DETECTING NILPOTENCE AND PROJECTIVITY OVER FINITE UNIPOTENT SUPERGROUP SCHEMES

DAVE BENSON, SRIKANTH B. IYENGAR, HENNING KRAUSE AND JULIA PEVTSOVA

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 ≥ 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 ⊆ 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 ⊆ 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.

Contents

1. Introduction  2
Part 1. Recollections  5
2. Finite supergroup schemes  5
3. Low degree cohomology  9
4. Steenrod operations  10
5. Super Quillen–Venkov  13
6. Nilpotence and projectivity  17
7. Inductive detection theorem  18
Part 2. The detection theorem  19
8. Witt elementary supergroup schemes  19
9. Cohomological calculations  25
10. Cohomological characterisation of elementary supergroups  32
11. The main detection theorem  37
12. The Steenrod algebra  40
Appendix A. Witt vectors and Dieudonné modules  42
References  45

Date: January 24, 2019. The work was supported by the NSF grant DMS-1440140 while DB, SBI and JP were in residence at the

MSRI. SBI was partly supported by NSF grant DMS-1700985 and JP was partly supported by NSF grants DMS-0953011 and DMS-1501146.

1. Introduction

There has been considerable research, some of recent vintage, aimed at understanding representations of finite group schemes through the lens of their support varieties; see [3, 4, 5, 6, 10, 12, 11, 27, 28]. The paradigm for these developments is the work on the modular representation theory of finite groups due to Alperin and Evens [1], Avrunin and Scott [2], Chouinard [18], Carlson [16], Dade [19], Quillen [38], among others. This paper is part of a project aimed at finding analogues of some of these results and techniques for finite supergroup schemes. The first step in this direction was taken by Drupieski [21, 22], who proved finite generation of cohomology for finite supergroup schemes, generalizing the theorem of Friedlander and Suslin for finite group schemes [29]. Drupieski and Kujawa [23, 24, 25] have initiated a study of support varieties for restricted Lie superalgebras.

A starting point for any theory of support varieties is the identification of a family of subgroups that detect nilpotence of cohomology classes and projectivity of representations. Once again, finite groups provide a model: Quillen [38] proved that a class in mod p cohomology of a finite group G is nilpotent if (and only if) its restriction to any elementary abelian p-subgroup E < G is nilpotent in H^{∗}(E, \overline{𝔽}_{p}); see also Quillen and Venkov [39]. This detection theorem is a key ingredient in the proof of Quillen’s stratification theorem that gives a complete description of the Zariski spectrum of H^{∗}(G, \overline{𝔽}_{p}). Around the same time, Chouinard [18] proved that a representation M of G is projective if (and only if) the restriction of M to any elementary abelian p-subgroups E < G is projective.

In this work we establish analogues of the results of Quillen and Chouinard for finite supergroup schemes. Throughout we fix a perfect field k of positive characteristic p ≥ 3. A finite supergroup scheme over k may be viewed either as a functor on the category of ℤ/2-graded commutative k-algebras with values in finite groups, or a finite dimensional ℤ/2-graded cocommutative Hopf algebra; see Section 2 for details. The focus will be on unipotent supergroup schemes, though some of the preliminary results apply more generally. Each finite supergroup scheme has an even part which is a finite group scheme. In turn any finite group or group scheme furnishes an example of a supergroup scheme, but there are many more. Notably, the odd version of the additive group 𝔾_{a}, denoted 𝔾_{a}^{−} and defined as a functor by 𝔾_{a}^{−}(R) = R_{1}^{+}, the additive group on the odd part of R. The corresponding Hopf algebra is k[σ]/(σ^{2}), where σ is in odd degree and a primitive element.

The notion of an “elementary” supergroup scheme is a lot more involved than in the case of finite groups. To begin with, we construct a two-parameter family of finite supergroup schemes related to the Witt vectors, denoted E_{m,n}^{−}, with m ≥ 2, n ≥ 1; see Construction 8.5. For example, E_{m,1}^{−} can be realised as an extension of 𝔾_{a}^{−} by W_{m,1}, the first Frobenius kernel of Witt vectors of length m, recalled in Appendix A. Also E_{1,n}^{−} ≅ 𝔾_{a(n)} × 𝔾_{a}^{−}, where 𝔾_{a(n)} is the nth Frobenius kernel of 𝔾_{a}.

Definition 1.1. A finite supergroup scheme E over k is elementary if it is isomorphic to a quotient of some E_{m,n}^{−} × (ℤ/p)^{×s}.

A special role is played by the quotients of E_{m,n}^{−} by an even subgroup scheme; these are the Witt elementary supergroup schemes, and described completely in Theorem 8.13. Besides the E_{m,n}^{−} themselves, one has also finite supergroup schemes that we denote E_{m,n,μ}^{−}, involving an element μ in k^{×}/(k^{×})^{p^{m+n}−1}. The Hopf algebra corresponding to E_{m,n,μ}^{−} is described in (8.10). Any elementary supergroup scheme is of the form E ≅ E^{0} × (ℤ/p)^{×s} where E^{0} is isomorphic to either 𝔾_{a(r)} or a Witt elementary supergroup scheme.

The group algebra kE of an elementary finite supergroup scheme E is isomorphic to a tensor product of algebras of the form

(i) k[s]/(s^{p})

(ii) k[σ]/(σ^{2}), and (iii) k[s, σ]/(s^{p^{n}}, σ^{2} − s^{p}), where n ≥ 1,

with |s| even and |σ| odd, and no more than one factor of types (ii) and (iii) combined is present. In particular, there is at most one generator of odd degree, and as an ungraded algebra kE is a commutative complete intersection, even though case (iii) is not graded commutative.

Our main detection theorem is proved in Section 11.

Theorem 1.2. Let G be a finite unipotent supergroup scheme over a field k of positive characteristic p ≥ 3. Then the following hold.

(i) An element x ∈ H^{∗,∗}(G, k) is nilpotent if and only if for every extension field K of k and every elementary sub-supergroup scheme E of G_{K}, the restriction of x_{K} ∈ H^{∗,∗}(G_{K}, K) to H^{∗,∗}(E, K) is nilpotent.

(ii) A kG-module M is projective if and only if for every extension field K of k and every elementary sub-supergroup scheme E of G_{K}, the restriction of M_{K} to E is projective.

We also prove two versions of (i) for arbitrary coefficients. Theorem 11.1(i) proves the detection of nilpotents for H^{∗,∗}(G, M) for any G-module M where nilpotents are understood in the sense of Definition 6.1. Theorem 11.2, which generalises a theorem of Bendel [3] for unipotent group schemes, gives detection of nilpotents for H^{∗,∗}(G, Λ) with coefficients in a unital G-algebra Λ.

We also formulate and prove ℤ-graded versions of our theorems, and apply them to finite dimensional subalgebras of the Steenrod algebra over 𝔽_{p}. The structure of the Steenrod algebra is well understood and the detection theorem in that case takes on a particularly simple form; see Theorem 12.4.

Looking ahead. Our results only cover unipotent supergroup schemes, and it would be interesting to understand what more needs to be done in order to cover the general case. Unlike the case of finite group schemes, for a general finite supergroup scheme it is not true that cohomology modulo nilpotents and projectivity of modules are detected on unipotent sub-supergroup schemes. Conversations with Chris Drupieski lead us to suspect that there is a mild generalisation of the Witt elementaries that are not unipotent, but which leads to a suitable detection family in this context.

In a different direction, the detection theorems are only the first steps towards developing a theory of support varieties. Again we turn to groups to show us the way: While Chouinard’s work highlights the role of elementary abelian groups, Dade [19] proved that to detect projectivity of a representation of an elementary abelian p-group E, one can restrict further to all cyclic shifted subgroups of the group algebra kE, which then becomes purely a problem in linear algebra. This detection theorem, now known as “Dade’s lemma”, is the foundation for the theory of rank varieties for modules for finite groups pioneered by Carlson [16], and further developed by Benson, Carlson, and Rickard [9]. Their work was absorbed and generalised to the theory of π-points for finite groups schemes by Friedlander and Pevtsova [27, 28].

Theorem 1.2 opens up the road to a theory of π-points for finite unipotent supergroup schemes. We take this up in a follow up paper [13], where it is used to establish a stratification theorem for the stable module category, akin to the one in [11].

Structure of the paper. The strategy of the proof of Theorem 1.2 is quite intricate and we found it expedient to divide the paper into two parts. Before delving into a summary of the parts, it would perhaps help to present a roadmap of the proof; it follows the one for finite unipotent group schemes given in [3], but a number of extra complications arise.

The simplest scenario is that there is a surjective map from G to either 𝔾_{a}^{−} × 𝔾_{a}^{−} or 𝔾_{a(1)} × 𝔾_{a(1)}, for then the argument in [3, Theorem 8.1] applies. Otherwise one reduces to the case where there is a surjective map f: G → 𝔾_{a(r)} × (𝔾_{a}^{−})^{ϵ} × (ℤ/p)^{s} with r, s ≥ 0 and ϵ = 0 or 1, such that H^{1,∗}(f) is an isomorphism. It is easy to tackle the case when f is itself an isomorphism. When it is not an isomorphism, a standard argument yields that H^{2,∗}(f) has a kernel. The situation when this kernel contains an element of odd degree, that is to say, when H^{2,1}(f) is not one-to-one, is dealt with in [14]. The difficulty arises when the kernel of H^{2,∗}(f) is concentrated in even degrees. Even here there are two cases, as elaborated on further below. The first one allows us to drop to proper subgroups and is easy to handle. The second one leads to elementary supergroup schemes. This is where the major deviation from [3] occurs, and requires the bulk of the work. It occupies Part II of this paper.

Here is a more detailed description of the paper: Part I, comprising Sections 2 to 7, provides background material on finite supergroup schemes and extensions of a number of techniques used in other contexts. Section 2 starts things off with main definitions, examples, and basic properties of supergroup schemes. Section 3 records some key facts on low degree cohomology modules. Section 4 describes the action of Steenrod operations on the cohomology of finite supergroup schemes. The central calculation there is Theorem 4.3 that establishes that a homogeneous ideal in H^{∗,∗}(𝔾_{a(r)} × 𝔾_{a}^{−} × (ℤ/p)^{×s}, k) stable under the Steenrod operations and containing an element from H^{2,0} must have an element of a specific form. The proof follows closely the proofs of the analogous result for (ℤ/p)^{×s}, due to Serre [40], for 𝔾_{a(r)}, due to Bendel, Friedlander, and Suslin [6], and for 𝔾_{a(r)} × (ℤ/p)^{×s} due to Bendel [3], but the conclusion is different. Whereas for finite group schemes, such an ideal always has an element that is a product of Bocksteins of elements in degree 1, in the super case we get either a product of Bocksteins or a mysterious element ζ^{2} − γx_{r} with |ζ| = (1, 1), |x_{r}| = (2, 0), γ ∈ k. This element is responsible for the work we have to do in Part II.

Part I culminates in Theorem 7.2 that asserts that if a finite unipotent supergroup satisfies certain conditions, laid out in Hypothesis 7.1, nilpotence (of cohomology elements) and projecitivty (of modules) are detected on proper sub-supergroup schemes after field extensions. For finite group schemes (not super ones) the calculation with the Steenrod operations in Section 4 would then yield that any unipotent group scheme that is not isomorphic to 𝔾_{a(r)} × (ℤ/p)^{×s} satisfies Hypothesis 7.1. And this is precisely the argument in Bendel [3]. Thus, up to the end of Part I we are mostly mimicking the techniques existing in the literature. Life in the super world turns out to be more complicated, all because of the cohomology class ζ^{2} − γx_{r} that cannot be eliminated with the help of the Steenrod operations. The task of the second part of the paper is to show that if a finite unipotent supergroup scheme does not satisfy Hypothesis 7.1, then, in fact, it must be elementary.

Part II begins in Section 8 with the construction of the elementary supergroup schemes featuring in the statement of Theorem 1.2. Their cohomology rings are calculated in Section 9. These calculations feed into the proof of Theorem 10.3 that is a cohomological criterion for recognising elementary supergroup schemes. Theorem 1.2 is proved as Theorem 11.1. Its consequences for the Steenrod algebra are described in Section 12. Appendix A provides background on Dieudonné modules needed to describe elementary supergroup schemes.

Acknowledgements. We gratefully acknowledge the support and hospitality of the Mathematical Sciences Research Institute in Berkeley, California where we were in residence during the semester on “Group Representation Theory and Applications” in the Spring of 2018. The American Institute of Mathematics in San Jose, California gave us a fantastic opportunity to carry out part of this project during intensive research periods supported by their “Research in Squares” program; our thanks to them for that. Dave Benson thanks Pacific Institute for Mathematical Science for its support during his research visit to the University of Washington in the Summer of 2016 as a distinguished visitor of the Collaborative Research Group in Geometric and Cohomological Methods in Algebra. Dave Benson and Julia Pevtsova have enjoyed the hospitality of City University while working on this project in the summers of 2017 and 2018. We are grateful to Chris Drupieski and Jon Kujawa for useful and informative conversations and their interest in our work.

Part 1. Recollections

2. Finite supergroup schemes

We give a compressed introduction to the terminology we shall employ in the paper referring the reader to a number of excellent sources on super vector spaces, super algebras and super groups schemes, such as, for example, a survey paper by A. Masuoka [34] or [23].

Throughout this manuscript k will be a field of positive characteristic p ≥ 3. We assume k is perfect since some of the structural results for supergroup schemes require that condition. It is clear that the main theorem holds for an arbitrary field k of characteristic p once it is proved for a perfect field of the same characteristic.

An affine supergroup scheme over k is a covariant functor from ℤ/2-graded commutative k-algebras (in the sense that yx = (−1)^{|x||y|}xy) to groups, whose underlying functor to sets is representable. If G is a supergroup scheme then its coordinate ring k[G] is the representing object. By applying Yoneda’s lemma to the group multiplication and inverse maps, it is a ℤ/2-graded commutative Hopf algebra. We denote the comultiplication on k[G] by Δ: k[G] → k[G] ⊗ k[G] and the counit map by ε: k[G] → k with I = ker ε being the augmentation ideal and note that these are degree-preserving (equivalently, even) algebra homomorphisms. The correspondence between affine supergroup schemes and their coordinate algebras gives a contravariant equivalence of categories between affine supergroup schemes and ℤ/2-graded commutative Hopf algebras.

A finite supergroup scheme G is an affine supergroup scheme whose coordinate ring is finite dimensional. In this case, the dual kG = Hom_{k}(k[G], k) is a finite dimensional ℤ/2graded cocommutative Hopf algebra called the group ring of G. This gives a covariant equivalence of categories between finite supergroup schemes and finite dimensional ℤ/2graded (equivalently, “super”) cocommutative Hopf algebras.

(2.1)  \{ \begin{matrix}finite super- \\ group schemes \end{matrix} \} ∼ \{ \begin{matrix}finite dimensional super- \\ cocommutative Hopf algebras \end{matrix} \}

We employ the notation V = V_{0} ⊕V_{1} for ℤ/2-graded (equivalently, “super”) vector spaces, where V_{0} are the even degree elements, and V_{1} are the odd degree elements. A kG-module is a ℤ/2-graded k-vector space on which kG acts respecting the grading in the usual way. As in the ungraded setting, a kG-module has an equivalent description as a rational representation of the supergroup G on the category of super vector spaces. We consider all modules including infinite dimensional ones. The trivial module k is the trivial one dimensional representation concentrated in the even degree.

If K is a field extension of k, and G is an affine supergroup scheme, we write K[G] for K ⊗_{k} k[G], which is a graded commutative Hopf algebra over K. This defines a supergroup scheme over K denoted G_{K}, and we have a natural isomorphism of Hopf superalgebras KG_{K} ≅ K ⊗_{k} kG.

For each kG-module M, we set

M_{K} := K ⊗_{k} M and M^{K} := Hom_{k}(K, M),

viewed as KG_{K}-modules.

The even part G_{ev} of an affine supergroup scheme G is the largest sub-supergroup scheme whose coordinate ring contains no odd degree elements (see [34]). It may be regarded as an affine group scheme. Its coordinate ring k[G_{ev}] is the quotient of k[G] by the ideal generated by the odd degree elements. This ideal is automatically a Hopf ideal, since the coproduct Δ applied to an odd degree element is necessarily a linear combination of tensors a ⊗ b where either a or b is odd. An even subgroup scheme of G is a subgroup scheme of G_{ev}.

Example 2.1. Any affine group scheme G may be thought of as an affine supergroup scheme with G = G_{ev}.

Another way to look at the assignment G ↦ G_{ev} is that it gives the right adjoint to the inclusion functor from the affine group schemes to affine supergroups schemes.

Definition 2.2. If G is an affine supergroup scheme, let G^{(1)} be the base change of G via the Frobenius map x ↦ x^{p} on k. Then the Frobenius map F: G → G^{(1)} corresponds to the map of coordinate rings k[G^{(1)}] → k[G] given by x → x^{p}. The rth Frobenius kernel G_{(r)} of G is defined to be the kernel of the iterate F^{r}: G → G^{(r)}.

Convention 2.3. By G_{(0)} we always mean the trivial group scheme.

Definition 2.4. A finite supergroup scheme G over k is said to be unipotent if k is the unique irreducible kG-module, which may be either in even or odd degree. A supergroup scheme G is connected if k[G] is local.

If G is a finite connected supergroup scheme then for some r ≥ 0 we have G = G_{(r)}. The least such value of r is called the height of G. Note that G has height zero if and only if G is the trivial supergroup scheme.

Lemma 2.5. Any finite supergroup scheme G is a semidirect product G^{0} ⋊ π_{0}(G) with G^{0} connected and π_{0}(G) the finite group of connected components.

Proof. See Lemma 5.3.1 of Drupieski [21]. The proof uses the fact that k is perfect and has odd prime characteristic. □

Theorem 2.6. Let G be a connected finite supergroup scheme. Then there exist odd degree elements y_{1}, …, y_{n} ∈ k[G] such that we have an isomorphism of ℤ/2-graded k-algebras

k[G] ≅ k[G_{ev}] ⊗ Λ(y_{1}, …, y_{n}).

In particular, if G_{ev} is non-trivial then G has the same height as G_{ev}.

Proof. Let I be the augmentation ideal of k[G]. Pick odd elements {y_{1}, …, y_{n}} such that their residues give a basis of the odd part of the super vector space I/I^{2}. Then the ideal (y_{1}, …, y_{n}) is a Hopf ideal, and we have an isomorphism k[G]/(y_{1}, …, y_{n}) ≅ k[G_{ev}]. Since k[G_{ev}] is a connected finite group scheme, we can find algebraic generators x′_{1}, …, x′_{m} ∈ k[G_{ev}] such that k[G_{ev}] is a truncated polynomial algebra on these generators ([43, 14.4]). Let x_{1}, …, x_{m} ∈ I be even liftings of x′_{1}, …, x′_{m} to k[G], and let B be the (even) subalgebra of k[G] generated by x_{1}, …, x_{m}. By construction {x_{1}, …, x_{m}} give a basis of the even part of I/I^{2}. Moreover, the odd elements y_{1}, …, y_{n} square to zero and (super) commute, hence, generate a copy of Λ(y_{1}, …, y_{n}) in k[G]. We therefore have a surjective map

f: B ⊗ Λ(y_{1}, …, y_{n}) → k[G]

We wish to show that this map is an isomorphism of algebras. The augmentation ideals are the same on both sides by constructions, and, hence, it suffices to show that f induces an isomorphism on the associated graded algebras. Note that gr k[G] ≅ ⨁ I^{i}/I^{i+1} inherits the structure of a Hopf algebra.

