Flat group schemes are local complete intersections

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

A group scheme over a field need not be smooth: in characteristic $p$ the kernel of the Frobenius endomorphism of the additive group is $\operatorname{Spec}k[x]/(x^p)$. It is nevertheless always a local complete intersection. This lesson proves that a group scheme which is flat and locally of finite presentation over an arbitrary base scheme is syntomic, so that its cotangent complex is locally a two-term complex of finite free modules in degrees $-1$ and $0$. Over a field of characteristic $p$ the proof removes Frobenius kernels one at a time. The first Frobenius kernel is a truncated polynomial algebra, the quotient by it needs one Frobenius kernel fewer to become smooth, and the quotient by a large enough Frobenius kernel is the reduced subgroup of a Frobenius twist, which is smooth. Over a general base the fibre criterion for syntomic morphisms finishes the proof. At the unit of a group scheme over a field, the cotangent complex measures the failure of smoothness: its cohomology in degree $-1$ has dimension $\dim\operatorname{Lie}(G)-\dim G$.

Prerequisites. From the programme course Group schemes: Group schemes over a field (Proposition 1.1, Theorem 2.3 and Proposition 4.1); Lie algebras and smoothness of group schemes (Theorem 3.2 and Theorem 4.3); Quotients and torsors (Theorem 11.1b, Corollary 11.10, Theorem 11.16 and Proposition C.6). From the programme lesson The cotangent complex: Theorem 7.2 with formula (7.3), and Proposition 10.1. From the AI-integrated Stacks project: syntomic morphisms [Stacks, Tag 01UC, Tag 01UE, Tag 01UF, Tag 01UH, Tag 01UI, Tag 01VD] and their descent [Tag 02VK]; differentials of smooth morphisms [Tag 02G1]; relative global complete intersections [Tag 00SP, Tag 00SV, Tag 07D2]; relative Frobenius morphisms [Tag 0CC7, Tag 0CC9, Tag 0CCA, Tag 0CCB, Tag 0CCG]; closed subschemes and reductions [Tag 01HP, Tag 0356]; the Nullstellensatz, local dimension and tangent spaces [Tag 00FV, Tag 00OP, Tag 00KW, Tag 0B2D].

Conventions

A morphism of schemes is syntomic in the sense of [Tag 01UC]: locally on source and target it is given by a syntomic ring map, that is, a flat ring map of finite presentation whose fibre rings are local complete intersections. A scheme $X$ over a field $k$ is a local complete intersection over $k$ if $X\to\operatorname{Spec}k$ is syntomic. Being syntomic is local on the source, stable under composition [Tag 01UH] and base change [Tag 01UI], and local on the base for the fpqc topology [Tag 02VK]; smooth morphisms are syntomic [Tag 01VD]. A morphism $X\to S$ is syntomic if and only if every point of $X$ has an affine open neighbourhood $\operatorname{Spec}B$ mapping into an affine open $\operatorname{Spec}A$ of $S$ with $B\cong A[x_1,\ldots,x_m]/(f_1,\ldots,f_c)$ a relative global complete intersection: every nonempty fibre of $\operatorname{Spec}B\to\operatorname{Spec}A$ has dimension $m-c$ [Tags 00SP, 01UE]. In such a presentation $f_1,\ldots,f_c$ is a Koszul-regular sequence in $A[x_1,\ldots,x_m]$ [Tag 07D2], and it is a regular sequence in the localization of $A[x_1,\ldots,x_m]$ at every prime containing it [Tag 00SV].

Group schemes $G\to S$ have multiplication $m$, unit $e\colon S\to G$ and inverse. For a group scheme $G$ over a field, $\operatorname{Lie}(G)$ is the tangent space of $G$ at $e$, and $\dim G$ denotes the dimension of the local ring $\mathcal O_{G,e}$, as in Theorem 3.2 of Lie algebras and smoothness of group schemes.

