commute with the simplicial maps, by the formula of that Lemma 1.3. \(\square\)

Definition 2.2. Let \(g\colon Y\to Z\) be a morphism of \(G\)-schemes. An equivariant \(Z\)-extension of \(Y\) is a triple \((Y',j,g')\) of a \(G\)-scheme \(Y'\) and morphisms of \(G\)-schemes \(j\colon Y\to Y'\) and \(g'\colon Y'\to Z\) with \(g'j=g\), such that

1. \(j\) is a first-order thickening, and
2. the ideal \(\mathcal J\) of the levelwise first-order thickening \(N(G,j)\colon N(G,Y)\to N(G,Y')\) is cartesian along every \(\alpha\colon[m]\to[n]\) with \(\alpha(0)=0\).

Here \(N_n(G,j)\) is the base change of \(j\) along the first vertex \(N_n(G,Y')\to Y'\) (Lemma 1.3 of the lesson on equivariant cotangent complexes), a closed immersion with ideal of square zero [Tag 01JU]. For a module \(J\) of \(B[G,Y]^{\mathrm o}\) whose underlying module is quasi-coherent, an equivariant \(Z\)-extension of \(Y\) by \(J\) is one with an isomorphism \(\mathcal J\cong\underline J\). Isomorphisms are isomorphisms of \(G\)-schemes compatible with \(j\), \(g'\) and these identifications.

Proposition 2.3. Let \(g\colon Y\to Z\) be a morphism of \(G\)-schemes.

1. If \((Y',j,g')\) satisfies condition 1 of Definition 2.2 and \(g\) and \(g'\) are Tor independent of \(G\), then it satisfies condition 2.
2. \(N(G,-)\) is an equivalence from the groupoid of equivariant \(Z\)-extensions of \(Y\) by \(J\) to the groupoid of \(N(G,Z)\)-extensions of \(N(G,Y)\) by \(\underline J\) (Definition 2.1 of the lesson on deformations of diagrams).

Proof. (1) Consider first the first vertex \(v_0\colon[0]\to[n]\). The square formed by \(N_n(G,j)\), \(j\) and the first vertices is cartesian, so by Lemma 3.2 (2) of the lesson on deformations of diagrams the structure map \(v_0^*\mathcal J_0\to\mathcal J_n\) is an isomorphism as soon as \(Y\) and \(N_n(G,Y')\) are Tor independent over \(Y'\) [Tag 0FXV]. This follows from Lemma 2.2 of the lesson on equivariant cotangent complexes, with \(Q=Z\), \(P=N_n(G,Z)\), \(R=Y'\) and \(R'=Y\): \(P\) is Tor independent of \(R\) and of \(R'\) over \(Q\), because \(g'\) and \(g\) are Tor independent of \(G\), and \(P\times_QR=N_n(G,Y')\). For a general \(\alpha\) with \(\alpha(0)=0\), the first vertex of \(N_n\) is the first vertex of \(N_m\) composed with \(N(\alpha)\), and the structure maps of the ideal compose (Lemma 3.4 (1) there); so the structure map along \(\alpha\) is an isomorphism.

(2) For an equivariant \(Z\)-extension, \((N(G,Y'),N(G,j),N(G,g'))\) is an \(N(G,Z)\)-extension of \(N(G,Y)\) with ideal \(\mathcal J\cong\underline J\). By Lemma 2.1 (1), morphisms of simplicial schemes between such nerves over \(N(G,Z)\) are the nerves of morphisms of \(G\)-schemes, so \(N(G,-)\) is fully faithful on these groupoids. Let \((N,\iota,q')\) be an \(N(G,Z)\)-extension of \(N(G,Y)\) by \(\underline J\). For every \(n\), consider the square formed by the first vertices \(N_n(G,Y)\to Y\) and \(N_n(G,Z)\to Z\) and the maps \(N_n(G,g)\) and \(g\); it is cartesian (Lemma 1.3 there). The thickenings \(N_n(G,Y)\to N_n\) and \(Y\to N_0\), with the first vertex \(N_n\to N_0\) of \(N\) and the maps \(q'_n\), \(q'_0\), form an extension of this square in which \(N_n(G,Z)\) and \(Z\) are not thickened, and the first vertex of \(N\) is a deformation of that of \(N(G,Y)\), since \(\underline J\) is cartesian along \(v_0\). By Corollary 3.6 of the lesson on deformations of diagrams, \(N_n=\) \(N_n(G,Z)\times_ZN_0=\) \(N_n(G)\times_{G_0}N_0\). So Lemma 2.1 (2) applies to \(N\to N(G,Z)\to N(G)\): \(N\cong N(G,Y')\) with \(Y'=N_0\), a scheme with an action of \(G\), and by Lemma 2.1 (1) \(\iota\) and \(q'\) are \(N(G,j)\) and \(N(G,g')\) for morphisms of \(G\)-schemes. Condition 1 holds at the level \(0\), and condition 2 because \(\underline J\) is cartesian along every \(\alpha\) with \(\alpha(0)=0\). \(\square\)

