The co-Lie complex of a group scheme

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

The co-Lie complex of a group scheme $G$ over $S$, flat and locally of finite presentation, is the derived pullback of its cotangent complex along the unit section. It is a perfect complex on $S$ with tor amplitude in $[-1,0]$. Its cohomology in degree $0$ is the module of invariant differential forms, its cohomology in degree $-1$ vanishes exactly when $G$ is smooth, and its dual, the Lie complex, governs the deformations of $G$. This lesson proves its basic properties: it is the co-Lie complex of $G$ viewed as a torsor over $S$, it depends contravariantly on $G$ and commutes with base change, and an exact sequence of group schemes $1\to G\to H\to K\to1$ gives a distinguished triangle $\ell_K\to\ell_H\to\ell_G\to\ell_K[1]$. The triangle computes the co-Lie complexes of the groups of roots of unity and of Frobenius kernels.

Prerequisites. Torsors, co-Lie complexes and the Atiyah class (Definition 1.1, Lemma 1.2, Proposition 1.3, Theorem 3.3, Propositions 3.4 and 3.5); Flat group schemes are local complete intersections (Theorem 3.1, Corollary 3.2); Deformations of diagrams of schemes (Conventions, Propositions 1.3 and 1.4). From the programme lesson The cotangent complex: Theorem 2.2, Theorem 6.1 and formula (7.3) of Theorem 7.2. From the AI-integrated Stacks project: duals of perfect complexes [Stacks, Tag 08DQ]; tor amplitude and pullback [Tag 08H5, Tag 0D6D]; smooth morphisms [Tag 01V8, Tag 02G1]; differentials at a rational point [Tag 0B2D]; relative Frobenius and differentials [Tag 0CCC]; triangles [Tag 05QR, Tag 05QT, Tag 0149].

Conventions

$S$ is a scheme and $G$ a group scheme over $S$, flat and locally of finite presentation, with structure morphism $\pi\colon G\to S$, unit $e\colon S\to G$ and multiplication $m$. Cotangent complexes of morphisms of schemes are those of Section 10 of The cotangent complex, on Zariski sites. A homomorphism $u\colon G\to H$ of group schemes over $S$ satisfies $ue_G=e_H$. The notions of torsor and co-Lie complex of a torsor are those of Torsors, co-Lie complexes and the Atiyah class; $G$ acts on itself by left multiplication, and $\pi\colon G\to S$ is then the trivial $G_S$-torsor, with the section $e$.

1. Definition and first properties

Definition 1.1. The co-Lie complex of $G$ is $\ell_G=Le^*L_{G/S}\in D(\mathcal O_S),$ and its Lie complex is $\ell_G^\vee=R\mathcal Hom(\ell_G,\mathcal O_S)$. The co-Lie module is $\omega_G=e^*\Omega_{G/S}$, and $\operatorname{Lie}(G)=\mathcal Hom(\omega_G,\mathcal O_S)$.

Proposition 1.2.

$\ell_G$ is perfect with tor amplitude in $[-1,0]$, and $\ell_G^\vee$ is perfect with tor amplitude in $[0,1]$. Locally on $S$, $\ell_G$ is a complex of finite free modules in degrees $-1$ and $0$.
$H^0(\ell_G)=\omega_G$.
$\ell_G$ is the co-Lie complex $\ell_{G/S}$ of the trivial torsor $\pi\colon G\to S$: the isomorphism $\ell_{G/S}\cong Le^*L_{G/S}$ of Theorem 3.3 (1) of the lesson on torsors, for the section $e$, identifies them. In particular $a^*\ell_G\cong L_{N(G,G)/N(G,S)}$ for the augmentation $a\colon N(G,G)\to S$, and $\ell_G\cong R\varepsilon_*f^G_*t_{[n}L^G_{G/S}$ for $n\leq-1$. The same holds for $G$ acting on itself by right multiplication, a torsor under the opposite group $G^{\mathrm o}$, with the same section $e$.

Proof. (1) By Theorem 3.1 of Flat group schemes are local complete intersections, $L_{G/S}$ is locally a complex of finite free modules in degrees $-1$ and $0$; its derived pullback along $e$ is locally such a complex [Tag 08H5], since every point of $S$ has an open neighbourhood $V$ with $e(V)$ inside an open of $G$ where $L_{G/S}$ has this form. The dual of a perfect complex is perfect [Tag 08DQ], and locally it is the dual complex of finite free modules, in degrees $0$ and $1$.

(2) $L_{G/S}\in D^{\leq0}$ with $H^0=\Omega_{G/S}$ (Theorem 6.1 of The cotangent complex), and for objects of $D^{\leq0}$, $H^0(Le^*K)=e^*H^0(K)$ ((0.1) of the lesson on deformations of diagrams).