If f is not an isomorphism, its kernel contains a nonzero polynomial involving both x_{i} and y_{i}. Choose one which involves the minimal number of the variables y_{i}, let r be the maximal index such that this polynomial involves y_{r}, and write it in the form

a + by_{r} = 0

where a and b only involve B and y_{1}, …, y_{r−1}. Apply the coproduct map Δ to obtain

Δ(a) + Δ(b)(y_{r} ⊗ 1 + 1 ⊗ y_{r}) = 0 .

Since Δ(b) = b⊗1+1⊗b+I ⊗I ([30, I.2], there is a term b⊗y_{r} in the sum which must vanish. We conclude that b = 0 and, hence, a = 0, contradicting the minimality of r. This proves that f is an isomorphism. In particular, B does not intersect the ideal (y_{1}, …, y_{n}), and so the projection map k[G] → k[G]/(y_{1}, …, y_{n}) ≅ k[G_{ev}] induces an isomorphism B ≅ k[G_{ev}]. □

Remark 2.7. (1) Masuoka [33, Theorem 4.5] proves, without the finiteness hypothesis,

that there is counital algebra isomorphism k[G] ≅ k[G_{ev}] ⊗ Λ((Lie G)_{odd}^{#}).

(2) Since k[π_{0}(G)] sits in even degree, Lemma 2.5 implies that the tensor decomposition of Theorem 2.6 holds for any finite supergroup scheme.

(3) The structure of the coordinate ring of an ungraded finite connected group scheme is known ([43, Theorem 14.4]). Putting it together with Theorem 2.6, we conclude that for any finite connected supergroup scheme G there exists a k-algebra isomorphism

k[G] ≅ k[x_{1}, …, x_{n}]/(x_{1}^{p^{i_{1}}}, …, x_{n}^{p^{in}}) ⊗ Λ(y_{1}, …, y_{m})

where x_{i} are even and y_{j} are odd.

(4) The Frobenius map F: k[G^{(1)}] → k[G] kills k[G^{(1)}]_{odd} since odd elements square to 0 by supercommutativity. Hence, the image of F lands in k[G_{ev}], that is, the composite k[G/G_{(1)}] → k[G] → k[G_{ev}] is injective.

Corollary 2.8. If G is a finite supergroup scheme then G = G_{ev}G_{(1)}.

Proof. It follows from Lemma 2.5 that G_{ev} = G_{ev}^{0} ⋊ π_{0}(G). So we may assume that G is connected. It then follows from Theorem 2.6 (see By Remark 2.7 the composite k[G/G_{(1)}] → k[G] → k[G_{ev}] is injective. Since this is an injective map of Hopf algebras, is it faithfully flat (see, for example, [43, Theorem 14.1]) and, therefore, the corresponding map on group schemes G_{ev} → G → G/G_{(1)} is surjective. Hence, G = G_{ev}G_{(1)}. □

Warning 2.9. The subgroup G_{(1)} is normal in G, but G_{ev} need not be normal.

Example 2.10. The additive (super)group scheme 𝔾_{a} is a purely even group scheme, given by the assignment

𝔾_{a}(R) = R_{0}^{+},

where R_{0}^{+} the additive group on the even part of a superalgebra R. We have k[𝔾_{a}] = k[T] with the T primitive in even degree. The Frobenius kernels 𝔾_{a(r)} are purely even connected unipotent supergroup schemes with k[𝔾_{a(r)}] = k[T]/T^{p^{r}}, and T primitive.

Example 2.11. We denote by 𝔾_{a}^{−} the finite supergroup scheme such that k𝔾_{a}^{−} = k[σ]/(σ^{2}) with σ primitive in odd degree. Then 𝔾_{a}^{−} is connected and unipotent. As a functor, 𝔾_{a}^{−} is defined by 𝔾_{a}^{−}(R) = R_{1}^{+}, the additive group on the odd part of a superalgebra R.

More generally, let V be a finite-dimensional vector space, and let Λ^{∗}(V) be the ℤ/2graded exterior algebra on V where the elements of V are primitive of odd degree. With this convention, Λ^{∗}(V) becomes a supercommutative Hopf algebra and, hence, is isomorphic to a group algebra of a product of copies of 𝔾_{a}^{−}, and hence corresponds to a connected unipotent finite supergroup scheme.

Example 2.12. Let W_{1,1}^{−} be the finite supergroup scheme such that kW_{1,1}^{−} = k[σ]/(σ^{2p}) with σ primitive in odd degree. Then W_{1,1}^{−} has height 1 and sits in a nonsplit short exact sequence

(2.2)  1 → 𝔾_{a(1)} → W_{1,1}^{−} → 𝔾_{a}^{−} → 1.

More generally, let W_{m,1}^{−} be the finite supergroup scheme with kW_{m,1}^{−} = k[σ]/(σ^{2p^{m}}) where σ is primitive in odd degree and m ≥ 1. Then W_{m,1}^{−} has height one, and it sits in a nonsplit short exact sequence

(2.3)  1 → W_{m,1} → W_{m,1}^{−} → 𝔾_{a}^{−} → 1

where W_{m,1} denotes the Witt vectors of length m and height one as described in Appendix A, whose group algebra is kW_{m,1} = k[s]/(s^{p^{m}}), and s = σ^{2} is primitive in even degree.

Example 2.13. A p-restricted Lie superalgebra 𝔤 = 𝔤_{0} ⊕𝔤_{1} is a ℤ/2-graded Lie algebra with a p-restriction map on the even part, and such that the odd part is a p-restricted module over the even part. The p-restricted enveloping algebra U^{[p]}(𝔤) is the group algebra of a connected finite supergroup scheme which is unipotent if and only if 𝔤 is unipotent.

Lemma 2.14. Let G be a finite supergroup scheme. Then the primitive elements in kG form a p-restricted Lie superalgebra 𝔤 = Lie(G) over k with Lie bracket given by commutator and p-restriction map given by the p-power map in kG. The natural map U^{[p]}(𝔤) → kG induces an isomorphism U^{[p]}(𝔤) → kG_{(1)}.

Proof. See Lemma 4.4.2 of Drupieski [21]. □

Example 2.15. Example 2.12 has height one, so is of the form U^{[p]}(𝔤). The p-restricted Lie superalgebra 𝔤 is generated by an element σ in odd degree with relation [σ, σ]^{[p^{m}]} = 0.

Remark 2.16. If G is a finite connected supergroup scheme of height 1 with the corresponding Lie algebra 𝔤, then 𝔤_{0} is an even (restricted) Lie algebra corresponding to G_{ev}, that is, U^{[p]}(𝔤_{0}) ≅ kG_{ev}.

Lemma 2.17. A finite unipotent supergroup scheme G with G_{ev}^{0} = 1 is isomorphic to (𝔾_{a}^{−})^{×r} ⋊ (ℤ/p)^{×s}.

Proof. The assumption G_{ev}^{0} = 1 implies that G_{0} has height 1, and, hence, corresponds to a Lie superalgebra 𝔤. By Remark 2.16, 𝔤_{0} = 0, therefore, U^{[p]}(𝔤) = Λ^{∗}(𝔤_{1}), and, hence, G_{0} = (𝔾_{a}^{−})^{× dim 𝔤_{1}}. The statement follows from Lemma 2.5. □

For sub-supergroup schemes H, H′ ⩽ G, the commutator sub-supergroup scheme is defined as in [20, II.5.4.8] as a representable functor. We need an analogue of the following standard result in group theory.

Lemma 2.18. Let G be a finite supergroup scheme, and H, H′ ⊴G be normal sub-supergroup schemes. Then [H, H′] is normal in G.

Proof. It suffices to check pointwise that for a ∈ H(R), b ∈ H′(R) and c ∈ G(R), we have that [a, b]^{c} ∈ [H(R), H′(R)], where the latter commutator is as of discrete groups. This follows from the obvious identity

c(aba^{−1}b^{−1})c^{−1} = cac^{−1}cbc^{−1}ca^{−1}c^{−1}cb^{−1}c^{−1}.  □

3. Low degree cohomology

The cohomology H^{∗,∗}(G, k) of a finite supergroup scheme G is isomorphic to Ext_{kG}^{∗,∗}(k, k). The first index is homological, and the second is the internal ℤ/2-grading. Drupieski [21, 22] has proved that H^{∗,∗}(G, k) is a finitely generated k-algebra, which is graded commutative in the sense that if x ∈ H^{m,α}(G, k) and y ∈ H^{n,β}(G, k) then

yx = (−1)^{mn}(−1)^{αβ}xy.

We start by identifying the first cohomology group of G.

Lemma 3.1. Let G be a finite supergroup scheme with the group of connected components π. Then we have H^{1,0}(G, k) = Hom_{Gr/k}(G, 𝔾_{a}) and H^{1,1}(G, k) = Hom_{sGr/k}(G^{0}, 𝔾_{a}^{−})^{π}. Moreover, Hom_{sGr/k}(G, 𝔾_{a}) ≅ Hom_{sGr/k}(G^{0}, 𝔾_{a})^{π} × Hom_{Gr/k}(π, 𝔾_{a}).

Proof. Identification of H^{1,∗} with Hom follows from the standard cobar resolution used to compute cohomology H^{∗,∗}(G, k). The last statement is proved as in [3, Lemma 5.1]. □

Lemma 3.2. If G is a non-trivial unipotent finite supergroup scheme then there is a nontrivial homomorphism from G to either 𝔾_{a(1)} or 𝔾_{a}^{−} or ℤ/p.

Proof. Since G is unipotent, the group of connected components π is a p-group. If there are no non-trivial maps to ℤ/p, then π is trivial and G is connected. For a finite connected supergroup scheme, if there are no non-trivial homomorphisms from G to 𝔾_{a(1)} then there are also none to 𝔾_{a}. So if there are also no non-trivial homomorphisms from G to 𝔾_{a}^{−}, Lemma 3.1 yields Ext_{kG}^{1,∗}(k, k) = 0. As kG is a local ring this implies G is trivial. □

If f: G → G′ is a group homomorphism then f^{∗}: H^{∗,∗}(G′, k) → H^{∗,∗}(G, k) preserves both the homological and the internal degree, and commutes with the Steenrod operations (to be discussed in Section 4). If N is a normal sub-supergroup scheme of G then there is the Lyndon-Hochschild–Serre spectral sequence

H^{∗,∗}(G/N, H^{∗,∗}(N, k)) ⇒ H^{∗,∗}(G, k)

in which the internal degrees are carried along, and preserved by all the differentials. The spectral sequence also gives the five-term exact sequence:

\xymatrix{1 \ar[r] & H^{1}(G/N, k) \ar[r] & H^{1}(G, k) \ar[r] & H^{1}(N, k)^{G/N} \ar[r]^{d_{2}} & H^{2}(G/N, k) \ar[r] & H^{2}(G, k).}

Lemma 3.3. Let L′ ⊂ L be Lie superalgebras such that L′ is odd and central in L and L/L′ is even. Then L ≃ L′ × L/L′.

Proof. Let L_{ev} be the p-restricted Lie sub-superalgebra of even elements in L. The assumption implies that it is normal and isomorphic to L/L′; hence, L ≃ L′ × L_{ev}. □

Lemma 3.4. If a unipotent finite supergroup scheme G has H^{1,1}(G, k) = 0, then G = G_{ev}.

Proof. Since G/G_{(1)} is even by Corollary 2.8, the Lyndon-Hochschild-Serre spectral sequence applied to the supergroup extension 1 → G_{(1)} → G → G/G_{(1)} → 1 implies that H^{∗,1}(G, k) = H^{∗,1}(G_{(1)}, k). Hence, the assumption together with Lemma 3.1 imply that there are no non-trivial maps from G_{(1)} to 𝔾_{a}^{−}. We need to show that G_{(1)} is purely even.

Let L be the unipotent Lie superalgebra associated with G_{(1)}. Since L is unipotent, we can choose a central series

L_{1} ⊂ L_{0} ⊂ L

such that L/L_{0} is purely even and L_{0}/L_{1} ≃ Lie 𝔾_{a}^{−}. By Lemma 3.3, we get that L/L_{1} has Lie 𝔾_{a}^{−} as a direct factor, so there is a surjective map from G_{(1)} to 𝔾_{a}^{−}, a contradiction. □

The five-term exact sequence can be used in exactly the same way as in the proof of [5, Lemma 1.2], to prove the following analogue.

Lemma 3.5. Let f: G → \overline{G} be a surjective homomorphism of unipotent supergroup schemes. If f^{∗}: H^{1,∗}(\overline{G}, k) → H^{1,∗}(G, k) is an isomorphism and f^{∗}: H^{2,∗}(\overline{G}, k) → H^{2,∗}(G, k) is injective then f is an isomorphism. □

Remark 3.6. Lemma 3.1 implies that the condition that f^{∗}: H^{1,∗}(\overline{G}, k) → H^{1,∗}(G, k) is an isomorphism guarantees that any homomorphism from G to 𝔾_{a(1)}, 𝔾_{a}^{−} and ℤ/p factors through \overline{G}.

4. Steenrod operations

The Steenrod algebra acts on the cohomology of any ℤ-graded cocommutative Hopf algebra, and hence also on the cohomology of any affine supergroup scheme ([35, Theorem 11.8], [44]). We recall how the Steenrod operations act using the re-indexing introduced in [14]. In order to make the indexing work for ℤ/2-graded algebra, we index with half-integers.

For p odd, there are natural operations

𝒫^{i}: H^{s,t}(G, k) → H^{s+2i(p−1),pt}(G, k)
β𝒫^{i}: H^{s,t}(G, k) → H^{s+1+2i(p−1),pt}(G, k),

defined in the following cases: when t is even, then i ∈ ℤ, and if t is odd, then i ∈ ℤ + \frac{1}{2}. Note that since p is odd, pt is congruent to t mod 2, so the operations preserve internal degree as elements of ℤ/2.

The Steenrod operations satisfy the following properties:

(i) 𝒫^{i} = 0 if either i < 0 or i > s/2, β𝒫^{i} = 0 if either i < 0 or i ≥ s/2;

(ii) Semi-linearity: 𝒫^{i}(ax) = a^{p}𝒫^{i}(x) for a ∈ k; (iii) 𝒫^{i}(x) = x^{p} if i = s/2; (iv) Cartan formula:

𝒫^{j}(xy) = ∑_{i} 𝒫^{i}(x)𝒫^{j−i}(y),

β𝒫^{j}(xy) = ∑_{i}(β𝒫^{i}(x)𝒫^{j−i}(y) + 𝒫^{i}(x)β𝒫^{j−i}(y));

(v) The 𝒫^{i} and β𝒫^{i} satisfy the Adem relations.

We record its action on H^{∗,∗}(𝔾_{a}^{−}, k) (see [14, Proposition 3.1]).

Proposition 4.1. One has H^{∗,∗}(𝔾_{a}^{−}, k) ≅ k[ζ], a polynomial ring on ζ in degree (1, 1). The

action of the Steenrod operations on H^{∗,∗}(𝔾_{a}^{−}, k) are given by 𝒫^{\frac{1}{2}} (ζ) = ζ^{p}, β𝒫^{\frac{1}{2}} (ζ) = 0. □

Next, we describe the analogue of Proposition 3.6 of [3] for G = 𝔾_{a(r)} × (𝔾_{a}^{−})^{ϵ} × (ℤ/p)^{×s} with r, s ≥ 0, ϵ = 0 or 1. If ϵ = 1 we have

H^{∗,∗}(G, k) = k[x_{1}, …, x_{r}] ⊗ Λ(λ_{1}, …, λ_{r}) ⊗ k[ζ] ⊗ k[z_{1}, …, z_{s}] ⊗ Λ(y_{1}, …, y_{s})

while if ϵ = 0 the term k[ζ] is missing. The degrees and action of the Steenrod algebra are as follows. We make the assumption that λ_{r+1} = 0 = x_{r+1}, that is, 𝒫^{0} kills λ_{r} and x_{r}.

degree 𝒫^{0} β𝒫^{0} 𝒫^{\frac{1}{2}} β𝒫^{\frac{1}{2}} 𝒫^{1}  𝒫^{i}  β𝒫^{i}
(i ≥ 2) (i ≥ 1)
λ_{i} (1, 0) λ_{i+1} −x_{i}  0  0  0
y_{i} (1, 0) y_{i}  z_{i}  0  0  0
ζ (1, 1)  ζ^{p}  0
x_{i} (2, 0) x_{i+1} 0  x_{i}^{p}  0  0
z_{i} (2, 0) z_{i}  0  z_{i}^{p}  0  0

Table 1. Steenrod operations

We recall the following theorem of Serre [40] which is a prototype for both Proposition 3.6 of [3] and Theorem 4.3 and will be used in the proof. The precise result we quote is a special case of Proposition 3.2 of [42]; it differs slightly from Serre’s original formulation since we need to consider arbitrary coefficients, not just 𝔽_{p}.

Theorem 4.2. Let I be a homogeneous ideal in H^{∗}((ℤ/p)^{×s}, k) stable under the Steenrod operations. If I contains a nonzero element of degree two, then there exists a finite family {u_{i}} ⊂ H^{2}((ℤ/p)^{×s}, k), each of which is a non-trivial linear combination of {z_{j}} with coefficients in 𝔽_{p} such that the product ∏ u_{i} ∈ H^{∗}((ℤ/p)^{×s}, k) lies in I. □

Let f: G → \overline{G} ≅ 𝔾_{a(r)} × (𝔾_{a}^{−})^{ϵ} × (ℤ/p)^{×s}, (r, s ≥ 0, ϵ = 0 or 1) be a surjective map of finite unipotent supergroup schemes. The proof of Theorem 11.1 uses in an essential way the description of the kernel of the induced map on degree 2 cohomology

(4.1)  f^{∗}: H^{2,∗}(\overline{G}, k) → H^{2,∗}(G, k)

under the assumption that

f^{∗}: H^{1,∗}(\overline{G}, k) →^{∼} H^{1,∗}(G, k)

is an isomorphism. There are two scenarios: Theorem 4.3 deals with the case when the kernel I = ker f^{∗} has an element of degree (2, 0) whereas Theorem 4.4 considers the case of degree (2, 1). We use extensively the observation that I is stable under the Steenrod operations.

The following theorem includes the case \overline{G} = (𝔾_{a}^{−})^{ϵ} × (ℤ/p)^{×s} in disguise; it corresponds to r = 0 per our convention that 𝔾_{a(0)} = 1.