Theorem 2.4 (classification of equivariant extensions). Let \(g\colon Y\to Z\) be a morphism of \(G\)-schemes, Tor independent of \(G\), and \(J\) a module of \(B[G,Y]^{\mathrm o}\) whose underlying module is quasi-coherent. The isomorphism classes of equivariant \(Z\)-extensions of \(Y\) by \(J\) are in bijection with \(\operatorname{Ext}^1(t_{[}L^G_{Y/Z},J)\), the trivial extension corresponding to \(0\), and the automorphism group of each is isomorphic to \(\operatorname{Ext}^0(t_{[}L^G_{Y/Z},J)\).

Proof. By Proposition 2.3 (2), these are the isomorphism classes and automorphism groups of \(N(G,Z)\)-extensions of \(N(G,Y)\) by \(\underline J\). By Theorem 2.4 of the lesson on deformations of diagrams they correspond to \(\operatorname{Ext}^1\) and \(\operatorname{Ext}^0\) of \(L_{N(G,Y)/N(G,Z)}\) into \(\underline J\), and by Proposition 3.2 (2) of the lesson on equivariant cotangent complexes these are \(\operatorname{Ext}^1(t_{[}L^G_{Y/Z},J)\) and \(\operatorname{Ext}^0(t_{[}L^G_{Y/Z},J)\). The trivial extension of \(N(G,Y)\) has class \(0\) and is the nerve of the trivial equivariant extension. \(\square\)

We write \(e(j)\in\operatorname{Ext}^1(t_{[}L^G_{Y/Z},J)\) for the class of \((Y',j,g')\), the image of the class \(e(N(G,Y'))\) of Theorem 2.4 of the lesson on deformations of diagrams.

3. Equivariant deformations of a morphism

Definition 3.1. Let \(f\colon X\to Y\) and \(g\colon Y\to Z\) be morphisms of \(G\)-schemes and \((Y',j,g')\) an equivariant \(Z\)-extension of \(Y\) by \(J\). An equivariant deformation of \(f\) over \(Y'\) is an equivariant \(Y'\)-extension \((X',i,f')\) of \(X\) for the morphism \(jf\colon X\to Y'\) (Definition 2.2) such that the map \(f^*\mathcal J_0\to\mathcal I_0\) to the ideal of \(i\), induced by \(f'\), is an isomorphism (Definition 4.1 of the lesson on deformations of diagrams, for the morphism of schemes \(f\)). Isomorphisms are isomorphisms of equivariant \(Y'\)-extensions.

Proposition 3.2.

1. If \((X',i,f')\) is an equivariant deformation of \(f\) over \(Y'\), then \((N(G,X'),N(G,i),N(G,f'))\) is a deformation of \(\varphi=N(G,f)\) over \(N(G,Y')\).
2. Conversely, every deformation of \(\varphi\) over \(N(G,Y')\) is isomorphic to the nerve of an equivariant deformation of \(f\) over \(Y'\), unique up to unique isomorphism. So \(N(G,-)\) is an equivalence between the groupoids of equivariant deformations of \(f\) over \(Y'\) and of deformations of \(\varphi\) over \(N(G,Y')\).

Proof. (1) The ideals \(\mathcal I\) of \(N(G,i)\) and \(\mathcal J\) of \(N(G,j)\) are cartesian along every \(\alpha\) with \(\alpha(0)=0\). At the level \(n\), the two composites \(N_n(G,X)\to X\to Y\) and \(N_n(G,X)\to N_n(G,Y)\to Y\) are equal, and the structure maps of the ideal of the thickened square compose (Lemma 3.4 (1) of the lesson on deformations of diagrams): the map \(\varphi_n^*\mathcal J_n\to\mathcal I_n\) composed with \(\varphi_n^*\) of the isomorphism \(v_0^*\mathcal J_0\to\mathcal J_n\) equals the isomorphism \(v_0^*\mathcal I_0\to\mathcal I_n\) composed with \(v_0^*\) of the isomorphism \(f^*\mathcal J_0\to\mathcal I_0\). So \(\varphi^*\mathcal J\to\mathcal I\) is an isomorphism at every level.

