PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0002-9939(XX)0000-0

PROJECTIVITY OF MODULES FOR INFINITESIMAL UNIPOTENT GROUP SCHEMES

CHRISTOPHER P. BENDEL

(Communicated by Lance W. Small)

Abstract. In this paper, it is shown that the projectivity of a rational module for an infinitesimal unipotent group scheme over an algebraically closed field of positive characteristic can be detected on a family of closed subgroups.

Let k be an algebraically closed field of characteristic p > 0 and G be an infinitesimal group scheme over k, that is an affine group scheme G over k whose coordinate (Hopf) algebra k[G] is a finite-dimensional local k-algebra. A rational G-module is equivalent to a k[G]-comodule and further equivalent to a module for the finitedimensional cocommutative Hopf algebra k[G]^{∗} ≡ Hom_{k}(k[G], k). Since k[G]^{∗} is a Frobenius algebra (cf. [Jan]), a rational G-module (even infinite-dimensional) is in fact projective if and only if it is injective (cf. [FW]). Further, for any rational G-module M and any closed subgroup scheme H ⊂ G, if M is projective over G, then it remains projective upon restriction to H (cf. [Jan]). We consider the question of whether there is a “nice” collection of closed subgroups of G upon which projectivity (over G) can be detected.

For an example of what we mean by a “nice” collection, consider the situation of modules over a finite group. Over a field of characteristic p > 0, a module over a finite group is projective if and only if it is projective upon restriction to a p-Sylow subgroup (cf. [Rim]). For a p-group (and hence for any finite group), L. Chouinard [Ch] showed that a module is projective if and only if it is projective upon restriction to every elementary abelian subgroup. If the module is assumed to be finite-dimensional, this result follows from the theory of varieties for finite groups (cf. [Ca] or [Ben]). Indeed, elementary abelian subgroups play an essential role in this theory.

In work of A. Suslin, E. Friedlander, and the author [SFB1], [SFB2], a theory of varieties for infinitesimal group schemes was developed. In this setting, subgroups of the form 𝔾_{a(r)} (the rth Frobenius kernel of the additive group scheme 𝔾_{a}) play the role analogous to that of elementary abelian subgroups in the case of finite groups. Not surprisingly then, for finite-dimensional modules, one obtains the following analogue of Chouinard’s Theorem.

Proposition 1 ([SFB2, Proposition 7.6]). Let k be an algebraically closed field of characteristic p > 0, r > 0 be an integer, G be an infinitesimal group scheme over k of height ≤ r, and M be a finite-dimensional rational G-module. Then M is projective as a rational G-module if and only if whenever H ⊂ G is a subgroup

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

1991 Mathematics Subject Classification. Primary 14L15, 20G05; Secondary 17B50.

scheme isomorphic to 𝔾_{a(s)} (with s ≤ r) the restriction of M to H is projective as a rational H-module.

Under some stronger hypotheses, E. Cline, B. Parshall, and L. Scott had previously shown that it suffices to consider a much smaller collection of subgroups (cf. [CPS, Main Theorem]). More precisely, their result applies to a group scheme G of the form N_{(r)} (the rth Frobenius kernel of N) where N is a connected (reduced) T -stable subgroup scheme of a connected, semisimple algebraic group scheme over k (and T is a maximal torus). If M is a finite-dimensional N_{(r)}T -module (i.e. M is an N_{(r)}-module which admits a compatible T -structure), then the projectivity of M may be detected by taking only those subgroup schemes of the form H = U_{α(r)} for each root subgroup U_{α} ⊂ N. However, this result does not hold in general for infinite-dimensional modules (cf. [CPS, Example (3.2)]), whereas Chouinard’s Theorem for finite groups does.