Theorem 4.3. Let \overline{G} = 𝔾_{a(r)}×(𝔾_{a}^{−})^{ϵ}×(ℤ/p)^{×s}, with r, s ≥ 0, ϵ = 0 or 1. Let I ⊆ H^{∗,∗}(\overline{G}, k) be a homogeneous ideal stable with respect to the action of the Steenrod operations. Suppose I contains a nonzero element of degree (2, 0). Then one of the following holds:

(i) Some element of the form x_{r}^{n}β𝒫^{0}(v_{1}) … β𝒫^{0}(v_{m}) (with n and m not both zero) lies in I, where v_{1}, …, v_{m} are nonzero elements of H^{1}((ℤ/p)^{×s}, 𝔽_{p}) ⊆ H^{1,0}(\overline{G}, k), or

(ii) I ∩ H^{2,0}(\overline{G}, k) is one dimensional, spanned by an element of the form ζ^{2} + γx_{r} with γ ∈ k.

Proof. We follow, for the most part, the notation and proof in Proposition 3.6 of [3], with adjustments as appropriate to deal with the extra factor, (𝔾_{a}^{−})^{ϵ}.

Any nonzero element u in I ∩ H^{2,0}(\overline{G}, k) has the form

u = αζ^{2} + ∑_{1≤i<j≤r} a_{i,j}λ_{i}λ_{j} + ∑_{1≤j≤r} b_{j}x_{j} + ∑_{1≤i≤r,1≤j≤s} c_{i,j}λ_{i}y_{j} + ∑_{1≤i<j≤s} d_{i,j}y_{i}y_{j} + ∑_{1≤j≤s} e_{j}z_{j}

for scalars α, a_{i,j}, b_{j}, c_{i,j}, d_{i,j}, e_{j} which are not all zero, and the term αζ^{2} only occurs if ϵ = 1.

First suppose that each such u has α ≠ 0. In this case I ∩ H^{2,0}(\overline{G}, k) is one dimensional and ϵ = 1. Furthermore, u has to be sent to a multiple of itself by 𝒫^{0}. The Cartan formula implies ζ^{2} is killed by 𝒫^{0}, and so is u. The condition 𝒫^{0}(u) = 0 forces u to be of the form

(4.2)  u = ζ^{2} + ∑_{1≤i<r} a_{i,r}λ_{i}λ_{r} + b_{r}x_{r} + ∑_{1≤j≤s} c_{r,j}λ_{r}y_{j}.

Assume on the other hand that there exists a u with α = 0. Repeated application of 𝒫^{0} to such a u results in an element of the form ∑_{1≤i<j≤s} d_{i,j}^{p^{ℓ}}y_{i}y_{j} + ∑_{1≤j≤s} e_{j}^{p^{ℓ}}z_{j}. So if at least one of the d_{i,j} or e_{j} is nonzero, we may apply Theorem 4.2, and this puts us in case (i) with m > 0 and n = 0. So we may assume

u = ∑_{1≤i<j≤r} a_{i,j}λ_{i}λ_{j} + ∑_{1≤j≤r} b_{j}x_{j} + ∑_{1≤i≤r,1≤j≤s} c_{i,j}λ_{i}y_{j}.

Repeatedly applying 𝒫^{0} and stopping just before we get zero, we can assume u has the form

(4.3)  u = ∑_{1≤i<r} a_{i,r}λ_{i}λ_{r} + b_{r}x_{r} + ∑_{1≤j≤s} c_{r,j}λ_{r}y_{j}.

So we are now in a situation where u has either the form (4.2) or (4.3), and in the first case I ∩ H^{2,0}(G, k) is one dimensional. In either case, if some c_{r,j} is nonzero, we apply β𝒫^{0} to get

β𝒫^{0}(u) = ∑_{1≤i<r} a_{i,r}^{p}λ_{i+1}x_{r} − ∑_{1≤j≤s} c_{r,j}^{p}x_{r}y_{j} ∈ I.

Applying β𝒫^{1}, we get

β𝒫^{1}β𝒫^{0}(u) = − ∑_{1≤i<r} a_{i,r}^{p^{2}}x_{i+1}x_{r}^{p} − ∑_{1≤j≤s} c_{r,j}^{p^{2}}x_{r}^{p}z_{j} ∈ I.

Now apply 𝒫^{p} to get

𝒫^{p}β𝒫^{1}β𝒫^{0}(u) = − ∑_{1≤i<r} a_{i,r}^{p^{3}}x_{i+2}x_{r}^{p^{2}} − ∑_{1≤j≤s} c_{r,j}^{p^{3}}x_{r}^{p^{2}}z_{j} ∈ I.

Successively applying 𝒫^{p^{2}}, 𝒫^{p^{3}}, …, we eventually conclude that I contains an element of the form ∑_{j} c_{r,j}^{p^{t}}x_{r}^{p^{t−1}}z_{j} = x_{r}^{p^{t−1}}(∑_{j} c_{r,j}^{p^{t}}z_{j}). The set of all such elements in I is stable under the Frobenius map (raising all the coefficients to the pth power), and therefore there is a nonzero element with coefficients in 𝔽_{p}. This puts us in case (i) with m = 1.

If every c_{r,j} = 0 but some a_{i,r} is nonzero, then

β𝒫^{1}β𝒫^{0}(u) = − ∑_{1≤i<r} a_{i,r}^{p^{2}}x_{i+1}x_{r}^{p}.

Now we apply 𝒫^{p}, then 𝒫^{p^{2}}, and so on, and just before we get zero, we get a multiple of a power of x_{r}. This gives case (i) with m = 0 and n > 1.

It remains to consider the case when all c_{r,j} and all a_{i,r} are zero. Then, if u has form (4.2) we are in case (ii), and if u has form (4.3) we are in case (i) with m = 0 and n = 1. □

To complete the description of the kernel of (4.1), we quote a result from [14] which describes what happens when the kernel of the map f^{∗}: H^{2,∗}(\overline{G}, k) → H^{2,∗}(G, k) has an element of degree (2, 1). Note that in this case we necessarily have ϵ = 1.

Theorem 4.4. Let G be a finite unipotent supergroup scheme and N a normal sub-super-group scheme with G/N ≅ 𝔾_{a}^{−} × 𝔾_{a(r)} × (ℤ/p)^{×s}. If the inflation H^{1,∗}(G/N, k) → H^{1,∗}(G, k) is an isomorphism and H^{2,1}(G/N, k) → H^{2,1}(G, k) is not injective then there exists a nonzero element ξ ∈ H^{1,1}(G, k) such that β𝒫^{0}(u)ξ^{p^{r+s−1}(p−1)} = 0 for all u ∈ H^{1,0}(G, k). □

5. Super Quillen–Venkov

We require an analogue of the Quillen–Venkov lemma ([39]). The proof in [39], and its later variants carry over to the present context; we adapt a purely representation–theoretic approach due to Kroll [32].

Remark 5.1. If H ≤ G is a maximal sub-supergroup scheme with G = G^{0} ⋊ π unipotent, then there are three possibilities for G/H, namely 𝔾_{a(1)}, ℤ/p, and 𝔾_{a}^{−}.

• If G/H ≅ 𝔾_{a(1)} then there is an element λ ∈ H^{1,0}(G^{0}, k)^{π} ⊆ H^{1,0}(G, k) corresponding to the homomorphism G → 𝔾_{a(1)} as in Lemma 3.1, and an associated element x = −β𝒫^{0}(λ) ∈ H^{2,0}(G, k).

• If G/H ≅ ℤ/p then there is an element y ∈ H^{1,0}(π, 𝔽_{p}) ⊆ H^{1,0}(G, k) corresponding to the homomorphism G → ℤ/p as in Lemma 3.1, and an associated element z = β𝒫^{0}(y) ∈ H^{2,0}(G, k).

• If G/H ≅ 𝔾_{a}^{−} then there is an element ζ ∈ H^{1,1}(G, k) ≅ Hom_{sGr/k}(G, 𝔾_{a}^{−}) corresponding to the homomorphism G → 𝔾_{a}^{−} as in Lemma 3.1.

For H < G, we denote by

ind_{H}^{G}: Mod H → Mod G

the induction functor which is the right adjoint to the restriction functor res_{H}^{G}: Mod G → Mod H (following the group scheme terminology here, as introduced, for example, in [30, I.3]). There is also the coinduction functor

coind_{H}^{G}: Mod H → Mod G

which is left adjoint to the restriction. In the unipotent case induction and coinduction are canonically isomorphic (see [30, I.3]) which we use implicitly in the proof below. If H ≤ G is a normal subgroup, then the kernel of the canonical map coind_{H}^{G} k → k is the relative syzygy Ω_{G/H}(k). The identity map on the trivial representation k induces a map

(5.1)  η: Ω(k) → Ω_{G/H}(k)

Similarly, we have a map

η′: Ω_{G/H}^{−1}(k) → Ω^{−1}(k).

We employ the same notation η′ for the shifts of this map.

Lemma 5.2. Let H ≤ G be a normal sub-supergroup scheme of a finite unipotent group scheme G with G/H isomorphic to ℤ/p or 𝔾_{a(1)}. Then z = β𝒫^{0}(y), respectively x = −β𝒫^{0}(λ) ∈ H^{2,0}(G, k) (cf. Remark 5.1), is represented by the composite

\xymatrix{Ω(k) \ar[r]^{η} & Ω_{G/H}(k) \ar[r]^{≅} & Ω_{G/H}^{−1}(k) \ar[r]^{η′} & Ω^{−1}(k)} .

Proof. We prove this in the case where G/H ≅ ℤ/p. The case G/H ≅ 𝔾_{a(1)} is proved by replacing z by x everywhere.

The cohomology class z = β𝒫^{0}(y) ∈ Ext_{G}^{2}(k, k) is represented by the extension

(5.2)  \xymatrix{0 \ar[r] & k \ar[r] & ind_{H}^{G} k \ar[r] & ind_{H}^{G} k \ar[r] & k \ar[r] & 0.}

This follows from the fact that ind_{H}^{G} k ≅ kG/H = kℤ/p, and this sequence is the inflation of the extension

\xymatrix{0 \ar[r] & k \ar[r] & kℤ/p \ar[r] & kℤ/p \ar[r] & k \ar[r] & 0}

for ℤ/p representing the corresponding cohomology class (see, for example, [7, I.3.4.2]).

Next consider the following commutative diagram with exact rows.

\xymatrix{0 \ar[r] & Ω^{2}(k) \ar[r] \ar[ddd]_{z} & P_{1} \ar[rr] \ar[dr] \ar[ddd] &  & P_{0} \ar[r] \ar[ddd] & k \ar[r] \ar[ddd]^{1} & 0 \\  &  &  & Ω(k) \ar[ur] \ar[d] \\  &  &  & Ω_{G/H}(k) \ar@{-->}[dr] \ar[dd]^{≅} \\ 0 \ar[r] & k \ar[r] \ar[ddd]_{1} & ind_{H}^{G} k \ar@{-->}[ur] \ar[rr] \ar@{-->}[dr] \ar[ddd] &  & ind_{H}^{G} k \ar[r] \ar[ddd] & k \ar[r] \ar[ddd]^{z} & 0 \\  &  &  & Ω_{G/H}^{−1}(k) \ar@{-->}[ur] \ar[d] \\  &  &  & Ω^{−1}(k) \ar@{-->}[dr] \\ 0 \ar[r] & k \ar[r] & P_{−1} \ar[rr] \ar@{-->}[ur] &  & P_{−2} \ar[r] & Ω^{−2}(k) \ar[r] & 0}

By (5.2) the sequence in the middle row represents z. So comparing with a projective resolution as in the top row, the comparison map Ω^{2}(k) → k represents z. Dually, comparing with an injective resolution as in the bottom row, the comparison map k → Ω^{−2}(k) also represents z. Therefore the vertical composite map in the middle of the diagram also represents z. □

Given ξ ∈ H^{s,t}(G, M), for each n ≥ 0 we write ξ^{n} for the class ξ^{⊗n} ∈ H^{ns,nt}(G, M^{⊗n}).

Proposition 5.3. Let H be a maximal sub-supergroup scheme of a finite supergroup scheme G, and let M be a G-module. Suppose that ξ ∈ H^{∗,∗}(G, M) restricts to zero on H. Then

(i) if G/H ≅ ℤ/p then ξ^{2} is divisible by the element z = β𝒫^{0}(y) ∈ H^{2,0}(G, k), (ii) if G/H ≅ 𝔾_{a(1)} then ξ^{2} is divisible by the element x = −β𝒫^{0}(λ) ∈ H^{2,0}(G, k), (iii) if G/H ≅ 𝔾_{a}^{−} then ξ is divisible by the element ζ ∈ H^{1,1}(G, k).

Proof. We shall start by proving (ii). Let ξ ∈ H^{n,∗}(G, M), and choose a map Ω^{n}(k) → M representing ξ; by abuse of notation we call this map ξ. We also use ξ to denote any shift of this map, as a map from Ω^{n+i}(k) to Ω^{i}(M) for i ∈ ℤ.

The exact sequence \xymatrix{k \ar[r]^{ε′} & ind_{H}^{G} k \ar[r] & Ω_{G/H}^{−1}(k)} induces a triangle in stmod(G)

M ⊗ Ω(Ω_{G/H}^{−1}(k)) ⟶^{1⊗η′} M ⟶^{1⊗ε′} M ⊗ ind_{H}^{G} k → M ⊗ Ω_{G/H}^{−1}(k).

The assumption that ξ restricts to zero on H means that the restriction of ξ: Ω^{n}(k) → M to H factors through a projective. Hence, so does the adjoint map

Ω^{n}(k) → ind_{H}^{G} M = M ⊗ ind_{H}^{G} k.

This adjoint factors as the composite of ξ with 1 ⊗ ε′. The fact that this composition factors through a projective implies that there exists a lifting ρ′: Ω^{n}(k) → Ω(Ω_{G/H}^{−1}(k)) making the following diagram commute:

\xymatrix{ & Ω^{n}(k) \ar@{-->}[dl]_{ρ′} \ar[d]^{ξ} \ar[dr] \\ M ⊗ Ω(Ω_{G/H}^{−1}(k)) \ar[r]^{1⊗η′} & M \ar[r]^{1⊗ε′} & M ⊗ ind_{H}^{G} k \ar[r] & M ⊗ Ω_{G/H}^{−1}(k).}

Shifting by Ω^{−1}, we get a commutative diagram

(5.3)  \xymatrix{ & Ω^{n−1}(k) \ar@{-->}[dl]_{ρ′} \ar[d]^{ξ} \\ M ⊗ Ω_{G/H}^{−1}(k) \ar[r]^{1⊗η′} & M ⊗ Ω^{−1}(k).}

Similarly, we can factor ξ: Ω(k) → M ⊗ Ω^{−n+1}(k) to obtain a commutative diagram

\xymatrix{Ω(k) \ar[r]^{η} \ar[d]^{ξ} & Ω_{G/H}(k) \ar@{-->}[dl]^{ρ} \\ M ⊗ Ω^{−n+1}(k)}

Tensoring with M, we get a commutative diagram

(5.4)  \xymatrix{M ⊗ Ω(k) \ar[r]^{1⊗η} \ar[d]^{1⊗ξ} & M ⊗ Ω_{G/H}(k) \ar@{-->}[dl]^{1⊗ρ} \\ M ⊗ M ⊗ Ω^{−n+1}(k)}

Putting (5.3) and (5.4) together, we get the following diagram, where the composite of the maps in the middle row is 1 ⊗ x by Lemma (5.2):

[figure: Ω^{n−1}(k); ρ′; ξ; M ⊗ Ω(k); 1⊗x; 1⊗η; 1⊗ξ; M ⊗ Ω_{G/H}(k); ≅; 1⊗ρ; M ⊗ Ω_{G/H}^{−1}(k); 1⊗η′; M ⊗ Ω^{−1}(k); M ⊗ M ⊗ Ω^{−n+1}(k)]

Completing the diagram, we see that ξ^{2} = (1 ⊗ ξ) ∘ ξ factors through x either on the left or on the right.

\xymatrix{Ω^{n+1}(k) \ar[d]^{ξ} \ar[rrr]^{x} &  &  & Ω^{n−1}(k) \ar@{-->}[dl]_{ρ′} \ar[d]^{ξ} \\ M ⊗ Ω(k) \ar[r]^{1⊗η} \ar[d]^{1⊗ξ} & M ⊗ Ω_{G/H}(k) \ar[r]^{≅} \ar@{-->}[dl]^{1⊗ρ} & M ⊗ Ω_{G/H}^{−1}(k) \ar[r]^{1⊗η′} & M ⊗ Ω^{−1}(k) \ar[d]^{1⊗ξ} \\ M ⊗ M ⊗ Ω^{−n+1}(k) \ar[rrr]^{1⊗1⊗x} &  &  & M ⊗ M ⊗ Ω^{−n−1}(k).}

The same argument works for part (i). Part (iii) is similar but easier. Namely, we have a short exact sequence of kG-modules

0 → k → ind_{H}^{G} k ⟶^{ε} k → 0.

We have failed to distinguish whether k is in even or odd degree, but the two ends are in opposite degrees. The connecting map for this in stmod(kG) is ζ, so we have a triangle

k → ind_{H}^{G} k ⟶^{ε} k ⟶^{ζ} Ω^{−1}(k).

If ξ: k → Ω^{−n}(M) restricts to the zero class on H then the composite with ind_{H}^{G} k ⟶^{ε} k is zero, and so ξ factors through ζ. □

6. Nilpotence and projectivity

We introduce the notion of nilpotence for cohomology classes and discuss its detection. This is closely related to the detection of projectivity.

Definition 6.1. Let G be a finite supergroup scheme and M be a G-module. We say that a class ξ ∈ H^{j,∗}(G, M) is nilpotent if there exists n ≥ 1 such that ξ^{n} ∈ H^{jn,∗}(G, M^{⊗n}) is zero.

In the remainder of the paper we employ the following terminology. Let G be a finite supergroup scheme, and let H be a family of subgroups after field extension, namely a family of pairs (H, K) where K is an extension field of k and H is a sub supergroup scheme of G_{K}. Note that the embeddings of H in G_{K} need not be defined over the ground field k.

We say that nilpotence of cohomology elements is detected on the family H if for any G-module M and cohomology class ξ ∈ H^{∗,∗}(G, M), we have that ξ is nilpotent if and only if res_{H}^{G_{K}}(ξ_{K}) ∈ H^{∗,∗}(H, M_{K}) is nilpotent for every (H, K) ∈ H.

Similarly, we say that projectivity of modules is detected on the family H if for any G-module M, we have that M is projective if and only if res_{H}^{G_{K}}(M_{K}) is projective as an H-module for every (H, K) ∈ H.

In particular, we say that nilpotence and projectivity are detected on proper subgroups of G after field extensions if the family H can be taken to be the family of all pairs (H, K) where K runs over all field extensions of k and H runs over all proper subgroups of G_{K}. In practice, it always suffices to take K to be an algebraically closed field of large enough finite transcendence degree over k.

Lemma 6.2. Let G be a finite supergroup scheme, M a G-module, and fix an element ξ ∈ H^{j,∗}(G, M) with j > 0. With ξ: k → Ω^{−j}(M) denoting also the corresponding map on modules, let X be the colimit

X = colim\{ \xymatrix{k \ar[r]^{ξ} & Ω^{−j}(M) \ar[r]^{1⊗ξ} & Ω^{−2j}(M^{⊗2}) \ar[r]^{1⊗1⊗ξ} & Ω^{−3j}(M^{⊗3}) \ar[r] & ⋯ \}}

Then ξ is nilpotent if and only if X is projective.