(2) Let \((N',\iota,\varphi')\) be a deformation of \(\varphi\) over \(N(G,Y')\). Its ideal is isomorphic to \(\varphi^*\mathcal J\), which is cartesian along every \(\alpha\) with \(\alpha(0)=0\) because \(\mathcal J\) is. So it is an \(N(G,Y')\)-extension of \(N(G,X)\), for the morphism \(N(G,jf)\), by the module \(\varphi^*\underline J\) of \(f_B^*J\) (Conventions), and Proposition 2.3 (2), applied to \(jf\colon X\to Y'\), gives an equivariant \(Y'\)-extension \((X',i,f')\) of \(X\) with nerve \(N'\), unique up to unique isomorphism; the map \(f^*\mathcal J_0\to\mathcal I_0\) is the level \(0\) of \(\varphi^*\mathcal J\to\mathcal I\), an isomorphism. \(\square\)

Theorem 3.3 (equivariant deformations). Let \(f\colon X\to Y\) be a morphism of \(G\)-schemes, Tor independent of \(G\), \(g\colon Y\to Z\) a morphism of \(G\)-schemes and \((Y',j,g')\) an equivariant \(Z\)-extension of \(Y\) by \(J\). Let \(\omega(f,j)\in\operatorname{Ext}^2(t_{[}L^G_{X/Y},f_B^*J)\) correspond to the obstruction \(\omega(\varphi,N(G,Y'))\) of Construction 4.2 of the lesson on deformations of diagrams, through Proposition 3.2 (2) of the lesson on equivariant cotangent complexes.

1. An equivariant deformation of \(f\) over \(Y'\) exists if and only if \(\omega(f,j)=0\).
2. If \(\omega(f,j)=0\), the isomorphism classes of equivariant deformations of \(f\) over \(Y'\) form a torsor under \(\operatorname{Ext}^1(t_{[}L^G_{X/Y},f_B^*J)\), and the automorphism group of each is isomorphic to \(\operatorname{Ext}^0(t_{[}L^G_{X/Y},f_B^*J)\).
3. If \(g\) and \(G\) are Tor independent, then \(\omega(f,j)\) is the product of \(f_B^*e(j)\) and the equivariant Kodaira–Spencer class \(K_G(X/Y/Z)\): under the identifications of Propositions 3.2 and 3.3 of the lesson on equivariant cotangent complexes, \(\omega(\varphi,N(G,Y'))=\) \(\varphi^*e(N(G,Y'))[1]\circ\) \(K(N(G,X)/N(G,Y)/N(G,Z))\).

Proof. By Proposition 3.2, equivariant deformations of \(f\) over \(Y'\), with their isomorphisms, are the deformations of \(\varphi\) over the extension \(N(G,Y')\) of \(N(G,Y)\), whose ideal \(\mathcal J\) is termwise quasi-coherent. Theorem 4.3 (1), (2) of the lesson on deformations of diagrams describes these with \(\operatorname{Ext}^i(L_{N(G,X)/N(G,Y)},\varphi^*\underline J)\), \(i=0,1,2\), which are the groups \(\operatorname{Ext}^i(t_{[}L^G_{X/Y},f_B^*J)\) by Proposition 3.2 (2) of the lesson on equivariant cotangent complexes and the Conventions. This proves (1) and (2). For (3), \(N(G,Y')\) is an \(N(G,Z)\)-extension of \(N(G,Y)\) through \(N(G,g')\), and Theorem 4.3 (3) of the lesson on deformations of diagrams gives the displayed formula; the identifications of Propositions 3.2 and 3.3 of the lesson on equivariant cotangent complexes carry \(e(N(G,Y'))\) to \(e(j)\) and the Kodaira–Spencer class of the nerves to \(K_G(X/Y/Z)\), and they are compatible with composition, being induced by the functors \(\operatorname{ner}\), \(L\varepsilon^*\) and truncations. \(\square\)

4. The spectral sequence of the obstruction

For a \(G\)-scheme \(X\), let \(\lambda_p\colon N_p(G,X)\to X\) be the last vertex, \((g_1,\ldots,g_p;x)\mapsto g_p\cdots g_1\cdot x\).

