Detecting nilpotence and projectivity over finite unipotent supergroup schemes

Abstract

This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme G over a perfect field k of positive characteristic $p\ge 3$. It is proved that an element x in the cohomology of G is nilpotent if and only if for every extension field K of k and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $x_K$ to E is nilpotent. It is also shown that a kG-module M is projective if and only if for every extension field K of k and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $M_K$ to E is projective. The statements are motivated by, and are analogues of, similar results for finite groups and finite group schemes, but the structure of elementary supergroups schemes necessary for detection is more complicated than in either of these cases. One application is a detection theorem for the nilpotence of cohomology, and projectivity of modules, over finite dimensional Hopf subalgebras of the Steenrod algebra.

Appendix A. Witt vectors and Dieudonné modules

Recall that finite commutative connected unipotent group schemes form an abelian category $\mathfrak {A}$ which is equivalent to an appropriate category of Dieudonné modules. This is described for example in Fontaine [25], but we give an outline here. What will interest us is the Dieudonné modules killed by p, which were classified by Koch [31].

We begin with a brief recollection concerning the Witt vectors. Define a polynomial $w_n$ in variables $Z_0,\dots ,Z_n$ with integer coefficients by

$$\begin{aligned} w_n(Z_0,\dots ,Z_n) = p^nZ_n + p^{n-1}Z_{n-1}^p + \dots + Z_0^{p^n}. \end{aligned}$$

Then the polynomials $S_i$ and $P_i$ in variables $X_0,\dots ,X_n,Y_0,\dots ,Y_n$, again with integer coefficients, are defined by

$$\begin{aligned} w_n(S_0,\dots ,S_n)&= w_n(X_0,\dots ,X_n)+w_n(Y_0,\dots ,Y_n), \\ w_n(P_0,\dots ,P_n)&= w_n(X_0,\dots ,X_n)w_n(Y_0,\dots ,Y_n). \end{aligned}$$

So for example $S_0=X_0+Y_0$, $P_0=X_0Y_0$,

$$\begin{aligned} S_1 = X_1 + Y_1 + \frac{(X_0+Y_0)^p - X_0^p-Y_0^p}{p}, \qquad P_1 = pX_1Y_1 + X_0^pY_1 + X_1Y_0^p, \end{aligned}$$

and so on.

Witt vectors W(k) over k are vectors $(a_0,a_1,\dots )$ with $a_i\in k$, where $S_i$ and $P_i$ give the coordinates of the sum and product:

$$\begin{aligned} (a_0,a_1,\dots )+(b_0,b_1,\dots )&= (S_0(a_0,b_0),S_1(a_0,a_1,b_0,b_1), \dots ) \\ (a_0,a_1,\dots )(b_0,b_1,\dots )&= (P_0(a_0,b_0),P_1(a_0,a_1,b_0,b_1), \dots ). \end{aligned}$$

Thus for example if $k=\mathbb {F}_p$ then W(k) is the ring of p-adic integers $\mathbb {Z}_p$. More generally, W(k) is a local ring of mixed characteristic p. The Frobenius endomorphism of k lifts to a ring endomorphism of W(k) denoted ${\varvec{\sigma }}$. It is defined by $(a_0,a_1\dots )^{\varvec{\sigma }}=(a_0^p,a_1^p,\dots )$.

More generally, if A is a commutative k-algebra then W(A) is the ring of Witt vectors over A, defined using the same formulae. This defines a functor from commutative k-algebras to rings. The additive part of this functor defines an affine group scheme over k denoted W, the additive Witt vectors. If we stop at length m vectors, we obtain $W_m$, and we write $W_{m,n}$ for the nth Frobenius kernel of $W_m$.