Let $k$ be a field of characteristic $p>0$ and $\varphi\colon k\to k$, $c\mapsto c^p$. For a scheme $X$ over $k$ and $n\geq0$, $X^{(p^n)}=X\times_{\operatorname{Spec}k,\operatorname{Spec}\varphi^n}\operatorname{Spec}k$, and $F_{X/k,n}\colon X\longrightarrow X^{(p^n)}$ is the $n$-fold relative Frobenius morphism [Tags 0CC9, 0CCG], with $F_{X/k,0}=\mathrm{id}$ and $F_{X/k}=F_{X/k,1}$. On an affine open $\operatorname{Spec}A\subseteq X$ and the affine open $\operatorname{Spec}(A\otimes_{k,\varphi^n}k)$ of $X^{(p^n)}$ above it, $F_{X/k,n}$ is given by the ring map $\begin{gathered}A\otimes_{k,\varphi^n}k\longrightarrow A,\\ a\otimes c\longmapsto c\,a^{p^n},\end{gathered} \tag{0.1}$ because $F_{X/k,n}$ is the pair of the $n$-th power of the absolute Frobenius of $X$ and the identity of $\operatorname{Spec}k$ [Tag 0CCG]. For a group scheme $G$ over $k$, $G^{(p^n)}$ is a group scheme by base change, and $G[F^n]$ denotes the kernel of $F_{G/k,n}$, the fibre of $F_{G/k,n}$ over the unit of $G^{(p^n)}$.

1. Frobenius kernels

Lemma 1.1. Let $k$ be a field of characteristic $p>0$, $G$ a group scheme locally of finite type over $k$, and $n\geq0$.

$F_{G/k,n}\colon G\to G^{(p^n)}$ is a homomorphism of group schemes, a universal homeomorphism and an integral morphism; in particular it is quasi-compact and surjective.
$G[F^n]$ is a closed normal subgroup scheme of $G$ whose underlying space is the point $e$, and it is the spectrum of a finite local $k$-algebra.
For $n\geq1$, $F_{G/k,n}=F_{G^{(p)}/k,n-1}\circ F_{G/k}$; in particular $G[F]\subseteq G[F^n]$.
If $H\subseteq G$ is a closed subgroup scheme, then $H[F^n]=H\times_GG[F^n]$.

Proof. (1) For schemes $X$ and $Y$ over $k$, [Tag 0CC7] applied to the two projections of $X\times_kY$ shows that the absolute Frobenius of $X\times_kY$ is the product $F_X\times F_Y$. Hence $(X\times_kY)^{(p)}=X^{(p)}\times_kY^{(p)}$ and $F_{X\times_kY/k}=F_{X/k}\times F_{Y/k}$, both sides being the unique morphism over $k$ with the property of [Tag 0CC9]. Relative Frobenius is natural for morphisms over $k$ [Tag 0CCA]. Applied to the multiplication, the unit and the inverse of $G$, this shows that $F_{G/k}$ is a homomorphism $G\to G^{(p)}$. By [Tag 0CCG], $F_{G/k,n}$ is the composite of the homomorphisms $F_{G^{(p^j)}/k}$, $0\leq j<n$. Each of them is a universal homeomorphism and integral [Tag 0CCB], and these properties are stable under composition. Integral morphisms are affine, hence quasi-compact.

(2) Group schemes over a field are separated (Proposition 1.1 of Group schemes over a field), so the unit of $G^{(p^n)}$ is a closed immersion and its fibre product $G[F^n]$ with $F_{G/k,n}$ is a closed subscheme of $G$. It is a normal subgroup scheme, being the kernel of a homomorphism. Since $F_{G/k,n}$ is injective on points, $G[F^n]$ has the single point $e$. A scheme with one point is affine, so $G[F^n]=\operatorname{Spec}A$ with $A$ a finitely generated $k$-algebra with a single prime ideal $\mathfrak m$. By the Nullstellensatz [Tag 00FV], $A/\mathfrak m$ is finite over $k$. The Noetherian ring $A$ has dimension zero, so $\mathfrak m^N=0$ for some $N$, and each $\mathfrak m^i/\mathfrak m^{i+1}$ is a finite $A/\mathfrak m$-vector space. So $A$ is finite over $k$.