Proposition 4.1. Let \(f\colon X\to Y\) be a morphism of \(G\)-schemes, Tor independent of \(G\), \(J\) a module of \(B[G,Y]^{\mathrm o}\) whose underlying module is \(\varepsilon^*J_0\) with \(J_0\) quasi-coherent, and \(n\) an integer.

1. There is a spectral sequence \[ \begin{aligned}E_1^{p,k}&=\operatorname{Ext}^k_{N_p(G,X)}\bigl(\lambda_p^*t_{[n}L\varepsilon^*L_{X/Y},\\&\qquad v_0^*\varepsilon^*f^*J_0\bigr)\\&\Longrightarrow\operatorname{Hom}\bigl(t_{[n}L^G_{X/Y},\\&\qquad f_B^*J[p+k]\bigr),\end{aligned} \] where \(E_1^{\bullet,k}\) is the cochain complex of a cosimplicial group.
2. Its edge map onto \(E_1^{0,k}\) is the forgetful map \(\operatorname{Hom}(t_{[n}L^G_{X/Y},f_B^*J[k])\to\) \(\operatorname{Ext}^k_X(t_{[n}L\varepsilon^*L_{X/Y},\varepsilon^*f^*J_0)\), and for \(k\leq-n\) the target is \(\operatorname{Ext}^k_X(L_{X/Y},f^*J_0)\).
3. For \(n\leq-2\) and an equivariant \(Z\)-extension \((Y',j,g')\) of \(Y\) by \(J\), the edge map carries \(\omega(f,j)\) to the obstruction \(\omega(f,Y')\) of Theorem 4.3 of the lesson on deformations of diagrams for the morphism of schemes \(f\) and the extension \(Y'\) of \(Y\) by \(J_0\).

Proof. (1) By Proposition 5.1 of Cohomology of equivariant modules, Corollary 3.4 of Mixed Hom complexes and Ext groups of equivariant modules applies to the simplicial topos of the nerve of \([G,X]^{\mathrm o}\), with \(E=t_{[n}L^G_{X/Y}\) and \(F=f_B^*J\). Its \(E_1\) term is formed with the inverse images along the first and the last vertex of the strings of \([G,X]^{\mathrm o}\), which are the last vertex \(\lambda_p\) and the first vertex \(v_0\) of the strings of \([G,X]\). The underlying complex of \(t_{[n}L^G_{X/Y}\) is \(t_{[n}L\varepsilon^*L_{X/Y}\) and that of \(f_B^*J\) is \(\varepsilon^*f^*J_0\) (Proposition 2.5 (1) and Lemma 3.1 of the lesson on equivariant cotangent complexes).

(2) The spectral sequence comes from the filtration of the total complex of (3.2) of the lesson on mixed Hom complexes by the cosimplicial degree, so its edge map onto \(E_1^{0,k}\) is induced by the projection to the column \(0\), which is \(\operatorname{Hom}^\bullet(UE,UI)\) for an injective resolution \(I\) of \(F\). The map (3.1) there sends \(u\) to a family whose component in degree \(0\) is \(u\). So the composite is the forgetful map. For \(k\leq-n\), (0.1) of the lesson on equivariant cotangent complexes and Corollary 2.2 (1) of the lesson on big fppf sites identify the target with \(\operatorname{Ext}^k_X(L_{X/Y},f^*J_0)\).

(3) The forgetful functor is the level \(0\) of \(\operatorname{ner}\), and the identifications of Proposition 3.2 (2) of the lesson on equivariant cotangent complexes are compatible with restriction to the level \(0\), because \(\varepsilon\), truncations and the adjunction of Lemma 4.2 (2) of the lesson on big fppf sites are computed levelwise. Construction 4.2 of the lesson on deformations of diagrams commutes with the exact restriction to the level \(0\), which carries the rings, the triangle and the map \(\tilde c\) for \(\varphi\) and \(N(G,Y')\) to those for \(f\) and \(Y'\) (Proposition 1.2 (1) there). So the level \(0\) of \(\omega(\varphi,N(G,Y'))\) is \(\omega(f,Y')\). \(\square\)

For a monoid scheme \(G\), \(N_p(G,X)=G^p\times_SX\), and the low-degree terms are \(E_1^{0,2}=\operatorname{Ext}^2_X(L_{X/Y},f^*J_0)\), \(E_1^{1,1}\) formed on \(G\times_SX\), and \(E_1^{2,0}\) formed on \(G\times_SG\times_SX\). By (3), if the image of \(\omega(f,j)\) in \(E_1^{0,2}\) is non-zero, \(f\) has no deformation over \(Y'\) even as a morphism of schemes. The interpretation of the components of \(\omega(f,j)\) in \(E_\infty^{1,1}\) and \(E_\infty^{2,0}\) as the obstruction to extending the action of \(G\) to a deformation of \(X\) and as the associativity defect of an extended action is not proved in this lesson.