The goal of this paper is to show that Proposition 1 also holds for infinitedimensional modules if G is assumed to be unipotent. (An affine group scheme G is said to be unipotent if it admits an embedding as a closed subgroup of U_{n}, the subgroup scheme in GL_{n} of strictly upper triangular matrices, for some postive integer n.) Although unnecessary for Chouinard’s Theorem, to handle arbitrary modules, we need a slightly stronger hypothesis. Indeed, consider the case that G is isomorphic to a product 𝔾_{a(1)}^{×n} of height one additive group schemes. As an algebra, k[G]^{∗} is isomorphic to the group algebra of an elementary abelian p-group and the desired theorem becomes equivalent to Dade’s Lemma [Dade] which says that a kΠ-module for an elementary abelian p-group Π is projective if and only if it is projective upon restriction to every “cyclic shifted subroup”. This result was originally proved for finite-dimensional modules and D. Benson, J. Carlson, and J. Rickard [BCR2] observe that one must consider a larger field in order for Dade’s Lemma to hold in general. As such, we must also consider field extensions. Specifically, in Section 2, we prove the following theorem using some of the ideas in [SFB2] and an argument similar to that of Chouinard [Ch] without appealing to the theory of varieties.

Theorem. Let k be an algebraically closed field of characteristic p > 0, r > 0 be an integer, and G be an infinitesimal unipotent group scheme over k of height ≤ r. For any rational G-module M, M is projective as a rational G-module if and only if for every field extension K/k and every (closed) K-subgroup scheme H ⊂ G⊗_{k} K with H ≃ 𝔾_{a(s)} ⊗_{k} K (with s ≤ r) the restriction of M ⊗_{k} K to H is projective as a rational H-module.

The essential property of a unipotent group scheme is that the trivial module k is the only simple module or equivalently that Hom_{G}(k, M) = M^{G} ≠ 0 for all non-zero rational G-modules M. (Indeed, this property is sometimes taken as the definition of a unipotent group scheme.) Observe that the theory of finitedimensional algebras shows that if k is the only simple module for such an algebra, then the algebra is indecomposable as a module over itself, and hence a module is in fact projective if and only if it is free. Hence, over an infinitesimal unipotent group scheme, the notions of projective, injective, and free are all equivalent.

In the sense of these properties, infinitesimal unipotent group schemes are analogues of finite p-groups. For finite groups, the key to reducing the projectivity of a G-module to the case of a p-group is that the restriction map in cohomology induced by the embedding of a p-Sylow subgroup is an injection. Unfortunately, there is no analogous result in the setting of group schemes.

We remind the reader that the restricted representation theory of a restricted Lie algebra 𝔤 over k is equivalent to the representation theory of a certain (height 1) infinitesimal group scheme (cf. [Jan]). As such, the theorem applies to p-nilpotent restricted Lie algebras, which may be embedded in a Lie algebra of strictly upper triangular matrices, and may be stated as follows.

Corollary. Let k be an algebraically closed field of characteristic p > 0, 𝔤 be a finite-dimensional, p-nilpotent restricted Lie algebra over k, and M be a u(𝔤)- module, where u(𝔤) denotes the restricted enveloping algebra of 𝔤. Then M is projective over u(𝔤) if and only if for every field extension K/k, M is projective upon restriction to each subalgebra u(⟨x⟩) ⊂ u(𝔤 ⊗_{k} K) for all x ∈ 𝔤 ⊗_{k} K with x^{[p]} = 0, where ⟨x⟩ ⊂ 𝔤 ⊗_{k} K denotes the one-dimensional restricted Lie subalgebra of 𝔤 ⊗_{k} K spanned by x, and x^{[p]} denotes the image of x under the restriction map on 𝔤 ⊗_{k} K.

In this context, the result of Cline, Parshall, and Scott asserts that it suffices to consider only those elements x which are root vectors. Consider the two-dimensional nilpotent Lie subalgebra 𝔤 = ⟨x_{α}, x_{α+β}⟩ ⊂ 𝔰𝔩_{3} generated by root vectors, where α and β are the simple roots. In Example (3.2) of [CPS], they exhibit an infinitedimensional non-projective 𝔤-module M which is free upon restriction to the root vectors x_{α} and x_{α+β}; contradicting an infinite-dimensional version of their result. On the other hand, the interested reader can readily check that this module M is not free upon restriction to the element x = cx_{α} + dx_{α+β} whenever c, d ∈ k are both non-zero and hence does not contradict the corollary.