(3) This is Theorem 3.3 (1) of the lesson on torsors, for the torsor $\pi$ and its section $e$, together with its Proposition 3.5 (1); the argument does not depend on which of the two actions is used. $\square$

Proposition 1.3 (functoriality). A homomorphism $u\colon G\to H$ of group schemes over $S$, both flat and locally of finite presentation, gives a map $\begin{aligned}&\ell(u) \colon\ell_H =Le_G^*Lu^*L_{H/S}\\&\quad \longrightarrow Le_G^*L_{G/S} =\ell_G,\end{aligned}$ $Le_G^*$ of the functoriality map $Lu^*L_{H/S}\to L_{G/S}$ of the square formed by $u$ and the identity of $S$, through $Le_G^*Lu^*=Le_H^*$ [Tag 0D6D]. Then $\ell(\mathrm{id})=\mathrm{id}$ and $\ell(vu)=\ell(u)\circ\ell(v)$ for $v\colon H\to K$. On $H^0$, $\ell(u)$ is the map $\omega_H\to\omega_G$ induced on differentials.

Proof. Functoriality maps compose (Theorem 2.2 of The cotangent complex), and so do the identifications of derived pullbacks. On $H^0$ the functoriality map is the map of differentials (Theorem 6.1 there). $\square$

Proposition 1.4 (base change). Let $g\colon S'\to S$ be a morphism, $G'=G\times_SS'$ and $g'\colon G'\to G$ the projection.

The functoriality map $Lg'^*L_{G/S}\to L_{G'/S'}$ is an isomorphism.
It induces an isomorphism $Lg^*\ell_G\cong\ell_{G'}$, through $Le_{G'}^*Lg'^*=Lg^*Le_G^*$.
In particular, for $s\in S$ and $i_s\colon\operatorname{Spec}\kappa(s)\to S$, $Li_s^*\ell_G\cong\ell_{G_s}$.

Proof. (1) The square formed by $g'$, $g$ and the structure morphisms is cartesian, and $G$ is flat over $S$, so this is Proposition 1.4 of the lesson on deformations of diagrams. (2) Apply $Le_{G'}^*$, with $g'e_{G'}=e_Gg$. (3) is (2) for $g=i_s$. $\square$

So $\ell_{G'}^\vee\cong Lg^*\ell_G^\vee$ as well, duals of perfect complexes commuting with derived pullback.

Proposition 1.5 (smoothness). $G$ is smooth over $S$ if and only if $\ell_G$ has tor amplitude in $[0,0]$, that is, $\ell_G\cong\omega_G[0]$ with $\omega_G$ finite locally free.

Proof. If $G$ is smooth, $L_{G/S}\cong\Omega_{G/S}[0]$ with $\Omega_{G/S}$ finite locally free (Propositions 5.1 and 10.1 of The cotangent complex, [Tag 02G1]), so $\ell_G=e^*\Omega_{G/S}[0]$. Conversely, let $\ell_G$ have tor amplitude in $[0,0]$. For $s\in S$, the fibre $G_s$ is a group scheme over $\kappa(s)$, and $\ell_{G_s}\cong Li_s^*\ell_G$ (Proposition 1.4 (3)) has no cohomology in degree $-1$. By Corollary 3.2 of the lesson on flat group schemes, $G_s$ is smooth over $\kappa(s)$. As $G\to S$ is flat and locally of finite presentation with smooth fibres, it is smooth [Tag 01V8]. $\square$

2. Exact sequences

A sequence $1\to G\xrightarrow{\iota}H\xrightarrow{\rho}K\to1$ of group schemes over $S$, flat and locally of finite presentation, is exact if $\iota$ identifies $G$ with the kernel $H\times_{K,e_K}S$ of $\rho$, and $\rho$ is an epimorphism of sheaves for the fpqc topology: for every scheme $V$ over $S$ and $k\in K(V)$ there is an fpqc covering $\{V_i\to V\}$ with points $h_i\in H(V_i)$, $\rho(h_i)=k|_{V_i}$.

Lemma 2.1. In an exact sequence, $\rho\colon H\to K$ is a $G_K$-torsor for the action of $G$ on $H$ by left multiplication, $g\cdot h=\iota(g)h$. In particular $\rho$ is syntomic and faithfully flat.