5. Morphisms between given equivariant extensions

Lemma 5.1. Let \(\mathcal Z\to\mathcal A\xrightarrow{u}\mathcal B\) be maps of rings of a topos, \(\mathcal E\) an extension of \(\mathcal Z\)-algebras of \(\mathcal A\) by an \(\mathcal A\)-module \(N\), \(\mathcal E'\) an extension of \(\mathcal Z\)-algebras of \(\mathcal B\) by a \(\mathcal B\)-module \(M\), and \(c\colon N\to M\) an \(\mathcal A\)-linear map. Call a lift of \(u\) a map of \(\mathcal Z\)-algebras \(\mathcal E\to\mathcal E'\) compatible with \(u\) and inducing \(c\) on the kernels.

1. Lifts of \(u\) correspond to isomorphisms between the two extensions of \(\mathcal A\) by \(M\): the push-out \(c_*\mathcal E\) and the pullback \(u^*\mathcal E'=\mathcal E'\times_{\mathcal B}\mathcal A\).
2. With \(\kappa\) the class (10.6) of The cotangent complex, \(\kappa(c_*\mathcal E)=c\circ\kappa(\mathcal E)\) and \(\kappa(u^*\mathcal E')=\kappa(\mathcal E')\circ\rho\) in \(\operatorname{Ext}^1_{\mathcal A}(L_{\mathcal A/\mathcal Z},M)\), where \(\rho\colon L_{\mathcal A/\mathcal Z}\to L_{\mathcal B/\mathcal Z}\) is the functoriality map and \(L_{\mathcal B/\mathcal Z}\), \(M\) are regarded as \(\mathcal A\)-modules.
3. A lift of \(u\) exists if and only if \(\kappa(\mathcal E')\circ\rho-c\circ\kappa(\mathcal E)=0\), and the lifts then form a torsor under \(\operatorname{Der}_{\mathcal Z}(\mathcal A,M)=\operatorname{Ext}^0_{\mathcal A}(L_{\mathcal A/\mathcal Z},M)\). Through \(\operatorname{Ext}^i_{\mathcal A}(L_{\mathcal A/\mathcal Z},M)=\) \(\operatorname{Ext}^i_{\mathcal B}(L_{\mathcal A/\mathcal Z}\otimes^{\mathbf L}_{\mathcal A}\mathcal B,M)\), the obstruction lies in \(\operatorname{Ext}^1_{\mathcal B}(L_{\mathcal A/\mathcal Z}\otimes^{\mathbf L}_{\mathcal A}\mathcal B,M)\).

Proof. (1) A lift \(\mathcal E\to\mathcal E'\) inducing \(c\) on the kernels is the same, by the universal property of the push-out \(c_*\mathcal E=(\mathcal E\oplus M)/\{(n,-c(n))\}\), as a map \(c_*\mathcal E\to\mathcal E'\) over \(u\) inducing the identity of \(M\), and, by that of the fibre product, as a map \(c_*\mathcal E\to u^*\mathcal E'\) of extensions of \(\mathcal A\) by \(M\) inducing the identity on \(M\) and on \(\mathcal A\). Such a map is an isomorphism by the five lemma.

(2) The map \(\mathcal E\to c_*\mathcal E\) is a map of extensions of \(\mathcal A\) with the coefficient map \(c\), and \(\kappa\) is natural in the coefficients (Theorem 10.3 of The cotangent complex). The projection \(u^*\mathcal E'\to\mathcal E'\) gives a map of triples \((\mathcal Z\to u^*\mathcal E'\to\mathcal A)\to(\mathcal Z\to\mathcal E'\to\mathcal B)\). The class (10.6) is the composite of the functoriality map \(L_{\mathcal A/\mathcal Z}\to L_{\mathcal A/u^*\mathcal E'}\) with the conormal truncation \(L_{\mathcal A/u^*\mathcal E'}\to M[1]\). Functoriality maps compose (Theorem 2.2 there), and the functoriality map \(L_{\mathcal A/u^*\mathcal E'}\to L_{\mathcal B/\mathcal E'}\) induces on \(H^{-1}\) the identity of \(M\), the kernel of both surjections; so it is compatible with the conormal truncations [Tag 06XS], and \(\kappa(u^*\mathcal E')=\kappa(\mathcal E')\circ\rho\).