Before turning to the proof, we note one potential application of the theorem (or preferably a generalization of it to arbitrary infinitesimal group schemes). Recently D. Benson, J. Carlson, and J. Rickard [BCR1], [BCR2] have developed a theory of varieties for infinitely generated modules over finite groups. While Chouinard’s Theorem for finite-dimensional modules follows from the theory of varieties, it turns out that knowing Chouinard’s Theorem in advance for arbitrary modules is necessary for certain results in the more general theory. Hence, an attempt to generalize the theory of varieties to arbitrary modules for infinitesimal group schemes may in part require the above theorem.

Acknowledgments. The author gratefully acknowledges the support of Northwestern University. He also thanks Eric Friedlander for the useful conversations, John Palmieri for the discussions which led to the discovery of an error in an earlier version, and the referee for the suggestions.

1. Cohomology Facts

In this section, we record two results about cohomology which will be used in the next section to prove the theorem. First, the following well known fact relates projectivity to vanishing of cohomology for unipotent group schemes. Indeed, the reader will readily observe that this result holds for modules over an arbitrary finite-dimensional k-algebra which admits k as the only simple module and has the property that the notions of injective and projective are equivalent.

Proposition 2. Let k be a field of characteristic p > 0 and G be an infinitesimal unipotent group scheme over k. For any rational G-module M, M is projective (= injective = free) if and only if H^{1}(G, M) = 0.

To prove Proposition 2, we need to consider a minimal injective resolution I_{∗} for M. Any rational G-module M has a unique up to isomorphism injective hull Q_{M} such that soc Q_{M} ≃ soc M (cf. [Jan, I 3.17]). We can form as usual an injective resolution of M:

I_{0} →^{δ_{0}} I_{1} →^{δ_{1}} I_{2} → ⋯

by taking I_{0} = Q_{M} and inductively I_{n} to be the injective hull of I_{n−1}/ im δ_{n−2} and δ_{n−1} : I_{n−1} ↠ I_{n−1}/ im δ_{n−2} ↪ I_{n} (with I_{−1} ≡ M and δ_{−1} : M ↪ Q_{M}). Evidently we have I_{n−1}/ im δ_{n−2} ≃ im δ_{n−1} = ker δ_{n}.

Lemma 1 (cf. [Ben, I 2.5.4]). Let I_{∗} be the minimal injective resolution of M constructed above. Then the differential in the complex Hom_{G}(k, I_{∗}) is trivial and hence we have H^{i}(G, M) = Hom_{G}(k, I_{i}) = I_{i}^{G} for all i ≥ 0.

Proof. Consider δ_{i} : I_{i} → I_{i+1} for any i ≥ 0 and let α : k → I_{i} be a nonzero map. We want to show that δ_{i}^{∘}α = 0. Since α is a nonzero map and k is simple, α is in fact injective and so α(k) is a simple submodule of I_{i}. Hence, α(k) is contained in the socle of I_{i}. By construction, soc I_{i} = soc(ker δ_{i}) ⊂ ker δ_{i}. Hence, α(k) ⊂ ker δ_{i}. In other words, δ_{i}^{∘}α = 0 as claimed.

Using Lemma 1, we now prove Proposition 2.

Proof of Proposition 2. If M is projective, then clearly H^{i}(G, M) = 0 for all i > 0. Conversely, suppose that H^{1}(G, M) = 0 and let I_{∗} be a minimal injective resolution of M as above. Lemma 1 shows in particular that H^{1}(G, M) = I_{1}^{G} and hence I_{1}^{G} = 0. Since G is unipotent, the G-fixed points of every non-zero module are nonzero. Hence, we must have I_{1} = 0. But, that means M was injective (equivalently projective) to begin with.

In the proof of the theorem, we will also need to make use of the structure of the cohomology algebra of 𝔾_{a(1)}, which is simply the same as the cohomology algebra of the finite cyclic group ℤ/p.