Proof. The action is over $K$, as $\rho(\iota(g)h)=\rho(h)$. The morphism $G\times_SH\to H\times_KH$, $(g,h)\mapsto(\iota(g)h,h)$, has the inverse $(h',h)\mapsto(h'h^{-1},h)$: a point $h'h^{-1}$ with $\rho(h'h^{-1})=e$ is a point of the kernel $G$. Applying the exactness to $V=K$ and $k=\mathrm{id}_K$ gives an fpqc covering $\{K_i\to K\}$ over which $\rho$ has sections. So $\rho$ is a torsor (Definition 1.1 of the lesson on torsors), and it is syntomic and faithfully flat by Proposition 1.3 (1) there. $\square$

Proposition 2.2 (the triangle of an exact sequence). For an exact sequence $1\to G\xrightarrow{\iota}H\xrightarrow{\rho}K\to1$ of group schemes over $S$, flat and locally of finite presentation, there is a distinguished triangle $\begin{aligned}&\ell_K \xrightarrow{\ \ell(\rho)\ }\ell_H \xrightarrow{\ \ell(\iota)\ }\ell_G\\&\quad \longrightarrow\ell_K[1]\end{aligned} \tag{2.1}$ in $D(\mathcal O_S)$, natural in the exact sequence.

Proof. Apply $Le_H^*$ to the transitivity triangle of $H\xrightarrow{\rho}K\to S$ (Proposition 1.3 of the lesson on deformations of diagrams): $\begin{aligned}&Le_H^*L\rho^*L_{K/S}\to Le_H^*L_{H/S}\\&\quad\to Le_H^*L_{H/K}\to Le_H^*L\rho^*L_{K/S}[1].\end{aligned}$ Since $\rho e_H=e_K$, the first term is $Le_K^*L_{K/S}=\ell_K$ [Tag 0D6D], and the first map is $\ell(\rho)$ (Proposition 1.3). The second term is $\ell_H$. The square formed by $\iota\colon G\to H$, $e_K\colon S\to K$, $\rho$ and $\pi_G$ is cartesian, $G$ being the kernel, and $\rho$ is flat (Lemma 2.1); so the functoriality map $L\iota^*L_{H/K}\to L_{G/S}$ is an isomorphism (Proposition 1.4 of the lesson on deformations of diagrams). With $\iota e_G=e_H$, the third term is $Le_G^*L\iota^*L_{H/K}\cong Le_G^*L_{G/S}=\ell_G$. The composite of the functoriality maps $L\iota^*L_{H/S}\to L\iota^*L_{H/K}\to L_{G/S}$ is the functoriality map of $\iota$ over $S$ (Theorem 2.2 of The cotangent complex), so the second map becomes $\ell(\iota)$. Naturality follows from the naturality of transitivity triangles and of functoriality maps. $\square$

Corollary 2.3. In an exact sequence as above, there is a long exact sequence $\begin{aligned}&0 \to H^{-1}(\ell_K) \to H^{-1}(\ell_H)\\&\quad \to H^{-1}(\ell_G) \to\omega_K \to\omega_H\\&\quad \to\omega_G \to0,\end{aligned}$ and $\ell(\rho)$ is an isomorphism if and only if $\ell_G=0$, that is, if and only if $G$ is étale over $S$.

Proof. The long exact sequence of (2.1) [Tag 0149], with $H^i=0$ outside $[-1,0]$. The third vertex of a distinguished triangle vanishes if and only if the first map is an isomorphism [Tag 05QR]. Finally $\ell_G=0$ if and only if $G$ is smooth with $\omega_G=0$ (Proposition 1.5), that is, smooth with $\Omega_{G/S}=\pi^*\omega_G=0$, which is étale. $\square$

3. Examples

Example 3.1 (smooth groups). For $\mathbf G_{a,S}=\operatorname{Spec}\mathcal O_S[x]$, $\ell=\mathcal O_S\,dx$ in degree $0$; for $\mathbf G_{m,S}=\operatorname{Spec}\mathcal O_S[t,t^{-1}]$, $\ell=\mathcal O_S\,dt$ in degree $0$, the value at $t=1$ of the invariant form $dt/t$.

Example 3.2 (roots of unity). Let $n\geq1$ and $\mu_n=\operatorname{Spec}\mathcal O_S[t]/(t^n-1)$. By formula (7.3) of The cotangent complex, $L_{\mu_n/S}$ is locally $[B\bar e\to B\,dt]$, $\bar e\mapsto nt^{n-1}dt$, with $\bar e$ the class of $t^n-1$, and pulling back along $t=1$, $\ell_{\mu_n}\cong\bigl[\mathcal O_S\xrightarrow{\ n\ }\mathcal O_S\bigr]$ in degrees $-1,0$. The exact Kummer sequence $1\to\mu_n\to\mathbf G_m\xrightarrow{n}\mathbf G_m\to1$ gives the same answer through Proposition 2.2: the $n$-th power map multiplies $dt/t$ by $n$, so $\ell(n)$ is multiplication by $n$ on $\mathcal O_S$ (Exercise 4.1).