(3) By (1) and Theorem 10.3 of The cotangent complex, a lift exists if and only if \(c_*\mathcal E\) and \(u^*\mathcal E'\) have the same class, and the lifts form a torsor under the automorphism group of \(c_*\mathcal E\), which is \(\operatorname{Der}_{\mathcal Z}(\mathcal A,M)=\operatorname{Ext}^0_{\mathcal A}(L_{\mathcal A/\mathcal Z},M)\). Use (2). The last identification is the adjunction between restriction of scalars and \(-\otimes^{\mathbf L}_{\mathcal A}\mathcal B\). \(\square\)

Proposition 5.2. Let \(X\xrightarrow{f}Y\xrightarrow{g}Z\) be morphisms of \(G\)-schemes with \(g\) and \(G\) Tor independent, \((Y',j,g')\) an equivariant \(Z\)-extension of \(Y\) by \(J\) and \((X',i,h')\) an equivariant \(Z\)-extension of \(X\) by \(f_B^*J\) (for \(gf\)). An equivariant deformation of \(f\) into \(f'\) is a morphism of \(G\)-schemes \(f'\colon X'\to Y'\) with \(f'i=jf\) and \(g'f'=h'\) whose induced map on ideals is the given identification \(f^*\mathcal J_0\cong\mathcal I_0\). There is a class \[ \omega\in\operatorname{Ext}^1\bigl(Lf_B^*t_{[}L^G_{Y/Z},f_B^*J\bigr), \] where the group is \(\operatorname{Hom}(t_{[n}Lf_B^*t_{[n}L^G_{Y/Z},f_B^*J[1])\) for \(n\leq-1\), whose vanishing is necessary and sufficient for the existence of an equivariant deformation of \(f\) into \(f'\). When \(\omega=0\), these deformations form a torsor under \(\operatorname{Ext}^0(Lf_B^*t_{[}L^G_{Y/Z},f_B^*J)\).

Proof. By Lemma 2.1 (1), the morphisms \(f'\) of the statement are the morphisms of simplicial schemes \(N(G,X')\to N(G,Y')\) with the corresponding properties. As in Proposition 2.3 (1) of the lesson on deformations of diagrams, applied to morphisms over \(\varphi\) instead of isomorphisms, these are the lifts of \(\varphi^\sharp\colon\varphi^{-1}\mathcal O_{N(G,Y)}\to\mathcal O_{N(G,X)}\) in the sense of Lemma 5.1, on \(\operatorname{Top}(N(G,X)_{\mathrm{Zar}})\), with \(\mathcal Z=(\varphi\circ N(G,g))^{-1}\mathcal O_{N(G,Z)}\), \(\mathcal A=\varphi^{-1}\mathcal O_{N(G,Y)}\), \(\mathcal B=\mathcal O_{N(G,X)}\), \(\mathcal E=\varphi^{-1}\mathcal O_{N(G,Y')}\), \(N=\varphi^{-1}\underline J\), \(\mathcal E'=\mathcal O_{N(G,X')}\), \(M=\varphi^*\underline J\) and \(c\) the canonical map: a map of rings over the identity of the underlying spaces is a morphism of locally ringed spaces at every level, since units are detected modulo the nilpotent ideals. By Lemma 5.1, the obstruction lies in \(\operatorname{Ext}^1(L_{\mathcal A/\mathcal Z}\otimes^{\mathbf L}_{\mathcal A}\mathcal B,\varphi^*\underline J)\), and \(L_{\mathcal A/\mathcal Z}\otimes^{\mathbf L}_{\mathcal A}\mathcal B=L\varphi^*L_{N(G,Y)/N(G,Z)}\) (proof of Proposition 1.3 of the lesson on deformations of diagrams), and the lifts form a torsor under the corresponding \(\operatorname{Ext}^0\). As in Proposition 3.2 (2) of the lesson on equivariant cotangent complexes, with Proposition 3.3 (1) there in place of (2.1), these groups are the groups of the statement. \(\square\)