Proposition 3 ([CPSvdK]). If p ≠ 2, then the cohomology algebra H^{∗}(𝔾_{a(1)}, k) is the tensor product of a polynomial algebra k[x_{1}] in one generator x_{1} of degree 2 and an exterior algebra Λ(λ_{1}) in one generator λ_{1} of degree 1. If p = 2, then H^{∗}(𝔾_{a(1)}, k) = k[λ_{1}] is a polynomial algebra in one generator λ_{1} of degree 1 (and in this case we set x_{1} = λ_{1}^{2}).

2. Proof of Theorem

We now proceed to prove the theorem. If M is projective as a rational G-module, then M ⊗_{k} K is projective over G ⊗_{k} K and, as already noted, remains so upon restriction to H. Conversely, suppose all restrictions are projective. Let k′/k be any (even trivial) algebraically closed field extension, G′ = G ⊗_{k} k′, and M′ = M ⊗_{k} k′. Clearly for any field extension K/k′ and K-subgroup scheme H ⊂ G′ ⊗_{k′} K with H ≃ 𝔾_{a(s)} ⊗_{k} k′ ⊗_{k′} K the restriction of M′ ⊗_{k′} K to H is projective as a rational H-module.

We proceed by induction on dim_{k′} k′[G′] to show the slightly stronger statment that M′ is projective over G′ for any such k′. If dim_{k′} k′[G′] = 1, G′ is the trivial group scheme and there is nothing to prove. So, assume now that the result holds for any k′ and any H ⊂ G′ with dim_{k′} k′[H] < dim_{k′} k′[G′]. If G′ ≃ 𝔾_{a(s)} ⊗_{k} k′ for some s ≤ r, then we are done by assumption.

So, we may assume from now on that G′ ≄ 𝔾_{a(s)} ⊗_{k} k′. By Proposition 2, to show that M′ is projective over G′, it suffices to show that H^{1}(G′, M′) = 0. Let ϕ : G′ ↠ 𝔾′_{a(1)} = 𝔾_{a(1)} ⊗_{k} k′ be any non-trivial homomorphism of group schemes. (As G is unipotent, there exists at least one such homomorphism.) Consider the short exact sequence of group schemes over k′

1 → N → G′ → 𝔾′_{a(1)} → 1

and the induced Hochschild-Serre spectral sequence

E_{2}^{p,q}(M′) = H^{p}(𝔾′_{a(1)}, H^{q}(N, M′)) ⇒ H^{p+q}(G′, M′).

Now, ϕ induces a map on cohomology ϕ^{∗} : H^{∗}(𝔾′_{a(1)}, k′) → H^{∗}(G′, k′). Let x_{ϕ} = ϕ^{∗}(x_{1}) where x_{1} ∈ H^{2}(𝔾′_{a(1)}, k′) is the canonical generator. Furthermore, the spectral sequence admits an action of the cohomology ring H^{∗}(𝔾′_{a(1)}, k′), with H^{∗}(𝔾′_{a(1)}, k′) acting on the abutment via ϕ^{∗}.

Since ϕ is non-trivial, dim_{k′} k′[N] < dim_{k′} k′[G′] and so by induction M′ is projective upon restriction to N. Thus, H^{i}(N, M′) = 0 for all i > 0 and the spectral sequence collapses to

E_{2}^{p,0} = H^{p}(𝔾′_{a(1)}, (M′)^{N}) ⇒ H^{p}(G′, M′)

giving an isomorphism H^{∗}(𝔾′_{a(1)}, (M′)^{N}) ≃ H^{∗}(G′, M′). We now recall that the action of x_{1} induces a periodicity isomorphism H^{i}(𝔾′_{a(1)}, Q) ≃ H^{i+2}(𝔾′_{a(1)}, Q) for all i > 0 and any rational 𝔾′_{a(1)}-module Q (cf. [SFB2, 2.3]). Hence, the action of x_{ϕ} ∈ H^{2}(G′, k′) also induces a periodicity isomorphism H^{i}(G′, M′) ≃ H^{i+2}(G′, M′) for all i > 0.