There are two endomorphisms V and F of W of interest to us. These are the Verschiebung V defined by

$$\begin{aligned} V(a_0,a_1,\dots )=(0,a_0,a_1,\dots ), \end{aligned}$$

and the Frobenius F given by

$$\begin{aligned} F(a_0,a_1,\dots )=(a_0^p,a_1^p,\dots ). \end{aligned}$$

These commute, and their product corresponds to multiplication by p on Witt vectors. Multiplication by a Witt vector $x\in W(k)$ also gives an endomorphism of W which we shall denote x by abuse of notation. These are related to V and F by the relations $Vx^{\varvec{\sigma }}=xV$ and $Fx=x^{\varvec{\sigma }}F$.

We write $W_{m}$ for the group scheme of Witt vectors of length m, corresponding to the quotient $W(k)/(p^m)$ of W(k). This is a group scheme with a filtration whose quotients are m copies of the additive group $\mathbb {G}_a$. We write $W_{m,n}$ for the nth Frobenius kernel of $W_m$. This is a finite group scheme with a filtration of length mn whose quotients are copies of $\mathbb {G}_{a(1)}$.

The Dieudonné ring $D_k$ is generated over W(k) by two commuting variables V and F satisfying the following relations:

$$\begin{aligned} FV=VF=p,\qquad Vx^{\varvec{\sigma }}= xV, \qquad Fx=x^{\varvec{\sigma }}F \end{aligned}$$

for $x\in W(k)$. Then W is a module over $D_k$, as are its quotients $W_m$ and their finite subgroup schemes $W_{m,n}$.

Recall that there is a duality on $\mathfrak {A}$ called Cartier duality, which corresponds to taking the k-linear dual of the corresponding Hopf algebras. We denote the Cartier dual of G by $G^\sharp$.

Now consider the subcategory $\mathfrak {A}_{m,n}$ of $\mathfrak {A}$ consisting of the those group schemes G in $\mathfrak {A}$ such that G has height at most n and the Cartier dual $G^\sharp$ has height at most m. Then there is a covariant equivalence of categories between $\mathfrak {A}_{m,n}$ and the category ${\mathsf {mod}}(D_k/(V^m,F^n))$ of finite length modules over the quotient ring $D_k/(V^m,F^n)$. This equivalence is given by the functor

$$\begin{aligned} {\text {Hom}}_\mathfrak {A}(W_{m,n},-):\mathfrak {A}_{m,n}\rightarrow {\mathsf {mod}}(D_k/(V^m,F^n)). \end{aligned}$$

Write $\hat{D}_k$ for the corresponding completion $\displaystyle \lim _\leftarrow D_k/(V^m,F^n)$ Then every $\hat{D}_k$-module of finite length is a module for some quotient of the form $D_k/(V^m,F^n)$, and these equivalences combine to give an equivalence between $\mathfrak {A}$ and the category ${\mathsf {fl}}(\hat{D}_k)$ of $\hat{D}_k$-modules of finite length. Let us write

$$\begin{aligned} \psi :{\mathsf {fl}}(\hat{D}_k)\rightarrow \mathfrak {A}\end{aligned}$$

for this equivalence. Thus for example

$$\begin{aligned}&\psi (D_k/(V^m,F^n))\cong W_{m,n}, \\&\psi (D_k/(V^m,F^n,p)) \cong W_{m,n}/W_{m-1,n-1} \cong E_{m,n} \end{aligned}$$

where the last notation is introduced in Definition 8.6.

Let $G = \psi (M)$ be a finite unipotent abelian group scheme, so that M is a finite length $D_k/(V^m,F^n)$-module for some $m,n\ge 1$. If we are only interested in the algebras structure of G, this means that we can ignore the action of F on M and just look at finite length modules for $D_k/(V^m,F)=W(k)[V]$ with $xV=Vx^{\varvec{\sigma }}$ ($x\in W(k)$). Such modules are always direct sums of cyclic submodules, and the cyclic modules are just truncations at smaller powers of V. Translating through the equivalence $\psi$, we have the following.

Lemma A.1

Let G be a finite unipotent abelian group scheme. Then kG is a isomorphic to a tensor product of algebras of the form $kW_{m,1} \simeq k[s]/s^{p^m}$.