6. Exercises

6.1. Let \(p\) be a prime. In Lemma 5.1 take the punctual topos, \(\mathcal Z=\mathbb Z\), \(\mathcal A=\mathcal B=\mathbb F_p\) with \(u\) the identity, \(\mathcal E=\mathbb Z/p^2\) (an extension of \(\mathbb F_p\) by \(N=p\mathbb Z/p^2\mathbb Z\)), \(\mathcal E'=\mathbb F_p[\epsilon]\), \(\epsilon^2=0\) (an extension by \(M=\mathbb F_p\epsilon\)), and \(c(p)=\epsilon\). Show directly that no lift exists, and identify the obstruction as \(-c\circ\kappa(\mathbb Z/p^2)\ne0\) in \(\operatorname{Ext}^1_{\mathbb F_p}(L_{\mathbb F_p/\mathbb Z},\mathbb F_p)\cong\mathbb F_p\).

6.2. Let \(G=S\) be the trivial monoid scheme. Show that equivariant \(Z\)-extensions of \(Y\) by \(J\) are the \(Z\)-extensions of \(Y\) by \(J_0\), that Theorem 2.4 is Theorem 2.4 of the lesson on deformations of diagrams for the morphism of schemes \(g\), and that the spectral sequence of Proposition 4.1 has \(E_1^{p,k}=\operatorname{Ext}^k_X(t_{[n}L\varepsilon^*L_{X/Y},\varepsilon^*f^*J_0)\) for every \(p\), with \(E_2^{p,k}=0\) for \(p>0\).

6.3. Let \(k\) be a field of characteristic \(\neq2\), \(Y=Z\times\mathbb A^1\) with \(Z=\operatorname{Spec}k\), and \(X=\operatorname{Spec}k[t]/(t^2)\subseteq Y\). Show that \(D_{X/Y/Z}\) is \((t^2)/(t^4)\to k[t]/(t^2)\,dt\), \(t^2\mapsto2t\,dt\), \(t^3\mapsto0\), and compute its cohomology. Let \(\theta\colon D\to D\) be the composite \(D\to H^0(D)=\Omega_{X/Z}\xrightarrow{\sigma}f^*\Omega_{Y/Z}\to D\), with \(\sigma(dt)=t\,dt\), and \(\iota\colon f^*\Omega_{Y/Z}\to D\), \(\pi\colon D\to\mathcal C_{X/Y}[1]\) the maps of its triangle. Show that \(\theta\neq0\), \(\theta\circ\iota=0\), \(\pi\circ\theta=0\) and \(\theta^2=0\) in \(D(\mathcal O_X)\). Deduce that if \(\psi\) is an isomorphism as in Proposition 1.2 (3), so is \((1+\theta)\psi\neq\psi\).

7. Complete solutions

6.1. A map of rings \(\mathbb Z/p^2\to\mathbb F_p[\epsilon]\) sends \(1\) to \(1\), hence \(p\) to \(p\cdot1=0\neq\epsilon=c(p)\); so no lift exists. Here \(u^*\mathcal E'=\mathbb F_p[\epsilon]\) is the split extension, with class \(0\), and \(c_*\mathcal E\) has class \(c\circ\kappa(\mathbb Z/p^2)\). The group \(\operatorname{Ext}^1_{\mathbb F_p}(L_{\mathbb F_p/\mathbb Z},\mathbb F_p)\) is \(\operatorname{Hom}(H^{-1}L_{\mathbb F_p/\mathbb Z},\mathbb F_p)=\) \(\operatorname{Hom}(p\mathbb Z/p^2\mathbb Z,\mathbb F_p)\cong\) \(\mathbb F_p\) [Tag 06XS], and \(\kappa(\mathbb Z/p^2)\) induces on \(H^{-1}\) the identification of the kernel \(p\mathbb Z/p^2\mathbb Z\) of \(\mathbb Z/p^2\to\mathbb F_p\) with the conormal module \(H^{-1}(L_{\mathbb F_p/\mathbb Z})\) (Theorem 6.1 of The cotangent complex), an isomorphism. So \(c\circ\kappa(\mathbb Z/p^2)\) induces the isomorphism \(p\mapsto\epsilon\) and is non-zero, and the obstruction \(0-c\circ\kappa(\mathbb Z/p^2)\) of Lemma 5.1 (3) is non-zero.

