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On the Cohomology of Modular Lie Algebras

In this paper we establish a connection between the cohomology of a modular Lie algebra and its p-envelopes. We also compute the cohomology of Zassenhaus algebras and their minimal p-envelopes with coefficients in generalized baby Verma modules and in simple modules over fields of characteristic p>2.


Introduction
This paper is mainly a survey on the cohomology of modular Lie algebras but there are also several new results. It is intended to be sufficiently self-contained to serve as a first introduction to this topic which plays an important role in modular representation theory. Another goal is to advertise the use of p-envelopes and truncated (co)induced modules in the representation theory of Lie algebras over fields of prime characteristic p. As a main example throughout the paper we consider a series of rank one Lie algebras of Cartan type, the Zassenhaus algebras (which are simple in characteristic p > 2). In the following we will describe the contents of the paper in more detail.
In the first section we give some background material on p-envelopes which will be useful for the rest of the paper. We refer the reader to [27,28,29] for more details and most of the proofs. The second section is devoted to establishing a connection between the cohomology of a modular Lie algebra and its p-envelopes which is very similar to a factorization theorem of Hochschild and Serre in characteristic zero. In particular, we show that the vanishing of the cohomology of a modular Lie algebra is equivalent to the vanishing of the cohomology of any of its p-envelopes. Parts of this section are contained in an unpublished diploma thesis of Katrin Legler [24]. In the third section we reformulate a cohomological vanishing theorem of Dzhumadil'daev in terms of the universal penvelope, a result that rests mainly on the trivial action of the universal p-envelopeL of L on the cohomology of L. We also derive several consequences which will be useful later in the paper. In particular, we generalize a cohomological vanishing theorem for Z-graded Lie algebras due to Chiu and Shen. In the fourth section we introduce truncated (co)induced modules which are important in the classification of simple modules for certain classes of modular Lie algebras. We recall the important fact that truncated induced modules and truncated coinduced modules coincide up to a twist. In particular, duals of truncated (co)induced modules are again truncated (co)induced modules, respectively. Moreover, we give a short proof of Shapiro's lemma for truncated induced modules. Mil'ner used his version of truncated induced modules in order to classify the simple modules of the Zassenhaus algebras [25]. In this paper we restrict our attention to certain truncated induced modules of the Zassenhaus algebras, the so-called generalized baby Verma modules which are sufficient for computing the cohomology of Zassenhaus algebras with coefficients in simple modules. In the final section we reprove Dzhumadil'daev's cohomological reduction theorem for Zassenhaus algebras by using Shapiro's lemma for truncated induced modules. Then we apply this to compute the 1-cohomology of Zassenhaus algebras with coefficients in generalized baby Verma modules and in simple modules over fields of characteristic p > 2 without using Mil'ner's classification of the simple modules for the Zassenhaus algebras. As a consequence we determine the central extensions of the Zassenhaus algebras. We also apply the main result of Section 2 in order to compute the 1-cohomology of the minimal p-envelopes of the Zassenhaus algebras with coefficients in generalized baby Verma modules and in simple modules as well as their central extensions. The results for the minimal p-envelopes of the Zassenhaus algebras seem to be new.
For the notation and fundamental results from the representation theory of modular Lie algebras we refer the reader to [29] and [28].