(3) By [Tag 0CCG], $F_{G/k,n}=F_{G^{(p^{n-1})}/k}\circ\cdots\circ F_{G^{(p)}/k}\circ F_{G/k}$, and the first $n-1$ factors compose to $F_{G^{(p)}/k,n-1}$, since $(G^{(p)})^{(p^j)}=G^{(p^{j+1})}$. So the kernel of $F_{G/k}$ is contained in that of $F_{G/k,n}$.

(4) Let $\iota\colon H\to G$ be the inclusion. Naturality [Tag 0CCA], applied $n$ times, gives $F_{G/k,n}\circ\iota=\iota^{(p^n)}\circ F_{H/k,n}$. The morphism $\iota^{(p^n)}$ is a closed immersion, as a base change of one, hence a monomorphism of group schemes. So a point $h$ of $H$ with values in a scheme lies in $H[F^n]$ if and only if $\iota(h)$ lies in $G[F^n]$. $\square$

Lemma 1.2 (the first Frobenius kernel). In the situation of Lemma 1.1, there is an integer $d\geq0$ with $\begin{aligned}&G[F]\\&\quad \cong\operatorname{Spec}k[X_1,\ldots,X_d]/(X_1^p,\ldots,X_d^p)\end{aligned}$ as schemes over $k$. In particular $G[F]$ is a local complete intersection over $k$.

Proof. The group scheme $H=G[F]$ is finite and connected over $k$ by Lemma 1.1 (2), and $H[F]=H\times_GG[F]=H$ by Lemma 1.1 (4). By Proposition C.6 of Quotients and torsors, the first Frobenius kernel $H[F]=H$ has the algebra $k[X_1,\ldots,X_d]/(X_1^p,\ldots,X_d^p)$, with $d=\dim_k\operatorname{Lie}(H)$. This algebra has dimension $0=d-d$, so it is a relative global complete intersection over $k$ [Tag 00SP], and $H\to\operatorname{Spec}k$ is syntomic [Tag 01UE]. $\square$

2. Group schemes over a field

Lemma 2.1. Let $k$ be a perfect field of characteristic $p>0$ and $G$ a group scheme of finite type over $k$. There is an integer $n\geq0$ such that $F_{G/k,n}$ factors as the quotient map $G\to G/G[F^n]$ followed by an isomorphism of $G/G[F^n]$ onto the reduced closed subgroup scheme $(G^{(p^n)})_{\mathrm{red}}$ of $G^{(p^n)}$. In particular $G/G[F^n]$ is smooth over $k$.

The quotient exists by Theorem 11.1b of Quotients and torsors, since $G[F^n]$ is a closed normal subgroup scheme.

Proof. Since $k$ is perfect, $\varphi^n$ is an automorphism of $k$. The projection $G^{(p^n)}\to G$ is the base change of the isomorphism $\operatorname{Spec}\varphi^n$, hence an isomorphism of schemes. Cover $G$ by finitely many affine opens $\operatorname{Spec}A_j$. The ring maps $A_j\to A_j\otimes_{k,\varphi^n}k$, $a\mapsto a\otimes1$, of this projection are isomorphisms, so the nilradical of $A_j\otimes_{k,\varphi^n}k$ consists of the elements $a\otimes1$ with $a$ nilpotent. The nilradical of the Noetherian ring $A_j$ is nilpotent; choose $n$ with $p^n$ at least all of its nilpotency indices. By (0.1), $F_{G/k,n}$ sends $a\otimes1$ to $a^{p^n}=0$ for nilpotent $a$. So on every chart the ring map of $F_{G/k,n}$ kills the nilradical, and $F_{G/k,n}$ factors through the closed subscheme $G'_{\mathrm{red}}$ of $G'=G^{(p^n)}$ [Tag 01HP].