6.2. For \(G=S\), \([G,Y]\) has only identity arrows, \(N(G,Y)\) is the constant simplicial scheme \(Y\), every structure map is the identity, and condition 2 of Definition 2.2 holds. A module of \(B[G,Y]^{\mathrm o}\) is a module on \(T/Y\) (Exercise 5.3 of the lesson on equivariant cotangent complexes), so \(J=\varepsilon^*J_0\). By Proposition 2.3 (2), equivariant \(Z\)-extensions by \(J\) are the \(N(G,Z)\)-extensions of the constant simplicial scheme \(Y\), and by Lemma 2.1 these are the \(Z\)-extensions of \(Y\) by \(J_0\). By that Exercise 5.3, \(\operatorname{Ext}^i(t_{[}L^G_{Y/Z},J)=\operatorname{Ext}^i_Y(L_{Y/Z},J_0)\), which is the group of Theorem 2.4 of the lesson on deformations of diagrams for \(g\). Every \(\lambda_p\) and \(v_0\) is the identity of \(X\), so \(E_1^{p,k}\) is constant in \(p\), and the cosimplicial group is constant: its cochain complex has the differentials \(\sum_i(-1)^i\) of identity maps, that is \(0\) from even to odd and the identity from odd to even degrees; its cohomology vanishes in degrees \(p>0\).

6.3. \(\mathcal C_{X/Y}=(t^2)/(t^4)\), free over \(R=k[t]/(t^2)\) on \(t^2\), and \(f^*\Omega_{Y/Z}=R\,dt\). The differential sends \(t^2\) to \(2t\,dt\) and \(t^3=t\cdot t^2\) to \(t\cdot2t\,dt=0\). So \(H^{-1}(D)=k\,t^3\) and \(H^0(D)=R\,dt/k\,t\,dt=k\,dt=\Omega_{X/Z}\). The map \(\sigma\) is \(R\)-linear, since \(t\,dt=0\) in \(\Omega_{X/Z}\) and \(t\cdot t\,dt=0\).

\(X\) is affine and \(D\) is a bounded complex of free \(R\)-modules, so morphisms from \(D\) in \(D(\mathcal O_X)\) are morphisms in \(D(R)\) [Tag 06Z0], which are homotopy classes of chain maps [Tag 064B]. The map \(\theta\) is represented by the chain map that is \(0\) in degree \(-1\) and multiplication by \(t\) in degree \(0\), \(a\,dt+b\,t\,dt\mapsto a\,t\,dt\); it is a chain map because \(t\cdot2t\,dt=0\). A homotopy \(h\colon R\,dt\to\mathcal C_{X/Y}\) with \(\theta=dh+hd\) would need \(d(h(dt))=t\,dt\) in degree \(0\), so \(h(dt)=\tfrac12t^2+b\,t^3\), and \(h(d(t^2))=0\) in degree \(-1\); but \(h(d(t^2))=h(2t\,dt)=2t\,h(dt)=t^3\neq0\). So \(\theta\neq0\). Next, \(\theta\circ\iota\) is \(\iota\circ(\sigma\circ\mathrm{pr})\), with \(\sigma\circ\mathrm{pr}\colon R\,dt\to R\,dt\), \(dt\mapsto t\,dt\), \(t\,dt\mapsto0\). This equals \(d\circ\tau\) for \(\tau(dt)=\tfrac12t^2\), and \(\iota\circ d=0\) in the derived category, \(d\) being the third map of the triangle shifted by \(-1\). So \(\theta\circ\iota=0\). The composite \(\pi\circ\theta\) contains \(\pi\circ\iota=0\). Finally \(\theta^2\) contains \(\mathrm{pr}\circ\sigma\colon\Omega_{X/Z}\to\Omega_{X/Z}\), \(dt\mapsto[t\,dt]=0\). So \(1+\theta\) is an automorphism of \(D\) with inverse \(1-\theta\), compatible with \(\iota\) and \(\pi\), and \((1+\theta)\psi\) has the properties of Proposition 1.2 (3) and differs from \(\psi\) by \(\theta\psi\neq0\).

Sources

The equivariant extensions and deformations of schemes with an action of a category scheme, their classification by the equivariant cotangent complex, the product formula for the equivariant obstruction and its spectral decomposition, and the computation of the equivariant cotangent complex of a closed immersion into a smooth scheme are due to Illusie, in his work on the cotangent complex.