Proof. If ξ^{n} = 0, then the composite of any n consecutive maps in the system defining X factors through a projective, and so X is projective. Conversely, if X is projective, then the map k → X factors through a projective. Since k is finite dimensional, it factors through a finite dimensional projective, and hence a finite composite of maps in the defining system factors through a projective. This implies that the corresponding power of ξ is zero. □

Lemma 6.2 immediately implies the following result.

Theorem 6.3. Let G be a finite supergroup scheme. If a family H of proper sub supergroup schemes after field extensions detects projectivity of G-modules, then it also detects nilpotence of cohomology elements.

Proof. Let M be a G-module and ξ ∈ H^{j,∗}(G, M) an element with j > 0. Represent it by a map ξ: k → Ω^{−j}(M), and consider the colimit X = colim Ω^{−jn}(M^{⊗n}) as in Lemma 6.2.

Our assumption is that ξ_{K}↓_{H} is nilpotent for each (H, K) ∈ H. That is, for some n depending on (H, K), the element (ξ_{K}↓_{H})^{⊗n} ∈ H^{jn,∗}(H, M_{K}^{⊗n}) is zero. Equivalently, the map K → ⋯ → Ω^{−jn}(M^{⊗n}) factors through a projective upon restriction to H. Hence, X_{K}↓_{H} is projective. Since we assumed that projectivity is detected on the family H, we conclude that X is a projective G-module. The statement now follows by Lemma 6.2. □

We omit the proof of the following lemma since the proof is similar to [11, Lemma 3.5] if one replaces π-support with the cohomological support. See also [15].

Lemma 6.4. Let G be a finite supergroup scheme, and M be a G-module. The following are equivalent:

(a) M is projective,

(b) any class ξ ∈ Ext_{G}^{>0,∗}(M, M) is nilpotent. □

Here is a partial converse to Theorem 6.3.

Proposition 6.5. Let G be a finite supergroup scheme. Suppose that nilpotence in cohomology of G-modules is detected on a family H of proper subgroups of G without field extension (i.e., each pair (H, K) ∈ H has K = k). Then projectivity of modules is also detected on H.

Proof. Let N be a G-module such that N↓_{H} is projective for all H ∈ H. Then Λ = End_{k}(N) is projective upon restriction to each H ∈ H so that for any cohomology class ξ ∈ H^{∗,∗}(G, Λ), we have ξ↓_{H} = 0. Since nilpotency is detected on H we deduce that all elements ξ ∈ H^{>0,∗}(G, Λ) ≅ Ext_{G}^{>0,∗}(N, N) are nilpotent. Now apply Lemma 6.4. □

Remark 6.6. The full converse to Theorem 6.3 is trickier. The argument above fails if we have to extend scalars. A deeper reason might be that it is not true that nilpotency of all elements in H^{∗}(G, M) implies that M is projective. We refer the reader to a cautionary example described in Proposition 5.1 of [8]: take G to be the Klein group ℤ/2 × ℤ/2, p = 2, and M be an infinite dimensional module represented by the following diagram:

[figure: ∘; ∘; ∘; ∘; ⋯; •; •; •; •]

As computed in [8, Proposition 5.1], all cohomology classes in H^{∗}(G, M) are nilpotent (of unbounded degree) whereas the module M is not projective.

7. Inductive detection theorem

We finish the first part of the paper with the inductive detection theorem. The point of Theorem 7.2 is to cover the cases of the detection that are straightforward, leaving the task of showing that the finite unipotent supergroup schemes not covered by Hypotheses 7.1 are precisely the elementary supergroup schemes from Definition 1.1; see Theorem 11.1. It is in the preparation work for that theorem that the degree 2 cohomology element of Theorem 4.3 becomes relevant.

We separate out the hypotheses since these will appear again in Section 11.

Hypothesis 7.1. The finite supergroup scheme G is unipotent and satisfies at least one of the following:

(a) There is a surjective map G → 𝔾_{a(1)} × 𝔾_{a(1)}.

(b) There is a surjective map G → 𝔾_{a}^{−} × 𝔾_{a}^{−}.

(c) There are nonzero elements

λ_{1}, … λ_{n} ∈ H^{1,0}(G^{0}, k)^{π} ⊆ H^{1,0}(G, k)
y_{1}, …, y_{s} ∈ H^{1,0}(π, 𝔽_{p}) ⊆ H^{1,0}(G, k)
ζ_{1}, …, ζ_{m} ∈ H^{1,1}(G, k) ≅ Hom(G, 𝔾_{a}^{−})

such that ∏ β𝒫^{0}(λ_{i}) ∏ β𝒫^{0}(y_{j}) ∏ ζ_{ℓ} = 0.

Theorem 7.2. If Hypothesis 7.1 hold for G, then

(i) nilpotence of elements of H^{∗,∗}(G, M) and (ii) projectivity of kG-modules

are detected on proper sub-supergroup schemes after field extension.

Proof. The argument that if G satisfies either condition (a) or (b), then projectivity of modules is detected on proper sub-supergroup schemes goes exactly as in the case 3(b) of the proof of [3, Theorem 8.1]; so we will not reproduce it here. The main ingredient of the proof is the Kronecker quiver lemma, see [9, Lemma 4.1]. Once we know detection of projectivity, the detection of nilpotents is implied by Theorem 6.3.

We now show that (c) implies detection of nilpotents in H^{∗,∗}(G, M) on sub-supergroup schemes, without any field extensions. Let ξ ∈ H^{n,∗}(G, M) be a cohomology class which restricts nilpotently to all proper subgroups of G, and let ∏ β𝒫^{0}(λ_{i}) ∏ β𝒫^{0}(y_{j}) ∏ ζ_{ℓ} = 0. Each of the elements λ_{i}, y_{j}, ζ_{ℓ} corresponds to a map from G to ℤ/p, 𝔾_{a(1)} or 𝔾_{a}^{−}, with ξ restricting nilpotently to the kernel of the corresponding map. Proposition 5.3 implies that ξ^{2i+2j+ℓ} is then divisible by ∏ β𝒫^{0}(λ_{i}) ∏ β𝒫^{0}(y_{j}) ∏ ζ_{ℓ}, and is therefore zero.

Finally, since the case (c) does not involve field extensions, Proposition 6.5 implies that we also have detection of projectivity in this case. □

Part 2. The detection theorem

8. Witt elementary supergroup schemes

In this section we introduce a family of Witt elementary supergroup schemes that play an essential role in our main detection theorem.

Notation 8.1. We shall make an extensive use of diagrams to depict many of the unipotent connected supergroup schemes to be introduced in this section. In these diagrams, ∘ denotes a composition factor isomorphic to 𝔾_{a(1)} and • denotes a composition factor isomorphic to 𝔾_{a}^{−}. A single bond represents an extension of 𝔾_{a(1)} by 𝔾_{a(1)} to make 𝔾_{a(2)} and the double bond represents an extension of 𝔾_{a(1)} by 𝔾_{a(1)} to make W_{2,1}. The dashed link denotes an extension of 𝔾_{a}^{−} by 𝔾_{a(1)} to make the supergroup scheme W_{1,1}^{−} discussed in Example 2.12.

[figure: ∘; ∘; •; ∘; ∘; ∘; 𝔾_{a(2)}; W_{2,1}; W_{1,1}^{−}]

Example 8.2. Let 𝔤 be the p-restricted Lie superalgebra described in Example 5.3.3 of Drupieski and Kujawa [23]. This is generated by an odd degree element σ and an even degree element s satisfying [σ, σ] = 2s^{[p]}. This is unipotent if and only if some s^{[p^{m}]} is zero. If n is minimal with this property then 𝔤 has a basis consisting of σ, s, s^{[p]}, …, s^{[p^{m−1}]}. The restricted enveloping algebra of 𝔤 is the group algebra of the finite supergroup scheme denoted E_{m,1}^{−} with

kE_{m,1}^{−} = k[s, σ]/(s^{p^{m}}, σ^{2} − s^{p})

where s and σ are primitive. Note that (E_{m,1}^{−})_{ev} ≅ W_{m,1}, the first Frobenius kernel of length m Witt vectors as introduced in Appendix A, so we have a short exact sequence

(8.1)  1 → W_{m,1} → E_{m,1}^{−} → 𝔾_{a}^{−} → 1.

For m ≥ 2, there are also short exact sequences

1 → W_{m−1,1}^{−} → E_{m,1}^{−} → 𝔾_{a(1)} → 1,

where kW_{m−1,1}^{−} = k[σ]/σ^{2p^{m−1}} (see Example 2.12), and

1 → W_{m−1,1} → E_{m,1}^{−} → 𝔾_{a(1)} × 𝔾_{a}^{−} → 1

where the group algebra of W_{m−1,1} is generated by s^{p} = σ^{2}. Using Notation 8.1, E_{m,1}^{−} is represented with the following diagram.‘

E_{m,1}^{−}: [figure: •; ∘; ∘; ∘; ∘; ∘]

As another example, we draw a diagram for W_{m,1}^{−} of Example 2.12.

•
∘

[figure]

∘

[figure]

∘

[figure]

∘

[figure]

∘

Lemma 8.3. If G is a finite supergroup scheme which sits in a short exact sequence

1 → 𝔾_{a(1)} × 𝔾_{a(1)} → G → 𝔾_{a}^{−} → 1

then there is a non-trivial homomorphism G → 𝔾_{a(1)}.

Proof. By Corollary 2.8, the height of G is one so it is of the form U^{[p]}(𝔤) with 𝔤 = Lie(G). Then 𝔤 has a two dimensional even part with trivial commutator and p-restriction map, and a one dimensional odd part. There is therefore a non-trivial homomorphism from 𝔤 to the one dimensional trivial Lie algebra Lie(𝔾_{a(1)}), and this induces a non-trivial homomorphism from G to 𝔾_{a(1)}. □

Next we classify all extensions of 𝔾_{a}^{−} by W_{m,1} complementing examples (2.3) and (8.1).

Lemma 8.4. Let G be a finite supergroup scheme fitting in an extension

1 → W_{m,1} → G → 𝔾_{a}^{−} → 1.

Then

kG ≅ k[s, σ]/(s^{p^{m}}, σ^{2} − s^{p^{j}})

for some 0 ≤ j ≤ m − 1, where σ is odd, s is even, and both are primitive. Hence, G can be represented by the following picture:

[figure: ∘^{s}; ∘; •^{σ}; ∘; ∘s^{[p]^{j}}; ∘; ∘_{s^{[p]^{m−1}}}]

Proof. By assumption, G_{ev} = W_{m,1}. Hence, G has height 1 by Theorem 2.6. By Lemma 2.14, there is a Lie superalgebra 𝔤 such that U^{[p]}(𝔤) ≅ kG. Let σ be a lifting to 𝔤 of the generator of k𝔾_{a}^{−}, and let s be an algebraic generator of kW_{m,1}, that is, s, s^{[p]}, …, s^{[p]^{m−1}} be a basis of the Lie algebra corresponding to W_{m,1}. Then we have

\frac{1}{2}[σ, σ] = ∑_{0}^{m−1} a_{i}s^{[p]^{i}}.

Let j be the minimal index such that a_{j} ≠ 0 and set s′ = ∑_{j}^{m−1} a_{i}s^{[p]^{i−j}}. The generators σ, s′

give the asserted presentation of U^{[p]}(𝔤) ≅ kG. □

Construction 8.5 (E_{m,n}^{−}). There is a homomorphism E_{m,1}^{−} → 𝔾_{a(1)} given by factoring out the ideal of kE_{m,1}^{−} generated by σ. There is also a surjective map 𝔾_{a(n)} → 𝔾_{a(1)} given by the (n − 1)st power of the Frobenius map. We define E_{m,n}^{−} to be the kernel of the map from the product to 𝔾_{a(1)}, so that there is a short exact sequence

1 → E_{m,n}^{−} → E_{m,1}^{−} × 𝔾_{a(n)} → 𝔾_{a(1)} → 1.

Its group ring is given by

kE_{m,n}^{−} = k[s_{1}, …, s_{n−1}, s_{n}, σ]/(s_{1}^{p}, …, s_{n−1}^{p}, s_{n}^{p^{m}}, σ^{2} − s_{n}^{p})

where s_{1}, …, s_{n} are in even degree and σ is in odd degree. The comultiplication is given by

Δ(s_{i}) = S_{i−1}(s_{1} ⊗ 1, …, s_{i} ⊗ 1, 1 ⊗ s_{1}, …, 1 ⊗ s_{i}) (i ≥ 1)
Δ(σ) = σ ⊗ 1 + 1 ⊗ σ

where the S_{i} are as defined in Appendix A, and come from the comultiplication in Dist(𝔾_{a}).

E_{m,n}^{−}: [figure: •^{σ}; ∘^{s_{n}}; ∘; ∘s_{n−1}; ∘; ∘; ∘; ∘s_{1}; ∘_{s_{n}^{p^{m−1}}}]

We define

E_{m,n}: = (E_{m,n})_{ev}^{−}

and observe that there is an isomorphism

E_{m,n} ≅ W_{m,n}/W_{m−1.n−1}.

Definition 8.6. A finite supergroup scheme is Witt elementary if it is isomorphic to a quotient of E_{m,n}^{−} by an even subgroup scheme.

Remark 8.7. For m = 1, E_{m,n}^{−} splits as a direct product:

E_{1,n}^{−} ≅ 𝔾_{a(n)} × 𝔾_{a}^{−}

Lemma 8.8. Let G be a finite supergroup scheme with the connected component G^{0} and the group of connected components π = π(G) which is a p-group. If G^{0} is an extension

1 → 𝔾_{a(1)} → G^{0} → 𝔾_{a}^{−} → 1

then G = G^{0} × π.

Proof. Since G^{0} has height, it corresponds to a 2-dimensional Lie superalgebra 𝔤 = 𝔤_{0} ⊕𝔤_{1} by Lemma 2.14. Each part is 1-dimensional and must be stabilised by π. Since π is a p-group, it centralises both 𝔾_{a}^{−} and 𝔾_{a(1)}; hence, centralises G. □

Lemma 8.9. If G^{0} = 𝔾_{a(r)} ×𝔾_{a}^{−}, and π(G) is a p-group, then the subgroup 𝔾_{a}^{−} is centralised by π(G).

Proof. We must have that G_{(1)} = 𝔾_{a(1)} × 𝔾_{a}^{−} is centralised by π. Now apply Lemma 8.8. □

Construction 8.10 (E_{m,n,μ}^{−}). The group algebra of E_{m+1,n+1}^{−} is described in Construction 8.5 except that we shift the indexing on the even generators s_{i} down by 1. With that shift, it has the form

kE_{m+1,n+1}^{−} = k[s_{0}, s_{1}, …, s_{n−1}, s_{n}, σ]/(s_{0}^{p}, s_{1}^{p}, …, s_{n−1}^{p}, s_{n}^{p^{m+1}}, σ^{2} − s_{n}^{p}).

Let k𝔾_{a(1)} = k[s]/s^{p} with s primitive in even degree. For μ ∈ k, define the supergroup scheme E_{m,n,μ}^{−} to be the quotient of E_{m+1,n+1}^{−} given by the embedding 𝔾_{a(1)} → E_{m,n,μ}^{−} which sends s to s_{0} − ms_{n}^{p^{m}}. Thus, there is a short exact sequence

1 → 𝔾_{a(1)} → E_{m+1,n+1}^{−} → E_{m,n,μ}^{−} → 1.

In the language of Dieudonné modules introduced in Appendix A, E_{m,n,μ}^{−} is quotient of E_{m+1,n+1}^{−} by the subgroup scheme of (E_{m+1,n+1}^{−})_{ev} ≅ ψ(D_{k}/(V^{m+1}, F^{n+1}, p)) given by applying ψ to the submodule of D_{k}/(V^{m+1}, F^{n+1}, p) spanned by F^{n} − μV^{m}. Explicitly, the group ring kE_{m,n,μ}^{−} is given by

kE_{m,n,μ}^{−} = k[s_{1}, …, s_{n−1}, s_{n}, σ]/(s_{1}^{p}, …, s_{n−1}^{p}, s_{n}^{p^{m+1}}, σ^{2} − s_{n}^{p})

where s_{1}, …, s_{n} are in even degree and σ is in odd degree. The comultiplication is given by

Δ(s_{i}) = S_{i}(μs_{n}^{p^{m}} ⊗ 1, s_{1} ⊗ 1, …, s_{i} ⊗ 1, 1 ⊗ μs_{n}^{p^{m}}, 1 ⊗ s_{1}, …, 1 ⊗ s_{i})
Δ(σ) = σ ⊗ 1 + 1 ⊗ σ.

E_{4,3,μ}^{−}: [figure: •^{σ}; ∘^{s_{3}}; ∘; ∘s_{2}; ∘; ∘; ∘s_{1}; μ; ∘_{s_{3}^{p^{4}}}]

We define

E_{m,n,μ} := (E_{m,n,μ}^{−})_{ev}.

Lemma 8.11. Let G be a finite unipotent supergroup scheme.

(1) If for some n ≥ 2, there is an extension

1 → E_{m,n} → G → 𝔾_{a}^{−} → 1,

then kG ≅ k[s_{1}, …, s_{n−1}, s_{n}, σ]/(s_{1}^{p}, …, s_{n−1}^{p}, s_{n}^{p^{m}}, σ^{2}−s_{n}^{p^{i}}−αs_{1}) for some 1 ≤ i ≤ m−1 and α ∈ k, where s_{1}, …, s_{n} are in even degree, σ is in odd degree, and comultiplication is given by the formulas in (8.5). Hence, G can be represented as follows:

G: [figure: ∘^{s_{n}}; •^{σ}; ∘; ∘s_{n−1}; ∘^{s_{n}^{p^{i}}}; ∘; ∘; ∘s_{1}; ∘_{s_{n}^{p^{m−1}}}]

(2) If G fits in the extension

1 → E_{m,n,μ} → G → 𝔾_{a}^{−} → 1,

then kG = k[s_{1}, …, s_{n−1}, s_{n}, σ]/(s_{1}^{p}, …, s_{n−1}^{p}, s_{n}^{p^{m+1}}, σ^{2} − s_{n}^{p^{i}}) for some 1 ≤ i ≤ m − 1 and degrees and comultiplication as in (8.10).

Proof. We handle only the first case; the second one is similar. We have (E_{m,n})_{(1)} = W_{m−1,1}× 𝔾_{a(1)}, and hence G_{(1)} fits into a short exact sequence:

1 → W_{m−1,1} × 𝔾_{a(1)} → G_{(1)} → 𝔾_{a}^{−} → 1.

Let 𝔤 = Lie(G), so that by Lemma 2.14 we have kG_{(1)} ≅ U^{[p]}(𝔤). Let σ be a lift to 𝔤 of a generator for Lie(𝔾_{a}^{−}). Then σ has odd degree, and \frac{1}{2}[σ, σ] is some element of Lie(W_{m−1,1} × 𝔾_{a(1)}), which is the linear span of the elements

s_{n}^{[p]}, s_{n}^{[p]^{2}}, …, s_{n}^{[p]^{m−1}}, s_{1} ∈ Lie(E_{m,r}).