The group scheme $G'$ is of finite type over the perfect field $k$, so $G'_{\mathrm{red}}$ is a smooth closed subgroup scheme (Proposition 4.1 of Group schemes over a field). By Lemma 1.1 (1), $F_{G/k,n}$ is a quasi-compact homomorphism, and its kernel $G[F^n]$ is flat over the field $k$. By Corollary 11.10 of Quotients and torsors, $F_{G/k,n}=i\circ q$ with $q\colon G\to G/G[F^n]$ the quotient map and $i$ a closed immersion whose image is the scheme theoretic image $Z$ of $F_{G/k,n}$. As $F_{G/k,n}$ factors through $G'_{\mathrm{red}}$, $Z\subseteq G'_{\mathrm{red}}$; as $F_{G/k,n}$ is surjective, $Z$ contains every point. The identity of the reduced scheme $G'_{\mathrm{red}}$ then factors through $Z$ [Tag 0356], so $Z=G'_{\mathrm{red}}$, and $i$ is an isomorphism of $G/G[F^n]$ onto the smooth group $G'_{\mathrm{red}}$. $\square$

Definition 2.2. For a group scheme $G$ of finite type over a perfect field $k$ of characteristic $p>0$, $n(G)$ is the least integer $n\geq0$ such that $G/G[F^n]$ is smooth over $k$. It exists by Lemma 2.1, and $n(G)=0$ if and only if $G$ is smooth.

Theorem 2.3. Let $k$ be a field and $G$ a group scheme locally of finite type over $k$. Then $G$ is a local complete intersection over $k$.

Proof. Step 1: reduction to an algebraically closed field. Let $\bar k$ be an algebraic closure of $k$. The morphism $\operatorname{Spec}\bar k\to\operatorname{Spec}k$ is an fpqc covering, and being syntomic is fpqc local on the base [Tag 02VK]; the base change $G_{\bar k}$ is a group scheme locally of finite type over $\bar k$. So we may assume $k$ algebraically closed.

Step 2: reduction to finite type. The identity component $G^0$ is an open subgroup scheme, and it is quasi-compact (Theorem 2.3 of Group schemes over a field), so it is of finite type over $k$. For $x\in G(k)$, left translation by $x$ is an automorphism of the $k$-scheme $G$, and it carries $G^0$ onto an open neighbourhood $xG^0$ of $x$. Every point $y$ of $G$ lies in such a neighbourhood. Indeed, choose an affine open $\operatorname{Spec}A$ containing $y$, with $A$ a finitely generated $k$-algebra, and a maximal ideal $\mathfrak m$ of $A$ containing the prime of $y$. Then $A/\mathfrak m=k$ by the Nullstellensatz [Tag 00FV], so $\mathfrak m$ is a point $x\in G(k)$ in the closure of $y$, and the open neighbourhood $xG^0$ of $x$ contains $y$. Since being syntomic is local on the source and $xG^0\cong G^0$ over $k$, it suffices to prove that $G^0$ is a local complete intersection. So we may assume $G$ of finite type.

Step 3: characteristic zero. If $k$ has characteristic $0$, then $G$ is smooth (Theorem 4.3 of Lie algebras and smoothness of group schemes), hence syntomic [Tag 01VD].

Step 4: characteristic $p>0$. The algebraically closed field $k$ is perfect. We prove the theorem for group schemes $G$ of finite type over $k$ by induction on $n(G)$ (Definition 2.2). If $n(G)=0$, then $G=G/G[F^0]$ is smooth, hence syntomic. Let $n=n(G)\geq1$, and let $q\colon G\to G_1=G/G[F]$ be the quotient by the closed normal subgroup $G[F]$. By Theorem 11.1b of Quotients and torsors, $G_1$ is a group scheme of finite type over $k$, and $q$ is a homomorphism, faithfully flat of finite presentation, and a $G[F]$-torsor.

(a) $q$ is syntomic. As $q$ is a $G[F]$-torsor, $(g,h)\mapsto(g,gh)$ is an isomorphism $G\times_kG[F]\to G\times_{G_1}G$ over the first factor. So the base change of $q$ along the fppf covering $q$ is isomorphic to the projection $G\times_kG[F]\to G$, which is the base change of $G[F]\to\operatorname{Spec}k$ and is syntomic by Lemma 1.2 and [Tag 01UI]. Hence $q$ is syntomic [Tag 02VK].