We consider the following two cases. CASE I: dim_{k′} Hom_{Gr/k′}(G′, 𝔾′_{a(1)}) = 1

By Theorem 1.6 of [SFB2] (which is an analogue of a characterization of pgroups in terms of cohomology by J.-P. Serre [Se]), x_{ϕ} ∈ H^{2}(G′, k′) is nilpotent and so we conclude from the above periodicity isomorphism that H^{1}(G′, M′) = 0 and hence M′ is projective. CASE II: dim_{k′} Hom_{Gr/k′}(G′, 𝔾′_{a(1)}) > 1

Let ϕ_{1} and ϕ_{2} be two linearly independent homomorphisms from G′ onto 𝔾′_{a(1)}. Further, let c_{1}, c_{2} ∈ k′. Then, by Corollary 1.5 of [SFB2],

c_{1}x_{ϕ_{1}} + c_{2}x_{ϕ_{2}} = x_{c_{1}^{1/p}ϕ_{1}} + x_{c_{2}^{1/p}ϕ_{2}} = x_{c_{1}^{1/p}ϕ_{1}+c_{2}^{1/p}ϕ_{2}} ∈ H^{2}(G′, k′).

If at least one of c_{1}, c_{2} is non-zero, the map c_{1}^{1/p}ϕ_{1} +c_{2}^{1/p}ϕ_{2} : G′ → 𝔾′_{a(1)} is non-zero and so the action of c_{1}x_{ϕ_{1}} + c_{2}x_{ϕ_{2}} induces a periodicity isomorphism H^{i}(G′, M′) ≃

H^{i+2}(G′, M′) for all i > 0. In particular, c_{1}x_{ϕ_{1}} +c_{2}x_{ϕ_{2}} : H^{1}(G′, M′) →^{∼} H^{3}(G′, M′).

Hence, if H^{1}(G′, M′) was a finite-dimensional space, then one would readily conclude from an eigenvalue argument that H^{1}(G′, M′) = 0. Since H^{1}(G′, M′) might be infinite-dimensional (as M′ might be), we make use of an infinite-dimensional substitute used in [BCR2]. For the reader’s convenience, we restate this result here.

Lemma 2 ([BCR2, Lemma 4.1]). Let V and W be vector spaces over an algebraically closed field k and K be a non-trivial field extension of k. Suppose that ϕ_{1}, ϕ_{2} : V → W are linear maps with the property that for every pair of scalars c_{1}, c_{2} ∈ K, not both zero, the linear map c_{1}ϕ_{1} + c_{2}ϕ_{2} : V ⊗_{k} K → W ⊗_{k} K is an isomorphism. Then both V and W are the zero vector space.

Continuing with the proof of the theorem, let K/k′ be any non-trivial algebraically closed field extension. Consider the base changes of G′ and M′ to K: G_{K} ≡ G′⊗_{k′} K and M_{K} ≡ M′⊗_{k′} K. Then we have H^{∗}(G′, k′)⊗_{k′} K ≃ H^{∗}(G_{K}, K) and H^{∗}(G′, M′) ⊗_{k′} K ≃ H^{∗}(G_{K}, M_{K}). Further, since ϕ_{1} and ϕ_{2} remain linearly independent after base change, for any c_{1}, c_{2} ∈ K with at least one non-zero, the map c_{1}ϕ_{1} +c_{2}ϕ_{2} : G_{K} → 𝔾′_{a(1)} ⊗_{k′} K is non-trivial. Since our inductive assumption applies to any algebraically closed field extension of k, it also applies to K and so as above we may conclude that

c_{1}x_{ϕ_{1}} + c_{2}x_{ϕ_{2}} = x_{c_{1}^{1/p}ϕ_{1}+c_{2}^{1/p}ϕ_{2}} : H^{1}(G_{K}, M_{K}) →^{∼} H^{3}(G_{K}, M_{K}).

Hence, applying Lemma 2 to K/k′, we again conclude that H^{1}(G′, M′) = 0 and so M′ is projective over G′ and the proof is complete.

References

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[SFB1] A. Suslin, E.M. Friedlander, C.P. Bendel, Infinitesimal 1-parameter subgroups and cohomology, Jour. Amer. Math. Soc. 10 (1997), 693-728.

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Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 Current address: Department of Mathematics, Statistics, and Computer Science, University

of Wisconsin-Stout, Menomonie, WI 54751

   E-mail address: bendelc@uwstout.edu