Arguing exactly as in the proof of Lemma 8.4, we can change the generator s_{n} so that \frac{1}{2}[σ, σ] = s_{n}^{[p]^{i}} + αs_{1} without changing the comultiplication on kE_{m,n}. □

Remark 8.12. The finite supergroup schemes E_{m,n}^{−} and E_{m,n,μ}^{−} also appear in the work of Drupieski and Kujawa [24], where they are denoted 𝕄_{n;m} and 𝕄_{n+1;m,−μ} respectively.

We also record the structure of the coordinate rings k[E_{m,n}^{−}] and k[E_{m,n,μ}^{−}]. For k[E_{m,n}^{−}] we have generators w, x_{1}, …, x_{p^{m}−1}, y with y odd and the remaining generators even. We have relations w^{p^{n−1}} = x_{1}, x_{i}x_{j} = \binom{i+j}{j}x_{i+j}, y^{2} = 0; which implies that as an algebra it is a truncated polynomial ring generated by w, x_{p}, x_{p^{2}}, …, x_{p^{m−1}}, y with relations w^{p^{n}}, x_{p}^{p}, x_{p^{2}}^{p}, …, x_{p^{m−1}}^{p}, y^{2}.

For the coalgebra structure, the elements w and y are primitive, while

Δ(x_{ℓ}) = ∑_{i+j=ℓ} x_{i} ⊗ x_{j} + ∑_{i+j+p=ℓ} x_{i}y ⊗ x_{j}y.

The antipode negates w and y, and sends x_{i} to (−1)^{i}x_{i}.

The coordinate ring k[E_{m,n,μ}^{−}] is the subalgebra of k[E_{m+1,n+1}^{−}] generated by the elements w − μx_{p^{m}}, x_{p}, x_{p^{2}}, …, x_{p^{m−1}}, y with the restriction of the comultiplication and antipode.

Theorem 8.13. Every Witt elementary supergroup scheme is isomorphic to one of the following:

(i) 𝔾_{a}^{−},

(ii) E_{m,n}^{−} with m, n ≥ 1,

(iii) E_{m,n,μ}^{−} with m, n ≥ 1 and 0 ≠ μ ∈ k.

The only isomorphisms between these are given by E_{m,n,μ}^{−} ≅ E_{m,n,μ′}^{−} if and only if μ/μ′ = a^{p^{m+n}−1} for some a ∈ k.

Note that E_{1,n}^{−} is isomorphic to 𝔾_{a(n)} × 𝔾_{a}^{−} for n ≥ 1.

Proof. The quotient of E_{m,n}^{−} by its entire even part is covered in part (i). The quotient by a proper subgroup of (E_{m,n}^{−})_{ev} uses Theorem A.3, and gives parts (ii) and (iii). □

We recall Definition 1.1 from the Introduction: a finite supergroup scheme is elementary if it is isomorphic to a quotient of E_{m,n}^{−} × (ℤ/p)^{×s}.

Remark 8.14. An elementary finite supergroup scheme is isomorphic to one of the following: (i) 𝔾_{a(n)} × (ℤ/p)^{×s} with n, s ≥ 0, (ii) 𝔾_{a(n)} × 𝔾_{a}^{−} × (ℤ/p)^{×s} with n, s ≥ 0, (iii) E_{m,n}^{−} × (ℤ/p)^{×s} with m ≥ 1, n ≥ 2, s ≥ 0, or (iv) E_{m,n,μ}^{−} × (ℤ/p)^{×s} with m, n ≥ 1, 0 ≠ μ ∈ k and s ≥ 0.

Definition 8.15. The rank of an elementary finite supergroup scheme is defined to be n + s in case (i), and n + s + 1 in cases (ii)–(iv) of the above remark.

9. Cohomological calculations

This section is dedicated to computing the cohomology rings of the supergroup schemes introduced in Section 8, and other preparatory results for use in the sequel.

Proposition 9.1. If G is a semidirect product (𝔾_{a(1)} × 𝔾_{a(1)}) ⋊ (ℤ/p)^{×s} with non-trivial action then there is an element 0 ≠ y ∈ H^{1}((ℤ/p)^{×s}, k) ⊆ H^{1,0}(G, k) whose product with 0 ≠ λ ∈ H^{1}(𝔾_{a(1)}, k) ⊆ H^{1,0}(G, k) is zero in H^{2,0}(G, k).

Proof. The non-triviality of the product of a pair of elements in H^{1}(G, k) ≅ Ext_{kG}^{1}(k, k) is the obstruction to producing a three dimensional module using these two extensions. So the proposition follows from the fact that G has a representation of the form

(\begin{matrix}1 & (ℤ/p)^{×s} & 𝔾_{a(1)} \\ 0 & 1 & 𝔾_{a(1)} \\ 0 & 0 & 1 \end{matrix}) .  □

We next discuss cohomology of abelian connected unipotent finite group schemes. Recall from Appendix A that as an augmented algebra, kG is isomorphic to a tensor product of algebras of the form kW_{m,1} = k[s]/(s^{p^{m}}). Since cohomology of a finite group scheme G in general only depends on the algebra structure of kG, not on the comultiplication, we get the following description of the cohomology ring.

Theorem 9.2. Let G be an abelian connected unipotent finite group scheme. The cohomology ring H^{∗}(G, k) is a tensor product of algebras of the form

H^{∗}(W_{m,1}, k) = k[x_{m}] ⊗ Λ(λ_{m})

where λ_{m} has degree one and x_{m} has degree two.

The surjective map W_{m,1} → W_{m−1,1} induces an inflation map

H^{∗}(W_{m−1,1}, k) → H^{∗}(W_{m,1}, k)

sending x_{m−1} to zero and λ_{m−1} to λ_{m}. On the other hand, the injective map W_{m−1,1} → W_{m,1} induces a restriction map

H^{∗}(W_{m,1}, k) → H^{∗}(W_{m−1,1}, k)

sending x_{m} to x_{m−1} and λ_{m} to zero.

Proof. The cohomology of the algebra k[s]/(s^{p^{m}}) and the restriction and inflation maps are well known from the cohomology theory of finite groups. See for example Chapter XII of Cartan and Eilenberg [17]. □

Proposition 9.3. The cohomology of the supergroup scheme W_{m,1}^{−} of Example 2.12 is given by

H^{∗,∗}(W_{m,1}^{−}, k) = k[x_{m}, ζ_{m}]/(ζ_{m}^{2})

with |x_{m}| = (2, 0) and |ζ_{m}| = (1, 1).

For m ≥ 2 the surjective map W_{m,1}^{−} → W_{m−1,1}^{−} induces an inflation map

H^{∗,∗}(W_{m−1,1}^{−}, k) → H^{∗,∗}(W_{m,1}^{−}, k)

sending x_{m−1} to zero and ζ_{m−1} to ζ_{m}.

Proof. The E_{2} page of the spectral sequence

H^{∗,∗}(𝔾_{a}^{−}, H^{∗,∗}(W_{m,1}, k)) ⇒ H^{∗,∗}(G, k)

has a polynomial generator ζ_{m} on the base in degree (1, 1), an exterior generator λ_{m} on the fibre in degree (1, 0) and a polynomial generator x_{m} on the fibre in degree (2, 0). The only differential is d_{2}, and this is determined by d_{2}(λ_{m}) = ζ_{m}^{2}, d_{2}(x_{m}) = 0. The inflation maps follow from Theorem 9.2. □

Proposition 9.4. If G is a nonsplit extension

1 → 𝔾_{a(r)} → G → 𝔾_{a}^{−} → 1

with r ≥ 1 then the inflation of ζ ∈ H^{1,1}(𝔾_{a}^{−}, k) to G squares to zero in H^{2,0}(G, k).

Proof. By Corollary 2.8, we have a nonsplit extension

1 → 𝔾_{a(1)} → G_{(1)} → 𝔾_{a}^{−} → 1.

Hence, G_{(1)} ≅ W_{1,1}^{−} by Lemma 8.3. This implies that ignoring the comultiplication, we have kG ≅ k𝔾_{a(r−1)} ⊗ kW_{1,1}^{−}. The result follows from the case m = 1 of Proposition 9.3. □

Lemma 9.5. If G is an extension

1 → W_{2,2} → G → 𝔾_{a}^{−} → 1,

then there exists a surjective map G → W_{2,1}.

Proof. Since G = G_{(1)}G_{ev} by Corollary 2.8, taking the first Frobenius kernels, we get an extension

1 → W_{2,1} → G_{(1)} → 𝔾_{a}^{−} → 1.

Hence, G/G_{(1)} ≅ W_{2,2}/W_{2,1} ≅ W_{2,1}. □

Lemma 9.6. Let G be a unipotent finite supergroup scheme, and f: G → \overline{G} = 𝔾_{a(r)} × 𝔾_{a}^{−} × (ℤ/p)^{×s} a surjective map of supergroup schemes. Assume that

(a) f^{∗}: H^{1,∗}(\overline{G}, k) → H^{1,∗}(G, k) is an isomorphism, and

(b) f^{∗} is one-to-one restricted to H^{2}((ℤ/p)^{×s}, k) ⊂ H^{2,0}(\overline{G}, k) Then π_{0}(G), the group of connected components of G, is isomorphic to (ℤ/p)^{×s}.

Proof. Set π = π_{0}(G) and let \overline{π} be the Frattini quotient for π, that is, the maximal quotient isomorphic to an elementary abelian p-group. Then the map f factors through G^{0} ⋊ \overline{π} and we have a commutative diagram

[figure: G^{0} ⋊ π; π; G^{0} ⋊ \overline{π}; \overline{π}; \overline{G}; (ℤ/p)^{×s}]

If π → \overline{π} is not an isomorphism, Lemma 3.5 implies that there exists an element u in H^{2}(\overline{π}, k) = H^{2}((ℤ/p)^{×s}, k) which pulls back to zero in H^{2}(π, k) and, hence, in H^{2,0}(G, k). Inflating the class u to H^{2,0}(\overline{G}, k), we get an element in Ker f^{∗}∩H^{2}((ℤ/p)^{×s}, k) contradicting assumption (b). Hence, π ≅ (ℤ/p)^{×s}. □

The result below is a denouement of the preceding developments. It’s import is that, in the situation of Theorem 4.3(ii), various finite (super)group schemes cannot be quotients of G, G^{0} and G_{ev}^{0}. Theorem 9.7 together with Theorem 4.4 are the major inputs in the proof of the detection Theorem 11.1.

Theorem 9.7. Let G be a unipotent finite supergroup scheme, and f: G → \overline{G} = 𝔾_{a(r)} × 𝔾_{a}^{−} × (ℤ/p)^{×s} a surjective map of supergroup schemes. Assume that

(a) f^{∗}: H^{1,∗}(\overline{G}, k) → H^{1,∗}(G, k) is an isomorphism, and

(b) I = Ker\{f^{∗}: H^{2,∗}(\overline{G}, k) → H^{2,∗}(G, k)\} is one dimensional, spanned by an element of the form ζ^{2} + γx_{r} with 0 ≠ γ ∈ k.

Then the following statements hold.

(I) G cannot have as a quotient the following supergroup schemes: (i) (𝔾_{a(1)} × 𝔾_{a(1)}) ⋊ (ℤ/p)^{×s}, (ii) (𝔾_{a}^{−} × 𝔾_{a}^{−}) ⋊ (ℤ/p)^{×s}.

(II) The restriction f_{0} = f↓_{G^{0}} : G^{0} → \overline{G}^{0} satisfies the following cohomological conditions: (a_{0}) f_{0}^{∗}: H^{1,∗}(\overline{G}^{0}, k) → H^{1,∗}(G^{0}, k) is an isomorphism, (b_{0}) Ker f_{0}^{∗} ∩ H^{2,0}(\overline{G}^{0}, k) is one dimensional, spanned by ζ^{2} + γx_{r}.

(III) The following connected supergroup schemes cannot be quotients of G^{0}: (i) H given by a nonsplit extension 1 → 𝔾_{a(r)} → H → 𝔾_{a}^{−} → 1, (ii) W_{m,1}^{−}, (iii) W_{2,1}.

(IV) G_{ev}^{0} cannot have W_{2,2} as a quotient.

Proof. (I). Let ρ: G → H be a surjective map of unipotent group schemes, and suppose that H surjects further on a group scheme H′ which is isomorphic to 𝔾_{a(1)}, 𝔾_{a}^{−} or ℤ/p. By Remark 3.6, we have a commutative diagram

(9.1)  \xymatrix{G \ar@{->>}[r]^{f} \ar@{->>}[d]_{ρ} & \overline{G} \ar@{->>}[d]^{\overline{ρ}} \\ H \ar@{->>}[r]^{χ} & H′}

Lemma 3.1 implies that \overline{ρ} : \overline{G} → H′ induces an injective map on H^{1,∗}. Moreover, the explicit calculation of cohomology for \overline{G} further implies that the map H^{∗,∗}(H′, k) → H^{∗,∗}(\overline{G}, k) is injective. Since H′ = 𝔾_{a(1)}, 𝔾_{a}^{−} or ℤ/p, we have that H^{1,∗}(H′, k) is a 1-dimensional vector space. Let α ∈ H^{1,∗}(H′, k) be a linear generator. Then the assumption (a) together with the commutativity of (9.1) imply that

0 ≠ (\overline{ρ} ∘ f)^{∗}(α) = (χ ∘ ρ)^{∗}(α) ∈ H^{1,∗}(G, k).

In Case (I.i), assume that there is a surjective map G → H where H = (𝔾_{a(1)} × 𝔾_{a(1)}) ⋊ (ℤ/p)^{×s}. There are maps χ : H → H′ with H′ = 𝔾_{a(1)}, ℤ/p. By Proposition 9.1, taking for α the elements y and λ, we obtain a relation

f^{∗}(\overline{ρ}^{∗}(y)\overline{ρ}^{∗}(λ)) = ρ^{∗}(χ^{∗}(y)χ^{∗}(λ)) = 0.

Hence, 0 ≠ \overline{ρ}^{∗}(y)\overline{ρ}^{∗}(λ) is in I which contradicts the assumption (b) completing the proof in that case.

In Case (I.ii), we assume there is a surjective map G → H where H = (𝔾_{a}^{−} × 𝔾_{a}^{−}) ⋊ (ℤ/p)^{×s}. Cohomology of H is computed explicitly in [14]; there exist non trivial elements λ_{1} ∈ H^{1,0}(H, k) and ζ ∈ H^{1,1}(H, k) such that λ_{1}ζ = 0. Arguing as in (I.i), we get a contradiction with the assumption (b) again.

(II.a_{0}). Let π = π_{0}(G) be the group of connected components of G. By Lemma 9.6, we have π ≅ (ℤ/p)^{×s}, which is the same as \overline{π} = π_{0}(\overline{G}). The map f: G → \overline{G} induces a commutative diagram of five-term sequences:

[figure: H^{1,∗}(π, k); H^{1,∗}(G, k); H^{1,∗}(G^{0}, k)^{π}; H^{2,∗}(π, k); H^{2,∗}(G, k); H^{1,∗}(\overline{π}, k); H^{1,∗}(\overline{G}, k); f^{∗}; H^{1,∗}(\overline{G}^{0}, k); f_{0}^{∗}; 0; H^{2,∗}(\overline{π}, k); H^{2,∗}(\overline{G}, k); f^{∗}]

Since f^{∗} is an iso on H^{1,∗}, we conclude that it induces an isomorphism H^{1,∗}(\overline{G}^{0}, k) ≅ H^{1,∗}(G^{0}, k)^{π}. It remains to show that π acts trivially on H^{1,∗}(G^{0}, k).

By Lemma 3.1, We have H^{1,∗}(G^{0}, k) ≅ H^{1,0}(G^{0}, k) ⊕ H^{1,1}(G^{0}, k) with H^{1,0}(G^{0}, k) ≅ Hom(G^{0}, 𝔾_{a}), H^{1,1}(G^{0}, k) ≅ Hom(G^{0}, 𝔾_{a}^{−}) by Lemma 3.1, and the action of π fixing the even and odd parts.

The assumption that f^{∗} is an isomorphism on H^{1} together with Lemma 3.1 imply that

dim Hom(G^{0}, 𝔾_{a}^{−})^{π} = 1 (9.2)

Hom(G^{0}, 𝔾_{a})^{π} = Hom(𝔾_{a(r)}, 𝔾_{a(r)}). (9.3)

Hence, to show that π acts trivially on H^{1,∗}(G_{0}, k), we need to show the same two equalities for Hom(G^{0}, 𝔾_{a}^{−}) and Hom(G^{0}, 𝔾_{a}).

We first show that dim_{k} Hom(G^{0}, 𝔾_{a}^{−}) = 1. Suppose dim_{k} Hom(G^{0}, 𝔾_{a}^{−}) ≥ 2. Since π is a p-group, there exists a two-dimensional π-invariant subspace of Hom(G^{0}, 𝔾_{a}^{−}) and, hence, a π-invariant quotient of the form 𝔾_{a}^{−} × 𝔾_{a}^{−}. But this implies that G has a quotient of the form H = (𝔾_{a}^{−} ×𝔾_{a}^{−})⋊(ℤ/p)^{×s} which is disallowed by (I.ii). Hence, dim_{k} Hom(G^{0}, 𝔾_{a}^{−}) = 1.

We now consider Hom(G^{0}, 𝔾_{a}). First, since G^{0} is finite, there exists a number n such that Hom(G^{0}, 𝔾_{a}) = Hom(G^{0}, 𝔾_{a(n)}). Pick the maximal n so that the map G^{0} → 𝔾_{a(n)} is surjective. The standard projection 𝔾_{a(n)}) → 𝔾_{a(1)} induces a map on Hom spaces Hom(G^{0}, 𝔾_{a(n)}) → Hom(G^{0}, 𝔾_{a(1)}); the action of π descends along this map since the Frobenius map is π-equivariant. If dim_{k} Hom(G^{0}, 𝔾_{a(1)}) > 1, then arguing just as in the case of 𝔾_{a}^{−} we deduce a contradiction with (I.i). Hence, Hom(G^{0}, 𝔾_{a(1)}) = 1. Therefore,

Hom(G^{0}, 𝔾_{a(n)}) ≅ Hom(𝔾_{a(n)}, 𝔾_{a(n)})

It remains to show that n = r. Note that Hom(𝔾_{a(n)}, 𝔾_{a(n)}) ≃ 𝔾_{a}^{×n} as a group scheme, with the action of π preserving the group scheme structure. Since 𝔾_{a}^{×n} is connected, the action of π must be trivial, hence, r = n.