(b) $n(G_1)\leq n-1$. By Corollary 11.10 of Quotients and torsors, applied to the quasi-compact homomorphism $F_{G/k}$ with kernel $G[F]$, there is a closed immersion of group schemes $i_1\colon G_1\to G^{(p)}$ with $i_1\circ q=F_{G/k}$. By Lemma 1.1 (4) for the closed subgroup $i_1(G_1)\cong G_1$ of $G^{(p)}$, $G_1[F^{n-1}]=i_1^{-1}(G^{(p)}[F^{n-1}])$. With Lemma 1.1 (3), $\begin{aligned}&q^{-1}\bigl(G_1[F^{n-1}]\bigr)\\ &\quad =\ker\bigl(F_{G^{(p)}/k,n-1}\circ F_{G/k}\bigr)\\ &\quad =\ker F_{G/k,n} =G[F^n].\end{aligned}$ Since $q$ is an epimorphism of fppf sheaves, $G_1[F^{n-1}]$ is the image of $G[F^n]$ in $G_1$, the quotient of $G[F^n]$ by $G[F]$. Theorem 11.16 of Quotients and torsors, applied to the closed normal subgroups $G[F]\subseteq G[F^n]$ of $G$, which are flat over $k$, identifies the quotient of $G_1$ by this image with $G/G[F^n]$. So $G_1/G_1[F^{n-1}]\cong G/G[F^n]$ is smooth, and $n(G_1)\leq n-1$.

3. Flat group schemes over a base

Theorem 3.1. Let $S$ be a scheme and $G$ a group scheme over $S$ which is flat and locally of finite presentation. Then $G\to S$ is syntomic, and every point of $G$ has an open neighbourhood $U$ on which $L_{G/S}|_U$ is isomorphic in $D(\mathcal O_U)$ to a complex of finite free $\mathcal O_U$-modules in degrees $-1$ and $0$. In particular $L_{G/S}$ is perfect with tor amplitude in $[-1,0]$. The same holds for every base change $G\times_SS'\to S'$.

Proof. For $s\in S$, the fibre $G_s$ is a group scheme locally of finite type over $\kappa(s)$, hence a local complete intersection over $\kappa(s)$ by Theorem 2.3. As $G\to S$ is flat and locally of finite presentation, it is syntomic [Tag 01UF]. Let $x\in G$. By [Tag 01UE] there are affine opens $x\in U=\operatorname{Spec}B$ and $V=\operatorname{Spec}A$ of $S$ containing the image of $U$, with $B=A[x_1,\ldots,x_m]/(f_1,\ldots,f_c)$ a relative global complete intersection. The sequence $f_1,\ldots,f_c$ is Koszul regular [Tag 07D2], so $A\to B$ is a local complete intersection map in the sense of Section 7 of The cotangent complex, and by its Theorem 7.2 and formula (7.3), with $I=(f_1,\ldots,f_c)$, $\begin{gathered}L_{B/A}\simeq\bigl[I/I^2\\ \xrightarrow{\ d\ }\Omega_{A[x]/A}\otimes_{A[x]}B\bigr].\end{gathered} \tag{3.1}$ Here $I/I^2$ is free with basis the classes of the $f_i$ [Tag 00SV], and $\Omega_{A[x]/A}\otimes B$ is free with basis $dx_1,\ldots,dx_m$. By Proposition 10.1 of that lesson, $L_{G/S}|_U\simeq\widetilde{L_{B/A}}$. A base change $G\times_SS'$ is a group scheme over $S'$, flat and locally of finite presentation, so the same argument applies to it. $\square$

Corollary 3.2 (the cotangent complex at the unit). Let $G$ be a group scheme locally of finite type over a field $k$, and $\ell_G=Le^*L_{G/k}\in D(k)$. Then $H^i(\ell_G)=0$ for $i\neq-1,0$, $H^0(\ell_G)=e^*\Omega_{G/k}$ is the dual of $\operatorname{Lie}(G)$, and $\dim_kH^{-1}(\ell_G)=\dim_k\operatorname{Lie}(G)-\dim G .$ In particular $H^{-1}(\ell_G)=0$ if and only if $G$ is smooth over $k$.