Example 3.3 (Frobenius kernel of the additive group). Let $S$ have characteristic $p$, and $\alpha_p=\operatorname{Spec}\mathcal O_S[x]/(x^p)$, the kernel of $F\colon\mathbf G_a\to\mathbf G_a$, $x\mapsto x^p$. The sequence $1\to\alpha_p\to\mathbf G_a\xrightarrow{F}\mathbf G_a\to1$ is exact: $F$ corresponds to $\mathcal O_S[y]\to\mathcal O_S[x]$, $y\mapsto x^p$, which makes $\mathcal O_S[x]$ free with basis $1,x,\ldots,x^{p-1}$, so $F$ is a finite locally free surjection, and every point of $\mathbf G_a$ has $p$-th roots fppf locally. Since $d(x^p)=0$, $\ell(F)=0$, and the triangle (2.1) splits [Tag 05QT]: $\ell_{\alpha_p}\cong\mathcal O_S\oplus\mathcal O_S[1].$ The same follows from formula (7.3) of The cotangent complex applied to $x^p$, whose derivative vanishes.

4. Exercises

4.1. Prove the claims of Example 3.2: that the Kummer sequence is exact for the fppf topology, hence for fpqc; that $\ell(n)$ is multiplication by $n$; and that (2.1) gives $\ell_{\mu_n}\cong[\mathcal O_S\xrightarrow{n}\mathcal O_S]$.

4.2. Let $G$ be a finite group, viewed as the constant group scheme $G_S$ over $S$. Show that $\ell_{G_S}=0$. Deduce from Corollary 2.3 that for an exact sequence $1\to G_S\to H\to K\to1$ with $G$ finite constant, $\ell(\rho)\colon\ell_K\to\ell_H$ is an isomorphism.

4.3. Let $k$ be a field. Show that $\dim_kH^{-1}(\ell_G)-\dim_kH^0(\ell_G)=$ $-\dim G$ for every group scheme $G$ locally of finite type over $k$, and compute both dimensions for $\mu_n$ when $\operatorname{char}k=p$ divides $n$.

5. Complete solutions

4.1. The $n$-th power map of $\mathbf G_m$ corresponds to $\mathcal O_S[s,s^{-1}]\to\mathcal O_S[t,t^{-1}]$, $s\mapsto t^n$, which makes $\mathcal O_S[t,t^{-1}]$ a free module with basis $1,t,\ldots,t^{n-1}$. So it is finite locally free and surjective, an fppf covering, and every point $c\in\mathbf G_m(V)$ has an $n$-th root after the base change $V\times_{\mathbf G_m}\mathbf G_m\to V$, an fppf covering. Its kernel is $\operatorname{Spec}\mathcal O_S[t,t^{-1}]/(t^n-1)=\mu_n$, as $t$ is invertible modulo $t^n-1$. By Example 3.1, $\ell_{\mathbf G_m}$ is free on the class of $dt$ at $t=1$, and the pullback of $ds$ is $d(t^n)=nt^{n-1}dt$, which is $n\,dt$ at $t=1$. So $\ell(n)$ is multiplication by $n$ on $\mathcal O_S$. The triangle (2.1) is $\mathcal O_S\xrightarrow{n}\mathcal O_S\to\ell_{\mu_n}\to\mathcal O_S[1]$, so $\ell_{\mu_n}$ is the cone of multiplication by $n$, the complex $[\mathcal O_S\xrightarrow{n}\mathcal O_S]$ in degrees $-1,0$.

4.2. $G_S$ is a disjoint union of copies of $S$, so $G_S\to S$ is étale and $L_{G_S/S}=0$ (Propositions 5.1 and 10.1 of The cotangent complex), hence $\ell_{G_S}=0$. Corollary 2.3 gives the rest.

4.3. By Proposition 1.2 (1), $\ell_G$ is a perfect complex of $k$-vector spaces with cohomology in degrees $-1,0$, and by Corollary 3.2 of the lesson on flat group schemes, $\dim H^0=\dim\operatorname{Lie}(G)$ and $\dim H^{-1}=\dim\operatorname{Lie}(G)-\dim G$; the difference is $-\dim G$. For $\mu_n$ with $p\mid n$, Example 3.2 gives $[k\xrightarrow{0}k]$: both dimensions are $1$, and $\dim\mu_n=0$.

Sources

The co-Lie complex of a finite flat group scheme over an affine base and the triangle of an exact sequence are due to Mazur and Roberts; the co-Lie complex of a flat group scheme locally of finite presentation over an arbitrary base and its interpretation through torsors are due to Illusie, in his work on the cotangent complex.