(II.b_{0}). The projection f: G → \overline{G} induces a map on spectral sequences making the following diagram commute:

(9.4)  \xymatrix{ & H^{*,*}(π, H^{*,*}(G^{0}, k)) \ar@{=>}[r] & H^{*,*}(G, k) \\ H^{*,*}(ℤ/p^{×s}, k) ⊗ H^{*,*}(\overline{G}^{0}, k) \ar[r]^{∼} & H^{*,*}(\overline{π}, H^{*,*}(\overline{G}^{0}, k)) \ar[u] \ar@{=>}[r] & H^{*,*}(\overline{G}, k) \ar[u]^{f^{∗}}}

Here, the star for the internal degree is preserved by the spectral sequence. The bottom sequence collapses at the E_{2} page giving an isomorphism H^{*,*}(ℤ/p^{×s}, k) ⊗ H^{*,*}(\overline{G}^{0}, k) ≅ H^{*,*}(\overline{G}, k). Since ζ^{2} + γx_{r} ∈ H^{2,0}(\overline{G}^{0}, k) = H^{2,0}(𝔾_{a(r)} × 𝔾_{a}^{−}, k), we conclude that it belongs to the kernel of f_{0}^{∗}. It remains to show that this class generates the kernel of f_{0}^{∗} on H^{2,0}.

Let

(9.5)  \xymatrix{H^{2,0}(G, k) & F^{1}H^{2,0}(G, k) \ar@{^{(}->}[l] \ar[d]^{≡} & F^{0}H^{2,0}(G, k) \ar@{^{(}->}[l] \ar[d]^{≡} \\  & F^{1}H^{2,0}(\overline{G}, k) & F^{0}H^{2,0}(\overline{G}, k) \ar@{^{(}->}[l]}

be the filtration on H^{2,0} with subquotients giving the E_{∞} term of the spectral sequences. We consider another diagram induced by f:

(9.6)  \xymatrix{H^{2,0}(\overline{G}^{0}, k) \ar[r]^{f_{0}^{∗}} \ar@{^{(}->}[d]^{ρ} & H^{2,0}(G^{0}, k)^{π} \\ H^{2,0}(\overline{G}, k) \ar[u] \ar[r]^{f^{∗}} & H^{2,0}(G, k) \ar[u]^{i}}

The left vertical map induced by the embedding \overline{G}^{0} < \overline{G} splits since

H^{2,0}(\overline{G}, k) ≅ H^{2}(π, k) ⊕ H^{1}(π, H^{1,0}(\overline{G}^{0}, k)) ⊕ H^{2,0}(\overline{G}^{0}, k).

The left vertical map ρ: H^{2,0}(\overline{G}^{0}, k) ↪ H^{2,0}(\overline{G}, k) is the identification of H^{2,0}(\overline{G}^{0}, k) with the last direct summand.

The right vertical map i: H^{2,0}(G, k) → H^{2,0}(G^{0}, k)^{π} is the edge homomorphism of the top row spectral sequence in (9.4), hence,

(9.7)  Ker i = F^{1}H^{2,0}(G, k).

Let α ∈ H^{2,0}(\overline{G}^{0}, k) be a class in the kernel of f_{0}^{∗}. Then f_{0}^{∗}(α) = if^{∗}ρ(α) = 0 implies that f^{∗}ρ(α) ∈ Ker i = F^{1}H^{2,0}(G, k). Since F^{1}H^{2,0}(G, k) ≅ F^{1}H^{2,0}(\overline{G}, k) by (9.5), there exists β ∈ F^{1}H^{2,0}(\overline{G}, k) = H^{2}(π, k) ⊕ H^{1}(π, H^{1,0}(\overline{G}^{0}, k)), such that f^{∗}(ρ(α)) = f^{∗}(β), that is,

f^{∗}(ρ(α) − β) = 0.

Assumption (b) now implies that ρ(α)−β is a multiple of ζ^{2}+γx_{r} and, hence, ρ(α)−β ∈ Im ρ. Therefore, β ∈ Im ρ. This implies that β = 0 since Im ρ∩F^{1}H^{2,0}(\overline{G}, k) = 0. We conclude that f^{∗}(ρ(α)) = 0, and, hence, α is a multiple of ζ^{2} + γx_{r}. Hence the kernel is one-dimensional.

(III). We apply the same argument as in Case (I) but to f_{0} : G^{0} → \overline{G}^{0}. Once again, we have a commutative diagram of surjective maps:

\xymatrix{G^{0} \ar@{->>}[r]^{f_{0}} \ar@{->>}[d]_{ρ} & \overline{G}^{0} \ar@{->>}[d]^{\overline{ρ}} \\ H \ar@{->>}[r]^{χ} & H′}

For (III.i), Proposition 9.4 gives an element ζ ∈ H^{1,1}(H′, k) such that χ^{∗}(ζ)^{2} = 0. Hence, commutativity of the diagram above implies that 0 ≠ (\overline{ρ}^{∗}(ζ))^{2} is in the kernel of f_{0}^{∗} contradicting the assumption II(b_{0}), and completing the proof in this case.

Case (III.ii) follows from Proposition 9.3 in a similar fashion taking H′ = 𝔾_{a}^{1} and α = ζ_{m}. If G^{0} has a quotient W_{2,1}, then β𝒫^{0}(λ_{2}), where λ_{2} is a degree (1, 0) cohomology generator

of H^{∗,∗}(W_{2,1}, k), is in the kernel of f_{0}^{∗}, contradicting II(b_{0}).

Finally, Case (IV) follows from Lemma 9.5 and case (II.iii). □

Corollary 9.8. Let G be a unipotent finite supergroup scheme satisfying the assumptions of Theorem 9.7. Let A = G/[G_{ev}, G_{ev}]. Then A_{ev}^{0} is isomorphic to a quotient of E_{m,n} = (E_{m,n}^{−})_{ev} for some m, n > 0.

Proof. First we claim that dim_{k} Hom_{Gr/k}(A_{ev}, 𝔾_{a(1)}) = 1. This is because if this dimension is two or greater then G, and, hence, G^{0}, has a quotient which is a nonsplit extension of 𝔾_{a}^{−} by 𝔾_{a(1)}, which is not allowed by Theorem 9.7.

Next, we claim that dim_{k} Hom_{Gr/k}(A_{ev}^{0}, 𝔾_{a(1)}) = 1. This is because if this dimension is two or greater then G has a quotient which is a semidirect product (𝔾_{a(1)} × 𝔾_{a(1)}) ⋊ (ℤ/p)^{×s} with non-trivial action. This is once again disallowed by Theorem 9.7.

By Theorem 9.7(III), A_{ev}^{0} does not have W_{2,2} as a quotient. Together with the condition dim_{k} Hom_{Gr/k}(A_{ev}^{0}, 𝔾_{a(1)}) = 1 this allows us to apply Lemma A.2, concluding that A_{ev}^{0} is isomorphic to a quotient of the group scheme E_{m,n}. □

Now for the promised computation of cohomology of Witt elementary supergroup schemes.

Theorem 9.9. The cohomology of the group E_{m,n}^{−} (as defined in (8.5)) (m ≥ 2, n ≥ 1) is given by

H^{∗,∗}(E_{m,n}^{−}, k) = k[x_{m,1}, …, x_{m,n}, ζ_{m}] ⊗ Λ(λ_{m,1}, …, λ_{m,n})

with |x_{m,i}| = (2, 0), |ζ_{m}| = (1, 1) and |λ_{m,i}| = (1, 0).

For m ≥ 3, the surjective map E_{m,n}^{−} → E_{m−1,n}^{−} induces an inflation map

H^{∗,∗}(E_{m−1,n}^{−}, k) → H^{∗,∗}(E_{m,n}^{−}, k)

sending x_{m−1,i} to x_{m,i} (1 ≤ i ≤ n − 1), x_{m−1,n} to zero, ζ_{m−1} to ζ_{m} and λ_{m−1,i} to λ_{m,i} (1 ≤ i ≤ n).

The surjective map E_{2,n}^{−} → E_{1,n}^{−} = 𝔾_{a(n)} × 𝔾_{a}^{−} induces an inflation map sending x_{i} to x_{2,i} (1 ≤ i ≤ n − 1), x_{n} to ζ^{2}, ζ_{2} to ζ and λ_{i} to λ_{2,i}. In particular, the kernel of

H^{2,0}(𝔾_{a(n)} × 𝔾_{a}^{−}, k) → H^{2,0}(E_{2,n}^{−}, k)

is one dimensional, spanned by ζ^{2} − x_{n}.

Proof. Again, we use the fact that the cohomology only depends on the algebra structure of the group algebra and not on the comultplication. The algebra structure is described in Definition 8.6, and is a tensor product k𝔾_{a(n−1)} ⊗kE_{m,1}^{−}. The first factor gives the generators λ_{m,1}, …, λ_{m,n−1}, x_{m,1} …, x_{m,n−1}, so we need to compute H^{∗,∗}(E_{m,1}^{−}, k). We do this using the spectral sequence

H^{∗,∗}(𝔾_{a}^{−}, H^{∗,∗}(W_{m,1}, k)) ⇒ H^{∗,∗}(E_{m,1}^{−}, k).

This has the same E_{2} page as the spectral sequence in the proof of Proposition 9.3, but all the differentials are zero. This accounts for the generators x_{m,n}, ζ_{m} and λ_{m,n}. The inflation maps again follow from Theorem 9.2. □

Theorem 9.10. The cohomology of the group E_{m,n,μ}^{−} of (8.10) is given by

H^{∗,∗}(E_{m,n,μ}^{−}, k) = k[x_{m,1,μ}, …, x_{m,n,μ}, ζ_{m,μ}] ⊗ Λ(λ_{m,1,μ}, …, λ_{m,n,μ})

with |x_{m,i,μ}| = (2, 0), |ζ_{m,μ}| = (1, 1) and |λ_{m,i,μ}| = (1, 0).

The surjective map E_{m+1,n+1}^{−} → E_{m,n,μ}^{−} induces an inflation map

H^{∗,∗}(E_{m,n,μ}^{−}, k) → H^{∗,∗}(E_{m,n}^{−}, k)

sending each element to the corresponding element without the subscript μ, except that it sends x_{m,n,μ} to zero.

Proof. The proof is essentially the same as for Theorem 9.9. □

Remark 9.11. The computation in Theorem 9.9 also appears in Proposition 3.2.1 (1) and (3) and Lemma 3.2.4 of Drupieski and Kujawa [24]. Similarly, Theorem 9.10 should be compared with Proposition 3.2.1 (4) and (5) of [24] and Lemma 3.1.1 (3) and Remark 2.2.3 (1) of [25].

Remark 9.12. We tabulate the action of the Steenrod operations on H^{∗,∗}(E_{m,n}^{−}, k), for use in the proof of Theorem 10.3. The table for H^{∗,∗}(E_{m,n,μ}^{−}, k) looks exactly the same after adding μ to all the indices; cf. Table 1.

degree 𝒫^{0} β𝒫^{0} 𝒫^{\frac{1}{2}} β𝒫^{\frac{1}{2}} 𝒫^{1} 𝒫^{i} β𝒫^{i}
(i ≥ 2) (i ≥ 1)
λ_{m,i} (1, 0) λ_{m,i+1} −x_{m,i}  0  0  0  1 ≤ i < n
λ_{m,n} (1, 0) 0  −ζ_{m}^{2}  0  0  0
ζ_{m} (1, 1)  ζ_{m}^{p}  0
x_{m,i} (2, 0) x_{m,i+1} 0  x_{i}^{p}  0  0  1 ≤ i < n
x_{m,n} (2, 0) 0  0  0  0  0

10. Cohomological characterisation of elementary supergroups

The purpose of this section is to show that elementary supergroups as introduced in Definition 1.1 can be characterised cohomologically. Recall that for \overline{G} = 𝔾_{a(r)}×𝔾_{a}^{−}×(ℤ/p)^{×s}, we employ the following notation for the standard generators in cohomology:

H^{∗,∗}(\overline{G}, k) = k[x_{1}, …, x_{r}] ⊗ Λ(λ_{1}, …, λ_{r}) ⊗ k[ζ] ⊗ k[z_{1}, …, z_{s}] ⊗ Λ(y_{1}, …, y_{s})

Theorems 9.9 and 9.10 show that if G is an elementary supergroup scheme equipped with a surjection G → \overline{G} which induces an isomorphism on H^{1,∗}, then either f is an isomorphism or Ker f^{∗} falls under the case (ii) of Theorem 4.3. Theorem 10.3 proves a partial converse to this statement, and is the key step in the proof of Theorem 1.2.

Lemma 10.1. Let 1 → Z ⟶^{f} G ⟶^{ψ} A → 1 be a central extension of group schemes with Z ≅ 𝔾_{a(1)} and A abelian. If the connecting homomorphism d_{2} : H^{1}(Z, k) → H^{2}(A, k) is zero then G is abelian.

Proof. The five-term sequence of the central extension shows that there is an element ũ ∈ H^{1}(G, k) whose restriction is f^{∗}(ũ) = u ∈ H^{1}(Z, k). Applying Lemma 3.1, we see that there is a homomorphism ϕ: G → 𝔾_{a} whose composite with Z → G is nonzero. Then (ψ, ϕ): G → A × 𝔾_{a} is an embedding, and G is a subgroup scheme of an abelian group scheme, hence abelian. □

The following proposition, which is the key observation necessary for the proof of Theorem 10.3, gives a cohomological criterion to establish that certain extensions of abelian finite group schemes are abelian themselves.

Proposition 10.2. Let 1 → Z → G → A → 1 be a central extension of group schemes with Z ≅ 𝔾_{a(1)} and A abelian. The following are equivalent:

(i) G is abelian.

(ii) There exists an abelian finite group scheme A′ and a surjective map A′ → A such that

the composition \xymatrix{H^{1}(Z, k) \ar[r]^{d_{2}} & H^{2}(A, k) \ar[r] & H^{2}(A′, k)} is zero. The induced map in cohomology sends d_{2}(λ) ∈ H^{2}(A, k) to zero in H^{2}(A′, k) for all 0 ≠ λ ∈ H^{1}(Z, k).

Proof. (i) ⇒ (ii): Take A′ = G and use the five-term sequence.

(ii) ⇒ (i): Let 1 → A′′ → A′ → A → 1 be the short exact sequence given by the surjection A′ → A. Form the pullback X of G → A and A′ → A:

[figure: 1; 1; Z; Z; 1; A′′; X; G; 1; 1; A′′; A′; A; 1; 1; 1]

If d_{2}(λ) goes to zero in H^{2}(A′, k) then the sequence

1 → Z → X → A′ → 1

satisfies the conditions of Lemma 10.1, and so X is abelian. Since G is a quotient of X, it follows that G is abelian. □

Theorem 10.3. Let G be a unipotent finite supergroup scheme, and f: G → \overline{G} = 𝔾_{a(r)} × 𝔾_{a}^{−} × (ℤ/p)^{×s} a surjective map of supergroup schemes. Assume that

(1) f^{∗}: H^{1,∗}(\overline{G}, k) → H^{1,∗}(G, k) is an isomorphism,

(2) Ker(f^{∗}) ∩ H^{2,0}(\overline{G}, k) is one dimensional, spanned by an element of the form ζ^{2} + γx_{r} with 0 ≠ γ ∈ k.

(3) Ker(f^{∗}) ∩ H^{2,1}(\overline{G}, k) = 0,

(4) There does not exist i ∈ ℤ_{≥0} and y ∈ H^{1}((ℤ/p)^{×s}, 𝔽_{p}) ⊂ H^{1,0}(\overline{G}, k) such that

ζ^{2p^{i}}β𝒫^{0}(y) or ζ^{2p^{i}+2} lie in Ker\{f^{∗} : H^{∗,∗}(\overline{G}, k) → H^{∗,∗}(G, k)\}. Then G is isomorphic to E_{m,r}^{−} × (ℤ/p)^{×s} or E_{m,r+1,μ}^{−} for some m ≥ 1, μ ∈ k.

Proof. The proof has three essential reduction steps: Step (1) The first step is to show that G_{ev} is normal in G, and G/G_{ev} ≅ 𝔾_{a}^{−}. Step (2) Let A = G/[G_{ev}, G_{ev}]. The second step is to show that A is isomorphic to either

E_{m,r}^{−} × (ℤ/p)^{×s} or E_{m,r+1,μ}^{−} for some m ≥ 1, μ ∈ k. Step (3) Finally, we show that G ≅ A.

By Lemma 9.6, π ≅ (ℤ/p)^{×s}.

Let ψ: \overline{G} → 𝔾_{a}^{−} be the projection map, and let H = Ker\{ψ ∘ f: G → \overline{G} → 𝔾_{a}^{−}\}. We now show that H = G_{ev}, proving Step (1).

We have the five-term sequence associated with the extension 1 → H → G → 𝔾_{a}^{−} → 1 of which we only need the odd internal degree part:

\xymatrix{0 \ar[r] & H^{1,1}(𝔾_{a}^{−}, k) \ar[r]^{∼}_{(ψ∘f)^{∗}} & H^{1,1}(G, k) \ar[r]^{res}_{=0} & H^{1,1}(H, k)^{𝔾_{a}^{−}} \ar[r]^{d_{2}}_{=0} & H^{2,1}(𝔾_{a}^{−}, k) \ar@{^{(}->}[r]_{f^{∗}} & H^{2,1}(G, k).}

The first map is an isomorphism since f^{∗} is an isomorphism by assumption (i), and ψ^{∗} is an isomorphism on H^{1,1} since we know cohomology of \overline{G} and 𝔾_{a}^{−} explicitly. Assumption (iii) implies that the last map is an embedding. Hence, H^{1,1}(H, k)^{𝔾_{a}^{−}} = 0 and, therefore, H^{1,1}(H, k) = 0 since 𝔾_{a}^{−} is unipotent. We conclude that H is even by Lemma 3.4. Since G/H ≅ 𝔾_{a}^{−}, H is the largest even subgroup scheme; hence, H = G_{ev} which proves the claim.

Assumption (1) implies that r is maximal such that there is a surjective π-invariant map G^{0} → 𝔾_{a(r)} since any π-invariant surjection G^{0} → 𝔾_{a(s)} induces an embedding in cohomology H^{1}(𝔾_{a(s)}, k) ↪ H^{1,0}(G, k) by Lemma 3.1. We claim that r is also maximal subject to the existence of a π-invariant surjective map G_{ev}^{0} → 𝔾_{a(r)}. Suppose, to the contrary, that there is a π-invariant surjective map G_{ev}^{0} → 𝔾_{a(r+1)}, and let N be the kernel. Since G_{ev}^{0} = G^{0} ∩ G_{ev}, we have that G^{0}/G_{ev}^{0} ≅ G/G_{ev} ≅ 𝔾_{a}^{−}. We have a commutative diagram of π-invariant homomorphisms:

[figure: N; N; 1; G_{ev}^{0}; G^{0}; 𝔾_{a}^{−}; 1; 1; 𝔾_{a(r+1)}; \overline{G}^{0} = G^{0}/N; 𝔾_{a}^{−}; 1]

If the extension on the bottom row splits, then it π-splits by Lemma 8.9. Hence, there is a π-invariant surjective map G^{0} → 𝔾_{a(r+1)} which contradicts maximality of r. On the other hand, if the map does not split, then the inflation of ζ ∈ H^{1,1}(𝔾_{a}^{−}, k) to \overline{G}^{0} is a non trivial cohomology class in H^{1,1}(\overline{G}^{0}, k) which squares to zero in H^{2,0}(G, k) by Proposition 9.4. Inflating ζ further to G^{0} via the projection G^{0} → \overline{G}^{0}, we get a non trivial π-invariant cohomology class in H^{1,1}(G^{0}, k) which squares to 0. Hence ζ^{2} is in the kernel of the map f^{∗} : H^{2,0}(\overline{G}, k) → H^{2,0}(G, k) which contradicts assumption (2). We therefore conclude that r is maximal such that there is a surjective map G_{ev}^{0} → 𝔾_{a(r)} as claimed.