Proof. Choose $U=\operatorname{Spec}B\ni e$ as in the proof of Theorem 3.1, with $A=k$. As (3.1) is a complex of free modules, $\ell_G$ is computed by pulling it back along the $k$-point $e$: it is a complex $k^c\to k^m$, given by the Jacobian matrix of $f_1,\ldots,f_c$ at $e$. So $H^i(\ell_G)=0$ for $i\neq-1,0$, and $\dim H^0(\ell_G)-\dim H^{-1}(\ell_G)=m-c$. The cokernel $H^0(\ell_G)$ is $e^*\Omega_{B/k}$, by right exactness of $e^*$, and it is the dual of the tangent space $\operatorname{Lie}(G)$ [Tag 0B2D]. Let $\mathfrak q$ be the maximal ideal of $k[x_1,\ldots,x_m]$ at the image of $e$. The local ring $k[x_1,\ldots,x_m]_{\mathfrak q}$ has dimension $m$ [Tag 00OP], and $f_1,\ldots,f_c$ is a regular sequence in it [Tag 00SV]. By [Tag 00KW], applied $c$ times, $\dim\mathcal O_{G,e}=m-c$. This gives the formula. The last assertion is Theorem 3.2 of Lie algebras and smoothness of group schemes: $\dim G\leq\dim\operatorname{Lie}(G)$, with equality exactly when $G$ is smooth. $\square$

Example 3.3. Let $k$ be a field of characteristic $p>0$ and $G=\mu_p=\operatorname{Spec}k[t]/(t^p-1)$. Here $c=m=1$, the derivative of $t^p-1$ is $pt^{p-1}=0$, so $\ell_{\mu_p}=[k\xrightarrow{0}k]$: $\dim\operatorname{Lie}(\mu_p)=1$ and $\dim\mu_p=0$. For $G$ smooth of dimension $r$, such as $\mathrm{GL}_r$ or an abelian variety, $\ell_G$ is $e^*\Omega_{G/k}$ in degree $0$.

4. Exercises

4.1. Let $k$ be a perfect field of characteristic $p>0$ and $G=\alpha_{p^2}=\operatorname{Spec}k[x]/(x^{p^2})$, the kernel of $F_{\mathbf G_a/k,2}$ on the additive group. Compute $G[F]$, the image of $F_{G/k}$ and $G/G[F]$; show that $n(G)=2$ (Definition 2.2), and compute $\ell_G$.

4.2. Let $p$ be a prime and $G=\mu_{p,\mathbb Z}=\operatorname{Spec}\mathbb Z[t]/(t^p-1)$. Show that $L_{G/\mathbb Z}$ is isomorphic to $\Omega_{G/\mathbb Z}$ placed in degree $0$, that $\Omega_{G/\mathbb Z}$ is not locally free, so that $G$ is syntomic but not smooth over $\mathbb Z$, and that $L_{G_{\mathbb F_p}/\mathbb F_p}$ has free cohomology of rank one in degrees $-1$ and $0$.

4.3. Let $k$ be a perfect field of characteristic $p>0$ and $G$ a finite connected group scheme over $k$. Show that $n(G)$ is the height of $G$, the least $n$ with $G[F^n]=G$.