Lemma A.2

Let G be a finite unipotent abelian group scheme. Assume that ${\text {dim}}_k{\text {Hom}}_{\mathsf {Gr}/k}(G,\mathbb {G}_{a(1)})=1$. If G does not have $W_{2,2}$ as a quotient, then G is isomorphic to a quotient of the group scheme $E_{m,n}$.

Proof

The condition ${\text {dim}}_k{\text {Hom}}_{\mathsf {Gr}/k}(G,\mathbb {G}_{a(1)})=1$ implies that the corresponding Dieudonné module is cyclic, $G_{\mathrm {ev}}\cong \psi (D_k/I)$ for some ideal I containing $V^m$ and $F^n$ for some m, n. Not having $W_{2,2}$ as a quotient implies that $p =FV$ kills $D_k/I$, and, hence, G is isomorphic to a quotient $D_k/(V^m, F^n, p)$. But the latter is precisely $E_{m,n}$.

The last thing we need is the classification of the quotients of the group scheme $E_{m,n}$. In terms of Dieudonné modules, we have

$$\begin{aligned} E_{m,n} = \psi (D_k/(V^m,F^n,p)). \end{aligned}$$

The isomorphism classes of quotients of $D_k/(V^m,F^n,p)$ were classified by Koch [31]. The main results of that paper may be stated as follows.

Theorem A.3

Every nonzero finite quotient of $\hat{D}_k/(p)$ as a left $\hat{D}_k$-module is isomorphic to either $M_{m,n}=D_k/(V^m,F^n,p)$ (of length $m+n-1$) or $M_{m,n,\mu }=D_k/(F^n-\mu V^m,p)$ (of length $m+n$) for some $m,n \ge 1$ and $0\ne \mu \in k$. The only isomorphisms among these modules are given by $M_{m,n,\mu }\cong M_{m,n,\mu '}$ if and only if $\mu /\mu '=a^{p^{m+n}-1}$ for some $a\in k$.

Outline of proof

Let M be a nonzero finite quotient of $\hat{D}_k/(p)$, let m be the height of $M^\sharp$ and n be the height of M. Then M is a finite quotient of $D_k/(V^m,F^n,p)$. So either M is isomorphic to $D_k/(V^m,F^n,p)$ or the kernel is at least one dimensional. If the kernel has length one, then it is in the socle, which has length two, and is the image of $V^{m-1}$ and $F^{n-1}$. By minimality of m and n, the kernel is then $(F^{n-1}-\mu V^{m-1})$ for some $0\ne \mu \in k$. If M is equal to this, we have $M \cong M_{m-1,n-1,\mu }$. Otherwise M is a proper quotient of $M_{m-1,n-1,\mu }$. But the socle of $M_{m-1,n-1,\mu }$ is one dimensional, spanned by the image of $V^{m-1}$, so in this case M is a quotient of $M_{m-1,n-1}$, which implies that m and n are not minimal. This contradiction proves that these are the only isomorphism types.

The dimensions of $M/F^iM$ and $M/V^iM$ distinguish all isomorphism classes, with the possible exception of isomorphisms between $M_{m,n,\mu }$ and $M_{m,n,\mu '}$. Such an isomorphism is determined modulo radical endomorphisms by a scalar $a\in k^\times \subseteq W(k)^\times$. The equation $(F^n-\mu V^m)a=b(F^n-\mu ' V^m)$ implies that $b=a^{{\varvec{\sigma }}^n}$ and $\mu a=b^{{\varvec{\sigma }}^m}\mu '$. Thus

$$\begin{aligned} \mu /\mu '=a^{{\varvec{\sigma }}^{m+n}}a^{-1}=a^{p^{m+n}-1}. \end{aligned}$$

$\square$

Remark A.4

Note that if $k=\mathbb {F}_p$ then this condition on $\mu$ and $\mu '$ is only satisfied if $\mu =\mu '$, so there are $p-1$ isomorphism classes of $M_{m,n,\mu }$. But if k is algebraically closed then the isomorphism type of $M_{m,n,\mu }$ is independent of $\mu$.