Since G_{ev} ⊴ G is a normal subgroup scheme, Lemma 2.18 implies that [G_{ev}, G_{ev}] ⊴ G. Let A = G/[G_{ev}, G_{ev}], so that A_{ev} is the abelianisation of G_{ev}.

Claim 10.3.1. We have that A is isomorphic to E_{m,r}^{−} × (ℤ/p)^{×s} or E_{m,r+1,μ}^{−} × (ℤ/p)^{×s} for some m ≥ 1.

Proof of the Claim. Corollary 9.8 implies that A_{ev}^{0} is isomorphic to a quotient of E_{m,n} for some m, n ≥ 1. By Theorem 8.13, this implies that A_{ev}^{0} is isomorphic either to E_{m,n} or to E_{m,n,μ} for some 0 ≠ μ ∈ k. We divide into two cases according to these two possibilities. Looking at homomorphisms from these to 𝔾_{a(r)}, we see that in the first case n = r, while in the second case n = r + 1.

Case I: A_{ev}^{0} ≅ E_{m,r}. This case splits further into two subcases.

(1) r = 1. Since E_{m,1} = W_{m,1}, we have that A^{0} = A_{(1)}^{0} fits in the extension

1 → W_{m,1} → A^{0} → 𝔾_{a}^{−} → 1,

and, hence, is described by Lemma 8.4. The cohomological restriction in the assumption (2) implies that the only allowed possibility is A^{0} ≅ E_{m,1}^{−} since in any other case A^{0}, and, hence, G^{0}, will have a quotient isomorphic to W_{2,1} or W_{1,1}^{−} which are disallowed by Theorem 9.7.III(iii). Hence, A ≅ E_{m,1}^{−} × (ℤ/p)^{×s}. In terms of the diagrams for the possibilities for A^{0} as in Lemma 8.4, the only way to attach the node σ to avoid quotients isomorphic to W_{2,1} and W_{1,1}^{−} is to the node marked with s^{[p]}.

(2) r > 1. In this case, kA^{0} is described by Lemma 8.11(1). The coefficient α in the relation σ^{2} − s^{[p]^{i}} − αs_{1} must be zero since G^{0}, and, hence, A^{0}, has a quotient isomorphic to 𝔾_{a(r)}. The parameter i must be 1 since for i > 1 there will be a quotient isomorphic to W_{2,1}. In terms of the picture in Lemma 8.11, the node σ can be connected only to the node s^{[p]}, for otherwise the top two nodes on the left arm will form a quotient isomorphic W_{2,1}. Hence, A^{0} ≅ E_{m,r}^{−}. Since A_{ev} is abelian, the group of connected components of A acts trivially on A_{ev}^{0}. Since (ℤ/p)^{×s} is a p-group, it also acts trivially on the quotient A^{0}/A_{ev}^{0} ≅ 𝔾_{a}^{−}. Therefore, A ≅ E_{m,r}^{−} × (ℤ/p)^{×s}.

Case II: A_{ev}^{0} ≅ E_{m,r+1,μ}. This case is similar. The possibilities for kA^{0} are given by Lemma 8.11(ii). All of them but one are disallowed by Theorem 9.7.III(iii). We conclude that A^{0} ≅ E_{m,r+1,μ}^{−}, and, therefore, we can identify A with E_{m,r+1,μ}^{−} × (ℤ/p)^{×s}. □

Now that we have identified A = G/[G_{ev}, G_{ev}], it remains to show that G = A, that is [G_{ev}, G_{ev}] = 1. We prove this by contradiction. Assume that G ≠ A.

Note that [G_{ev}, G_{ev}] ⊆ [G, G] ⊆ G^{0} since the group of connected components of G is abelian. Hence, [G_{ev}, G_{ev}] is a connected unipotent finite group scheme. Therefore, there exists a maximal proper subgroup N of [G_{ev}, G_{ev}] such that [G_{ev}, G_{ev}]/N ≅ 𝔾_{a(1)} giving rise to a central extension

(10.1)  1 → 𝔾_{a(1)} → G/N → A → 1.

Let ψ: G → A, ϕ : A → \overline{G} be the projection maps; we factor ψ as \xymatrix{G \ar[r]^{ψ_{2}} & G/N \ar[r]^{ψ_{1}} & A} . The map f: G → \overline{G} then factors as follows:

\xymatrix{f: G \ar[r]^{ψ_{2}} & G/N \ar[r]^{ψ_{1}} & A \ar[r]^{ϕ} & \overline{G} = 𝔾_{a(r)} × 𝔾_{a}^{−} × (ℤ/p)^{×s}}

Since ϕ, ψ are surjective, the induced maps on H^{1} are injective. Since the composition

\xymatrix{f: H^{1,∗}(\overline{G}, k) \ar@{^{(}->}[r]^{ϕ^{∗}} & H^{1,∗}(A, k) \ar@{^{(}->}[r]^{ψ^{∗}} & H^{1,∗}(G, k)}

is an isomorphism, we conclude that

\xymatrix{ψ^{∗}: H^{1,∗}(A, k) \ar[r]^{∼} & H^{1,∗}(G, k)}

is also an isomorphism.

We again consider two cases: A ≅ E_{m,r}^{−} × (ℤ/p)^{×s} and A ≅ E_{m,r+1,μ}^{−} × (ℤ/p)^{×s}. Case I: A ≅ E_{m,r}^{−} × (ℤ/p)^{×s}. Assume that m ≥ 2. By Theorem 9.9,

(1) H^{∗,∗}(A, k) ≅ k[x_{m,1}, …, x_{m,n}, ζ_{m}] ⊗ Λ(λ_{m,1}, …, λ_{m,n}) ⊗ k[z_{1}, …, z_{s}] ⊗ Λ(y_{1}, …, y_{s}),

(2) ϕ^{∗} : H^{1,∗}(\overline{G}, k) → H^{1,∗}(A, k) is an isomorphism,

(3) Ker\{ϕ^{∗} : H^{2,∗}(\overline{G}, k) → H^{2,∗}(A, k)\} = k⟨ζ^{2} + γx_{r}⟩. In particular, ϕ^{∗} and f^{∗} have the same kernel on H^{2,∗}(\overline{G}, k). Therefore, ψ_{1}^{∗}: H^{2,∗}(A, k) → H^{2,∗}(G/N, k) is one-to-one restricted to ϕ^{∗}(H^{2,∗}(\overline{G}, k)).

Consider the five-term exact sequence induced by (10.1):

\xymatrix{H^{1,∗}(A, k) \ar[r] & H^{1,∗}(G/N, k) \ar[r] & H^{1,∗}(𝔾_{a(1)}, k) \ar[r]^{d_{2}} & H^{2,∗}(A, k) \ar[r]^{ψ_{1}^{∗}} & H^{2,∗}(G/N, k)}

Let λ ∈ H^{1,0}(𝔾_{a(1)}, k) be any linear generator. Since G_{ev}/N is non-abelian, d_{2}(λ) ∈ H^{2,0}(A, k) is a nonzero element in the kernel of ψ_{1}^{∗} by Lemma 10.1, and, hence, a nonzero element in the kernel of ψ^{∗}. By Theorem 9.9, the only linear generator of H^{2,0}(A, k) which is not in the image of ϕ^{∗}, is x_{m,r}. Hence, replacing λ by a nonzero multiple if necessary, we may assume that

d_{2}(λ) = x_{m,r} + u

where u ∈ ϕ^{∗}(H^{2,∗}(\overline{G}, k)).

Claim 10.3.2. u = αζ^{2} for some α ∈ k.

Proof. We prove the claim by consecutive application of Steenrod operations, similarly to Theorem 4.3. Since any element in Ker{ψ^{∗} : H^{2,∗}(A, k) → H^{2,∗}(G, k)} must have the form ax_{m,r} + v with a ≠ 0 and v ∈ ϕ^{∗}(H^{2,0}(\overline{G}, k)), we have

dim_{k} Ker{ψ^{∗}: H^{2,∗}(A, k) → H^{2,∗}(G, k)} = 1.

Let

u = αζ_{m}^{2} + ∑_{1≤i<j≤r} a_{i,j}λ_{m,i}λ_{m,j} + ∑_{1≤j<r} b_{j}x_{m,j} + ∑_{1≤i≤r,1≤j≤s} c_{i,j}λ_{m,i}y_{j} + ∑_{1≤i<j≤s} d_{i,j}y_{i}y_{j} + ∑_{1≤j≤s} e_{j}z_{j}

for some constants α, a_{i,j}, b_{j}, c_{i,j}, d_{i,j}, e_{j} ∈ k which are not all zero. Since Ker ψ^{∗} is stable under the Steenrod operations and 𝒫^{0}(x_{m,r}) = 0, we conclude that 𝒫^{0}(u) = 0, which forces u to be of the form

u = αζ_{m}^{2} + λ_{m,r}( ∑_{1≤i<r} a_{i}λ_{m,i}) + λ_{m,r}( ∑_{1≤j≤s} c_{j}y_{j}).

Proposition 4.1 together with the Cartan formula imply (by induction) that

𝒫^{i}(ζ_{m}^{2i}) = ζ_{m}^{2pi}

and all other Steenrod operations vanish on ζ_{m}^{2i}.

Since β𝒫^{0}(λ_{m,r}) = ζ_{m}^{2}, applying β𝒫^{0} to (10), we get

ζ_{m}^{2}( ∑_{1≤i<r} a_{i}^{p}λ_{m,i+1}) + ζ_{m}^{2}( ∑_{1≤j≤s} c_{j}^{p}z_{j}).

Applying β𝒫^{1}, we get

ζ_{m}^{2p}( ∑_{1≤i<r−1} a_{i}^{p^{2}}λ_{m,i+2}) + ζ_{m}^{2p+2} + ζ_{m}^{2p}( ∑_{1≤j≤s} c_{j}^{p^{2}}z_{j}).

If there is a nonzero coefficient c_{j}, then applying 𝒫^{p}, 𝒫^{p^{2}}, … and then taking invariants under the Frobenius map as in the proof of Theorem 4.3, we eventually get that the kernel of the map ψ^{∗}: H^{2,∗}(A, k) → H^{2,∗}(G, k) contains an element ζ_{m}^{2p^{i}}β𝒫^{0}(y) with y ∈ H^{1}((ℤ/p)^{×s}, 𝔽_{p}) ⊂ H^{1,0}(\overline{G}, k) ≅ H^{1,0}(A, k). This means that f^{∗} vanishes on ζ^{2p^{i}}β𝒫^{0}(y), contradicting assumption (4). Hence, we can assume that all coefficients c_{j} are zero.

Suppose there is a coefficient a_{i} ≠ 0. Then (10) has the form

(10.2)  ζ_{m}^{2p}( ∑_{1≤i<r−1} a_{i}^{p^{2}}λ_{m,i+2}) + ζ_{m}^{2p+2}.

Applying 𝒫^{p}, 𝒫^{p^{2}}, … and stopping right before everything annihilates, we conclude that ζ_{m}^{2p^{i}+2} ∈ Ker ψ^{∗}, once again contradicting assumption (4).

Hence, all coefficients, except for possibly α, are zero. This proves the claim. □

Since G/G_{ev} ≅ A/A_{ev} ≅ 𝔾_{a}^{−}, the extension (10.1) restricts to an extension on the even subgroup schemes.

(10.3)  [figure: 1; 𝔾_{a(1)}; G/N; A; 1; 1; 𝔾_{a(1)}; G_{ev}/N; A_{ev}; 1]

This gives rise to a commutative diagram of the corresponding 5-term sequences:

[figure: H^{1,∗}(A, k); H^{1,∗}(G/N, k); H^{1,∗}(𝔾_{a(1)}, k); d_{2}; H^{2,∗}(A, k); res; ψ_{1}^{∗}; H^{2,∗}(G/N, k); H^{1,∗}(A_{ev}, k); H^{1,∗}(G_{ev}/N, k); H^{1,∗}(𝔾_{a(1)}, k); d_{2}^{ev}; H^{2,∗}(A_{ev}, k); ψ_{1,ev}^{∗}; H^{2,∗}(G_{ev}/N, k)]

By Claim 10.3.2, d_{2}(λ) = x_{m,r}+αζ^{2}. Since ζ goes to 0 under the restriction map H^{∗,∗}(A, k) → H^{∗}(A_{ev}, k), we get that d_{2}^{ev}(λ) = x_{m,r} ∈ H^{2}(A_{ev}, k) = H^{2}(E_{m,r}, k).

Consider the standard surjection map: E_{m+1,r} × (ℤ/p)^{×s} → E_{m,r} × (ℤ/p)^{×s}. By Theorem 9.9, d_{2}^{ev}(λ) = x_{m,r} vanishes when inflated to H^{2}(E_{m+1,r}, k). Proposition 10.2 now implies that G_{ev}/N is abelian. This contradicts the choice of N, and completes the proof that G = A in this case.

It remains to consider the case m = 1, that is, when A ≅ E_{1,r}^{−} × (ℤ/p)^{×s} = \overline{G}. In this case d_{2}(λ) = ζ^{2} + γx_{r}, and, hence, d_{2}^{ev}(λ) = γx_{r}. Considering the surjective map E_{2,r} × (ℤ/p)^{×s} → 𝔾_{a(r)} × (ℤ/p)^{×s}, we conclude by Proposition 10.2 that G_{ev}/N is abelian, getting a contradiction again. Hence, G = A in the case m = 1.

Case II: A ≅ E_{m,r+1,μ}^{−} × (ℤ/p)^{×s}. The proof is very similar, replacing x_{m,r} with x_{m,r+1,μ} from Theorem 9.10. The corresponding abelian cover which plays the role of A′ in Proposition 10.2 in this case is the canonical map E_{m,r+1} → E_{m,r+1,μ}. □

11. The main detection theorem

The proof of the main detection Theorem 1.2 effectively splits into two parts. The first part covers the case when G satisfies Hypothesis 7.1. The techniques needed to deal with this case are mostly adaptations of what was done for finite group schemes (without the grading) and are summarised in Part I of the paper. The only, but significant, exception is Theorem 4.4 which requires extensive new calculations for cohomology of supergroup schemes done in [14]. In the ungraded case the only group schemes which fail Hypotheses 7.1 are the elementary ones, that is, finite groups schemes isomorphic to 𝔾_{a(r)} × (ℤ/p)^{×s}, which form the detection family. Hence, the inductive detection Theorem 7.2 gives the full detection theorem in the ungraded case.

In the super case however we have to deal with case (ii) of Theorem 4.3 when the kernel of the map on cohomology induced by f: G → \overline{G} has an element of the form ζ^{2} − γx_{r}. The new technology developed in Part II culminating in the cohomological characterization of the elementary supergroup schemes in Theorem 10.3 is what we need to deal with this case.

Theorem 1.2 is an immediate consequence of the following theorem. We employ terminology of a detection family introduced in the beginning of Section 6.

Theorem 11.1. Suppose that G is a finite unipotent supergroup scheme which is not isomorphic to a quotient of some E_{m,n}^{−} × (ℤ/p)^{×s}. Then

(i) nilpotence of elements in cohomology of modules and (ii) projectivity of kG-modules

are detected on proper sub-supergroup schemes after field extension.

Proof. Let G = G^{0} ⋊ π with G^{0} connected and π finite. Since G is unipotent, so is G^{0}, and π is a finite p-group. If π is not elementary abelian, then by Theorem 4.2, G satisfies case (c) of Hypothesis 7.1, and we are done. So we now assume that π = (ℤ/p)^{×s} is elementary abelian. By Lemma 3.1,

H^{1,0}(G, k) ≅ Hom_{sGr/k}(G^{0}, 𝔾_{a})^{π} × Hom(π, 𝔾_{a})
H^{1,1}(G, k) ≅ Hom_{sGr/k}(G^{0}, 𝔾_{a}^{−})^{π}.

We examine the dimensions

δ = dim_{k} Hom_{sGr/k}(G^{0}, 𝔾_{a(1)})^{π}
ϵ = dim_{k} Hom_{sGr/k}(G^{0}, 𝔾_{a}^{−})^{π}.

Since π is unipotent, if δ = 0 then we have Hom_{sGr/k}(G^{0}, 𝔾_{a(1)}) = 0, and if ϵ = 0 then Hom_{sGr/k}(G^{0}, 𝔾_{a}^{−}) = 0. Thus, δ = ϵ = 0, then G^{0} is trivial by Lemma 3.2, hence, G ≅ (ℤ/p)^{×s}, and we are done. We may therefore assume that one of them is nonzero. If either δ or ϵ is greater than one then we are in case (a) or (b) of Hypothesis 7.1, and we are done by Theorem 7.2. So each is either zero or one, and they are not both zero.

The action of the Frobenius map F: 𝔾_{a} → 𝔾_{a} induces a map

F: Hom_{sGr/k}(G^{0}, 𝔾_{a}) → Hom_{sGr/k}(G^{0}, 𝔾_{a})

which commutes with the action of π. A π-invariant map G^{0} → 𝔾_{a} lands in 𝔾_{a(1)} ≤ 𝔾_{a} if and only if it is in the kernel of F. So there exists r ≥ 0 and a surjective map

ξ ∈ Hom_{sGr/k}(G^{0}, 𝔾_{a(r)})^{π}

such that ξ, F (ξ), …, F^{r−1}(ξ) is a k-basis for Hom_{Gr/k}(G^{0}, 𝔾_{a})^{π}. The map ξ extends to a surjective map

f: G → \overline{G} ≅ 𝔾_{a(r)} × (𝔾_{a}^{−})^{ϵ} × (ℤ/p)^{×s}

and f^{∗}: H^{1,∗}(\overline{G}, k) → H^{1,∗}(G, k). This construction accounts both for the case δ = 0 (with r = 0 so that \overline{G} = 𝔾_{a}^{−} × (ℤ/p)^{×s}) and ϵ = 0 (with \overline{G} = 𝔾_{a(r)} × (ℤ/p)^{×s}, r ≥ 1).

If f is an isomorphism then G ≅ \overline{G} is isomorphic to a quotient of E_{1,r}^{−} × (ℤ/p)^{×s} contradicting the assumption of the theorem. Otherwise, by Lemma 3.5,

f^{∗}: H^{2,∗}(\overline{G}, k) → H^{2,∗}(G, k)

is not injective. If the kernel contains an element of degree (2, 1), then by Theorem 4.4 we are in case (c) of Hypothesis 7.1, so we are done by Theorem 7.2. Therefore, we may assume that the kernel contains an element of degree (2, 0) and we have two cases according to Theorem 4.3. In the first case, it contains an element of the form

x_{r}^{n}β𝒫^{0}(v_{1}) … β𝒫^{0}(v_{m}),

which again puts us in case (c) of Hypothesis 7.1, and we again apply Theorem 7.2. In the second case, the kernel is generated by ζ^{2} + γx_{r}. If γ = 0, then we can apply Theorem 7.2 once again, since Hypothesis 7.1 is satisfied by the image of ζ^{2}.