5. Complete solutions

4.1. The group $G$ is a closed subgroup scheme of $\mathbf G_a$, and $\mathbf G_a[F]=\operatorname{Spec}k[x]/(x^p)$, since $F_{\mathbf G_a/k}$ is $y\mapsto x^p$ on $k[y]\to k[x]$, after identifying $\mathbf G_a^{(p)}$ with $\mathbf G_a$ through the coordinate. By Lemma 1.1 (4), $G[F]=$ $G\times_{\mathbf G_a}\mathbf G_a[F]=$ $\operatorname{Spec}k[x]/(x^p)=$ $\alpha_p$. The relative Frobenius of $G$ is the ring map $k[y]/(y^{p^2})\to k[x]/(x^{p^2})$, $y\mapsto x^p$, whose kernel is spanned by the $y^j$ with $pj\geq p^2$, that is $j\geq p$. So its scheme theoretic image is $\operatorname{Spec}k[y]/(y^p)\cong\alpha_p$, and by Corollary 11.10 of Quotients and torsors, $G/G[F]\cong\alpha_p$. This is not smooth, so $n(G)\geq2$; and $G[F^2]=G$, since $F_{G/k,2}$ is $y\mapsto x^{p^2}=0$ on the augmentation ideal, so $G/G[F^2]=\operatorname{Spec}k$ is smooth and $n(G)=2$. The proof of Theorem 2.3 passes from $G$ to $G_1=\alpha_p$, with $n(\alpha_p)=1$, and from $\alpha_p$ to the trivial group. Finally $G=\operatorname{Spec}k[x]/(f)$ with $f=x^{p^2}$ and $f'=p^2x^{p^2-1}=0$, so $\ell_G=[k\xrightarrow{0}k]$, in accordance with $\dim\operatorname{Lie}(G)=1$ and $\dim G=0$.

4.2. Put $B=\mathbb Z[t]/(t^p-1)$, free over $\mathbb Z$ with basis $1,t,\ldots,t^{p-1}$, and $I=(t^p-1)$. The element $t^p-1$ is a nonzerodivisor of $\mathbb Z[t]$, so (3.1) gives $L_{B/\mathbb Z}\simeq[B\,e\to B\,dt]$, $e\mapsto pt^{p-1}dt$, with $e$ the class of $t^p-1$. Since $t$ is a unit of $B$ and multiplication by $p$ is injective on the free $\mathbb Z$-module $B$, the differential is injective, so $H^{-1}=0$ and $L_{B/\mathbb Z}\simeq\Omega_{B/\mathbb Z}=(B/pB)\,dt$ in degree $0$. The module $B/pB$ is nonzero and killed by $p$, while $B$ is torsion free; a nonzero finite locally free $B$-module would be torsion free, so $\Omega_{B/\mathbb Z}$ is not locally free and $G$ is not smooth over $\mathbb Z$ [Tag 02G1]; $G$ is syntomic by Theorem 3.1. Over $\mathbb F_p$, $\bar B=\mathbb F_p[t]/(t^p-1)$ and the same presentation gives $[\bar B\,e\xrightarrow{0}\bar B\,dt]$, with $H^{-1}=\bar B\,e$ and $H^0=\bar B\,dt$ free of rank one.

4.3. If $G[F^n]=G$, then $G/G[F^n]=\operatorname{Spec}k$ is smooth, so $n(G)\leq n$. Conversely, let $n=n(G)$. By Corollary 11.10 of Quotients and torsors, the quotient $Q=G/G[F^n]$ is a closed subscheme of the finite $k$-scheme $G^{(p^n)}$, so it is finite; it is smooth by the choice of $n$, hence étale. It is connected, as the image of the connected $G$, and it has the rational point $e$. A connected finite étale $k$-scheme with a rational point is $\operatorname{Spec}k$. So $q\colon G\to Q$ is trivial, and $G[F^n]=G$. Hence $n(G)$ is the height.

Sources

That flat group schemes locally of finite presentation are local complete intersections follows from the local structure theory of algebraic groups of Demazure and Gabriel; its use for the perfect amplitude of co-Lie complexes is due to Illusie, in his work on the cotangent complex. The proof here reduces it, by the quotients of the programme course on group schemes, to the structure of the first Frobenius kernel. The structure of all infinitesimal group schemes over a perfect field is proved in J. S. Milne's freely available notes Algebraic Groups (Theorem 11.29 of the 2022 edition).