The upshot of this is that we may assume that we are in case (ii) of Theorem 4.3 with γ ≠ 0 and that f^{∗} induces an isomorphism on H^{2,1}. Hence, G satisfies the hypotheses (1), (2) and (3) of Theorem 10.3. If it fails hypothesis (4) of Theorem 10.3, then we are in case (c) of Hypothesis 7.1 one last time. Otherwise, G is isomorphic to a quotient of E_{m,r}^{−} × (ℤ/p)^{×s} for some m ≥ 2 by Theorem 10.3. □

There is another notion of nilpotency for elements of H^{∗,∗}(G, Λ) where Λ is a unital G-algebra. Namely, ξ ∈ H^{i,∗}(G, Λ) is nilpotent if for some n > 0, the image of ξ^{⊗n} ∈ H^{in,∗}(G, Λ^{⊗n}) in H^{in,∗}(G, Λ) is zero. The following analogue of Theorem 11.1 for this notion of nilpotents has both a weaker hypothesis and a weaker conclusion.

Theorem 11.2. Let G be a finite unipotent supergroup scheme over a field k, and Λ be unital G-algebra. Then an element x ∈ H^{i,∗}(G, Λ) is nilpotent, that is x^{n} ∈ H^{in,∗}(G, Λ) is zero for some n > 0, if and only if for every extension field K of k and every elementary sub-supergroup scheme E of G_{K}, the restriction of x_{K} ∈ H^{∗,∗}(G_{K}, Λ_{K}) to H^{∗,∗}(E, Λ_{K}) is nilpotent, that is some power of x_{K} vanishes in H^{∗,∗}(E, Λ_{K}).

Proof. First, we claim that the analogue of Theorem 7.2 holds for H^{∗,∗}(G, Λ) with this notion of nilpotency. Indeed, If we take M = Λ in Proposition 5.3 then the conclusion clearly holds for ξ^{2} ∈ H^{∗,∗}(G, Λ). Hence, if G satisfies Hypothesis 7.1(c), the proof of Theorem 7.2 carries over to this case.

If we assume that Hypotheses 7.1 (a) or (b) hold, then the proof is identical to that of Case II(b) in [3, Theorem 6.1] (see also [6, Theorem 2.5]) so we will not reproduce it here.

With these observations, the proof of the analogue of Theorem 11.1 is again identical to the one we give above. □

In [10], we show that projectivity for modules of finite group schemes is detected on the family of elementary subgroup schemes after coextension of scalars. In the following theorem we state that this also holds for finite unipotent supergroup schemes.

Theorem 11.3. Let G be a finite unipotent supergroup scheme over a field k of positive characteristic p > 2, and M be a kG-module. Then the following hold.

(i) An element x ∈ H^{∗,∗}(G, M) is nilpotent if and only if for every extension field K of k and every elementary sub-supergroup scheme E of G_{K}, the restriction of x_{K} ∈ H^{∗,∗}(G_{K}, M^{K}) to H^{∗,∗}(E, M^{K}) is nilpotent.

(ii) A kG-module M is projective if and only if for every extension field K of k and every elementary sub-supergroup scheme E of G_{K}, the restriction of M^{K} to E is projective.

Proof. The proof of Theorem 11.1 carries over to this case almost without change. The only difference occurs when G satisfies (a) or (b) of Hypothesis 7.1. Then we still proceed exactly as in [3, Theorem 8.1] but appeal to [10, Lemma 4.1] for the main ingredient which is the appropriate version of the Kronecker quiver lemma for M^{K}. □

12. The Steenrod algebra

Algebraic topology often deals with ℤ-graded commutative or ℤ-graded cocommutative Hopf algebras, where the ℤ-grading introduces the usual sign conventions.

Let k be a ℤ-graded field of odd characteristic p. This means that either k is a field in the usual sense, concentrated in degree zero, or it is a ring of Laurent polynomials k = k_{0}[u, u^{−1}], where u has nonzero degree, which is even if k_{0} does not have characteristic two. Over a ℤ-graded field, every graded module is free. This means that it is isomorphic to a direct sum of shifts of k.

An affine ℤ-graded group scheme over k is a covariant functor from ℤ-graded commutative k-algebras (again, the convention is that yx = (−1)^{|x||y|}xy) to groups, whose underlying functor to sets is representable. If G is an affine ℤ-graded group scheme over k then its coordinate ring k[G] is the representing object. It is a ℤ-graded commutative Hopf algebra. This gives a contravariant equivalence of categories between affine ℤ-graded group schemes and ℤ-graded commutative Hopf algebra.

An affine ℤ-graded group scheme G has finite type if each graded piece is finite dimensional. In this case, the graded dual kG^{i} = Hom_{k}(k[G]^{−i}, k) is a ℤ-graded cocommutative Hopf algebra of finite type. This gives a covariant equivalence of categories between ℤ-graded group schemes of finite type and ℤ-graded cocommutative Hopf algebras of finite type.

We are interested in particular in the finite ℤ-graded group schemes; these are the ones for which not only is each graded piece finite dimensional, but the total rank as a k-module is finite. In other words, if k is a field in degree zero, this means that there are only finitely many degrees in which k[G] is nonzero, whereas if k is a ring of Laurent polynomials, it is equivalent to finite type.

Theorem 12.1. Let G be a finite ℤ-graded unipotent group scheme over a ℤ-graded field k, and M be a kG-module. Then the following hold.

(i) An element ξ of H^{∗,∗}(G, M) is nilpotent if and only if for every extension ℤ-graded field K of k, and every elementary subgroup scheme E of G_{K}, the restriction of ξ_{K} ∈ H^{∗,∗}(G_{K}, M_{K}) to H^{∗,∗}(E, M_{K}) is nilpotent.

(ii) A kG-module M is projective if and only if for every extension ℤ-graded field K of k, and every elementary subgroup scheme E of G_{K}, the restriction of M_{K} to E is projective.

Proof. If K = K_{0}[u, u^{−1}] with |u| = 2, there is a one to one correspondence between finite ℤ-graded group schemes G over K and finite supergroup schemes \overline{G} over K_{0}. This is given by identifying the even graded pieces using powers of u, and similarly with the odd graded pieces. Similarly if M is a KG-module then \overline{M} is a K_{0}\overline{G}-module. Furthermore, the space of KG-module homomorphisms Hom_{KG}^{i}(M, N) of degree i is isomorphic to Hom\overline{ı} ; K_{0}\overline{G}(\overline{M}, \overline{N}) where \overline{ı} is the reduction of i modulo two. So if we use powers of u to identify an even shift of M with M, thus giving Mod(KG) a mod two internal shift, there is an equivalence of categories between Mod(KG) and Mod(K_{0}\overline{G}) preserving the internal shift modulo two. Projectivity is therefore preserved, and StMod(KG) is equivalent to StMod(K_{0}\overline{G}). Similarly, H^{∗,∗}(\overline{G}, K_{0}) is obtained by reducing the internal grading modulo two on H^{∗,∗}(G, K), again using powers of u.

Since every extension ℤ-graded field of k can be extended to one of the type K = K_{0}[u, u^{−1}] above, the theorem follows from Theorem 1.2. □

Recall from Milnor [36], Steenrod and Epstein [41] that for p odd, the graded dual A^{∗} of the Steenrod algebra A is a tensor product

k[ξ_{1}, ξ_{2}, …] ⊗ Λ(τ_{0}, τ_{1}, …)

of a polynomial ring in generators ξ_{n} of degree 2p^{n} − 2 and an exterior algebra in generators τ_{n} of degree 2p^{n} − 1. We also set ξ_{0} = 1. With this notation, the comultiplication is given by

Δ(ξ_{n}) = ∑_{i=0}^{n} ξ_{n−i}^{p^{i}} ⊗ ξ_{i}, Δ(τ_{n}) = τ_{n} ⊗ 1 + ∑_{i=0}^{n} ξ_{n−i}^{p^{i}} ⊗ τ_{i}.

If A is a finite dimensional Hopf subalgebra of A then the graded dual A^{∗} is a finite dimensional quotient of A^{∗}. Let G be the finite ℤ-graded group scheme corresponding to A, so that 𝔽_{p}G ≅ A and 𝔽_{p}[G] ≅ A^{∗}. Then 𝔽_{p}[G_{(1)}] is a quotient of A^{∗} by a Hopf ideal containing ξ_{1}^{p}, ξ_{2}^{p}, . . .. Letting ξ̄_{n} and τ̄_{n} be the images of ξ_{n} and τ_{n} in this quotient, for n ≥ 1 we have

Δ(ξ̄_{n}) = ξ̄_{n} ⊗ 1 + 1 ⊗ ξ̄_{n}, Δ(τ̄_{n}) = τ̄_{n} ⊗ 1 + 1 ⊗ τ̄_{n} + ξ̄_{n} ⊗ τ̄_{0}

while Δ(τ̄_{0}) = τ̄_{0} ⊗1+1⊗τ̄_{0}. In other words, ξ̄_{n} (n ≥ 1) and τ̄_{0} are primitive, and τ̄_{n} (n ≥ 1) are primitive modulo τ̄_{0}.

If we isolate a single n, and dualise these relations for ξ̄_{n}, τ̄_{n} and τ̄_{0} we get the restricted universal enveloping algebra of a three dimensional restricted Lie superalgebra consisting of the matrices

( \begin{matrix}0 & ∗ & ∗ \\ 0 & 0 & ∗ \\ 0 & 0 & 0 \end{matrix} )

in GL(2|1). The dual elements ξ̄_{n}^{∗} and τ̄_{n}^{∗} to ξ̄_{n} and τ̄_{n} are in the top row, and the dual element τ̄_{0}^{∗} to τ̄_{0} is in the second row. The only non-trivial commutator relation is [ξ̄_{n}^{∗}, τ̄_{0}^{∗}] = τ̄_{n}^{∗}.

Dualising, we get a homomorphism G_{(1)} → 𝔾_{a}^{−}, and the kernel is isomorphic to a subgroup scheme of (𝔾_{a(1)})^{×s} × (𝔾_{a}^{−})^{r}. Every subgroup scheme again has this form, so we have proved the following lemma.

Lemma 12.2. Let A be a finite dimensional Hopf subalgebra of the Steenrod algebra, and let G be the corresponding ℤ-graded group scheme. Then there is a (possibly trivial) homomorphism G_{(1)} → 𝔾_{a}^{−} whose kernel is isomorphic to (𝔾_{a(1)})^{×s} × (𝔾_{a}^{−})^{r} for some r, s ≥ 0. The (𝔾_{a}^{−})^{r} is normal, and the quotient is commutative. In particular, there is no sub-supergroup scheme isomorphic to W_{m,1}^{−} for m ≥ 1. □

Conceptually, what we have done amounts to showing that the first Frobenius kernel of the Steenrod algebra is an extension of 𝔾_{a}^{−} by an infinite product of copies of 𝔾_{a(1)} × 𝔾_{a}^{−}, with gradings tending to infinity, in such a way that over each factor the extension is the one described by the above subgroup of GL(2|1).

Proposition 12.3. Let A be a finite dimensional sub Hopf algebra of the Steenrod algebra A over 𝔽_{p}, and let G be the corresponding finite unipotent connected ℤ-graded group scheme. If E is an elementary ℤ-graded subgroup scheme of G then E ≅ 𝔾_{a(n)} × 𝔾_{a}^{−}.

Proof. By Theorem 8.13, we have that E ≅ E_{m,n}^{−} or E ≅ E_{m,n,μ}^{−}. We need to show that m = 1. But if m ≥ 2, the statement follows from the observation that (E_{m,n}^{−})_{(1)} and (E_{m,n,μ}^{−})_{(1)} both contain W_{m−1,1}^{−} as a subgroup scheme. But (𝔾_{a(1)})^{×s} × (𝔾_{a}^{−})^{×r} does not, and therefore by Lemma 12.2 neither does G_{(1)}. □

The detection theorem for the finite dimensional subalgebras of the Steenrod algebra now follows from Theorem 12.1 and Proposition 12.3.

Theorem 12.4. Let A be a finite dimensional sub-Hopf algebra of the Steenrod algebra A over 𝔽_{p}. Then A is the group algebra of a ℤ-graded finite group scheme. The following hold:

(1) For an A-module M, an element ξ in H^{∗,∗}(A, M) is nilpotent if and only if for every extension ℤ-graded field K of k, the restriction of ξ_{K} ∈ H^{∗,∗}(A_{K}, M_{K}) to every subgroup scheme of A_{K} isomorphic to 𝔾_{a(r)}, 𝔾_{a}^{−}, or 𝔾_{a(r)} × 𝔾_{a}^{−} is nilpotent.

(2) An A-module M is projective if and only if for every extension ℤ-graded field K of k, the restriction of M_{K} to every subgroup scheme of A_{K} isomorphic to 𝔾_{a(r)}, 𝔾_{a}^{−}, or 𝔾_{a(r)} × 𝔾_{a}^{−} is projective. □

Nakano and Palmieri [37] also considered the problem of finding a detecting family for the mod p Steenrod algebra. They do not consider field extensions, and arrive at a larger family of detecting subalgebras, which they call ”quasi-elementary.”

Appendix A. Witt vectors and Dieudonné modules

Recall that finite commutative connected unipotent group schemes form an abelian category 𝔄 which is equivalent to an appropriate category of Dieudonné modules. This is described for example in Fontaine [26], 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}, …, Z_{n} with integer coefficients by

w_{n}(Z_{0}, …, Z_{n}) = p^{n}Z_{n} + p^{n−1}Z_{n−1}^{p} + ⋯ + Z_{0}^{p^{n}}.

Then the polynomials S_{i} and P_{i} in variables X_{0}, …, X_{n}, Y_{0}, …, Y_{n}, again with integer coefficients, are defined by

w_{n}(S_{0}, …, S_{n}) = w_{n}(X_{0}, …, X_{n}) + w_{n}(Y_{0}, …, Y_{n}),
w_{n}(P_{0}, …, P_{n}) = w_{n}(X_{0}, …, X_{n})w_{n}(Y_{0}, …, Y_{n}).

So for example S_{0} = X_{0} + Y_{0}, P_{0} = X_{0}Y_{0},

S_{1} = X_{1} + Y_{1} + \frac{(X_{0} + Y_{0})^{p} − X_{0}^{p} − Y_{0}^{p}}{p} , P_{1} = pX_{1}Y_{1} + X_{0}^{p}Y_{1} + X_{1}Y_{0}^{p},

and so on.

Witt vectors W (k) over k are vectors (a_{0}, a_{1}, …) with a_{i} ∈ k, where S_{i} and P_{i} give the coordinates of the sum and product:

(a_{0}, a_{1}, …) + (b_{0}, b_{1}, …) = (S_{0}(a_{0}, b_{0}), S_{1}(a_{0}, a_{1}, b_{0}, b_{1}), …)
(a_{0}, a_{1}, …)(b_{0}, b_{1}, …) = (P_{0}(a_{0}, b_{0}), P_{1}(a_{0}, a_{1}, b_{0}, b_{1}), …).

Thus for example if k = 𝔽_{p} then W (k) is the ring of p-adic integers 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 σ. It is defined by (a_{0}, a_{1} …)^{σ} = (a_{0}^{p}, a_{1}^{p}, …).

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

V (a_{0}, a_{1}, …) = (0, a_{0}, a_{1}, …),

and the Frobenius F given by

F (a_{0}, a_{1}, …) = (a_{0}^{p}, a_{1}^{p}, …).

These commute, and their product corresponds to multiplication by p on Witt vectors. Multiplication by a Witt vector x ∈ 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 V x^{σ} = xV and F x = x^{σ}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 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 𝔾_{a(1)}.

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

F V = V F = p,  V x^{σ} = xV,  F x = x^{σ}F

for x ∈ 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 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^{♯}.

Now consider the subcategory 𝔄_{m,n} of 𝔄 consisting of the those group schemes G in 𝔄 such that G has height at most n and the Cartier dual G^{♯} has height at most m. Then there is a covariant equivalence of categories between 𝔄_{m,n} and the category 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

Hom_{𝔄}(W_{m,n}, −): 𝔄_{m,n} → mod(D_{k}/(V^{m}, F^{n})).

Write D̂_{k} for the corresponding completion lim_{←} D_{k}/(V^{m}, F^{n}) Then every D̂_{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 𝔄 and the category fl(D̂_{k}) of D̂_{k}-modules of finite length. Let us write

ψ: fl(D̂_{k}) → 𝔄

for this equivalence. Thus for example ψ(D_{k}/(V^{m}, F^{n})) ≅ W_{m,n} and ψ(D_{k}/(V^{m}, F^{n}, p)) ≅ W_{m,n}/W_{m−1,n−1} ≅ E_{m,n}, where the last notation is introduced in Defintion 8.6.

Let G = ψ(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 ≥ 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 = V x^{σ} (x ∈ 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 ψ, 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} ≃ k[s]/s^{p^{m}}.

Lemma A.2. Let G be a finite unipotent abelian group scheme. If dim_{k} Hom_{Gr/k}(G, 𝔾_{a(1)}) = 1 and 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 dim_{k} Hom_{Gr/k}(G, 𝔾_{a(1)}) = 1 implies that the corresponding Dieudonné module is cyclic, G_{ev} ≅ ψ(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 = F V 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

E_{m,n} = ψ(D_{k}/(V^{m}, F^{n}, p)).

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 D̂_{k}/(p) as a left D̂_{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,μ} = D_{k}/(F^{n} − μV^{m}, p) (of length m + n) for some m, n ≥ 1 and 0 ≠ μ ∈ k. The only isomorphisms among these modules are given by M_{m,n,μ} ≅ M_{m,n,μ′} if and only if μ/μ′ = a^{p^{m+n}−1} for some a ∈ k.

Outline of proof. Let M be a nonzero finite quotient of D̂_{k}/(p), let m be the height of M^{♯} 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} − μV^{m−1}) for some 0 ≠ μ ∈ k. If M is equal to this, we have M ≅ M_{m−1,n−1,μ}. Otherwise M is a proper quotient of M_{m−1,n−1,μ}. But the socle of M_{m−1,n−1,μ} 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^{i}M and M/V^{i}M distinguish all isomorphism classes, with the possible exception of isomorphisms between M_{m,n,μ} and M_{m,n,μ′}. Such an isomorphism is determined modulo radical endomorphisms by a scalar a ∈ k^{×} ⊆ W (k)^{×}. The equation (F^{n} − μV^{m})a = b(F^{n} − μ′V^{m}) implies that b = a^{σ^{n}} and μa = b^{σ^{m}}μ′. Thus

μ/μ′ = a^{σ^{m+n}}a^{−1} = a^{p^{m+n}−1}.  □

Remark A.4. Note that if k = 𝔽_{p} then this condition on μ and μ′ is only satisfied if μ = μ′, so there are p − 1 isomorphism classes of M_{m,n,μ}. But if k is algebraically closed then the isomorphism type of M_{m,n,μ} is independent of μ.

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Dave Benson, Institute of Mathematics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland U.K.

Srikanth B. Iyengar, Department of Mathematics, University of Utah, Salt Lake City, UT 84112, U.S.A.

Henning Krause, Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany.

Julia Pevtsova, Department of Mathematics, University of Washington, Seattle, WA 98195, U.S.A.