p-Envelopes
Let L be a Lie algebra over a field of prime characteristic p. A triple (L, (·) [p] , ι) is called a p-envelope of L if L is a restricted Lie algebra with p-mapping (·) [p] and ι is a Lie algebra monomorphism from L into L such that the restricted subalgebra ι(L) p of L generated by ι(L) coincides with L. Proposition 1.1 Let L be a Lie algebra over a field of prime characteristic p and let (L, (·) [p] , ι) be a p-envelope of L. Then the following statements hold: (1) [L, L] ⊆ ι(L).
(2) If I is a subspace of L such that ι(L) ⊆ I, then I is an ideal of L.  Moreover, the following structural features are preserved by p-envelopes. In particular, p-envelopes of solvable (nilpotent, abelian) Lie algebras are solvable (nilpotent, abelian, respectively). Proposition 1.2 Let L be a Lie algebra over a field of prime characteristic p and let (L, (·) [p] , ι) be a p-envelope of L. Then the following statements hold: (1) L is abelian if and only if L is abelian.
(2) L is nilpotent if and only if L is nilpotent.
Proof. By definition ι(L) p = L and therefore the assertions are immediate consequences of [ A p-envelope (L min , (·) [p] , ι) of a finite-dimensional modular Lie algebra L is called minimal if its dimension is minimal among the dimensions of all p-envelopes of L. Any two minimal p-envelopes are isomorphic as ordinary Lie algebras (cf. [29, Theorem 2.5.8(1)]). Of importance is the well-known fact that a p-envelope (L, (·) [p] , ι) of L is minimal if and only if the center C(L) of L is contained in ι(L) (cf. [29,Theorem 2.5.8(3)]).
It turns out that minimal p-envelopes of simple Lie algebras are simple as restricted Lie algebras (but in general not simple as ordinary Lie algebras since a non-restrictable Lie algebra is isomorphic to a non-zero proper ideal of any of its p-envelopes) and minimal p-envelopes of semisimple Lie algebras are semisimple. Proposition 1.3 Let L be a finite-dimensional Lie algebra over a field of prime characteristic p and let (L min , (·) [p] , ι) be a minimal p-envelope of L. Then the following statements hold: (1) If L is simple, then L min has no non-zero proper p-ideals.
(2) If L is semisimple, then L min is semisimple.
In the second case one has ι(L) ⊆ I. Since I is a p-ideal of L min , one obtains that L min = ι(L) p ⊆ I, i.e., that I = L min .
As in the proof of (1), one has I ⊆ C(L min ) ⊆ ι(L). Since ι(L) is semisimple and I is an abelian ideal of ι(L), it readily follows that I = 0. ✷ We conclude this section by introducing the Zassenhaus algebras and their minimal p-envelopes. Let F be a field of prime characteristic p, let m be a positive integer, and let A(m) denote the subalgebra of the algebra of divided powers over F generated by   [26, (3.2.2)]). In particular, t := Fe 0 is a one-dimensional torus of b(m). Since e Fe i is p-unipotent so that b(m) = t⊕u(m) is strongly solvable as are the Borel subalgebras of Lie algebras of reductive groups.
Suppose that L is a modular Lie algebra with C(L) = 0. Let L min denote the p-subalgebra of gl(L) generated by ad(L). Then (L min , (·) p , ad) is a minimal p-envelope of L (cf. [29, p. 97]).
Let us apply this to the Zassenhaus algebras W (m). It is clear from the Z-grading of W (m) that (ad e −1 ) p m = 0. Hence is a minimal p-envelope of W (m). Here e i is identified with ad e i for every −1 ≤ i ≤ p m −2 while e is identified with (ad e −1 ) p r for every 1 ≤ r ≤ m − 1. We propose to call the restricted Lie algebra W(m) a restricted Zassenhaus algebra. It follows from [29, Theorem 4.2.4(1)] and Proposition 1.3 that W(m) is semisimple and has no non-zero proper p-ideals if p > 2. In particular, the above p-mapping of W(m) is the only one possible (cf. [29, Corollary 2.2.2(1)]). Note also that W(m) is the semidirect product of a cyclic p-unipotent subalgebra with the ideal W (m).

Cohomology of p-Envelopes
In this section we investigate the relationship between the ordinary cohomology of finite-dimensional Lie algebras and their finite-dimensional p-envelopes. The first lemma we need is a consequence of a long exact sequence due to Dixmier [ Remark. Lemma 2.1 can also be obtained from the Hochschild-Serre spectral sequence (see [24, pp. 30-33]). Our direct proof has the advantage that the mappings in the short exact sequence ( * ) are known explicitly.
Let (L, (·) p ,ι) denote the universal p-envelope of a Lie algebra L over a field of prime characteristic p (cf. It will be crucial for the proof of the main result of this section that the universal p-envelope of a modular Lie algebra L acts trivially on the cohomology of L (cf. [24, Lemma 2.10]). Slightly more generally, we will need the following result: Lemma 2.2 Let L be a Lie algebra over a field of prime characteristic p and let M be an L-module. If ι(L) ⊆ H ⊆ K are subalgebras of a p-envelope L of L, then H n (H, M ) is a trivial K-module for every non-negative integer n.
Proof. Let n be a non-negative integer. It is a consequence of [2, Chapter I, §3, Exercise 12(b), (3)] that H n (L, M ) is a trivial L-module for every non-negative integer n. Since M is a re-strictedL-module and the adjoint representation is restricted, one obtains that H n (L, M ) is a restrictedL-module (cf. [2, Chapter I, §3, Exercise 12(a)] and [29,Theorem 5.2.7]). It follows that AnnL(H n (L, M )) is a p-subalgebra ofL that containsι(L). HenceL = ι(L) p = AnnL(H n (L, M )), i.e., H n (L, M ) is a trivialL-module. Now let (L, (·) [p] , ι) be an arbitrary p-envelope of L. Since the pullback functor fromL-modules to L-modules induced by φ commutes with tensor products and taking duals, the L-action on H n (L, M ) factors through that ofL and therefore is also trivial.
Finally, because of H p = ι(L) p = L, one can assume without loss of generality that H = L, hence that K ⊆ L acts trivially on H n (H, M ). ✷ Now we derive dimension formulas for the cohomology of a finite-dimensional modular Lie algebra in terms of the cohomology of any of its finite-dimensional p-envelopes and vice versa (see [24,Satz 3.9]). Theorem 2.3 Let L be a finite-dimensional Lie algebra over a field F of prime characteristic p, let M be a finite-dimensional L-module, and let L be a p-envelope of L with k := dim F L/L < ∞. Then the following statements hold: Let n be a non-negative integer. It follows from Lemma 2.1 and Lemma 2.2 that Set a n,l := dim F H n (I l , M ) for every non-negative integer n and every integer 0 ≤ l ≤ k. Then (1) can be written as a n,k = k l=0 k l a n−l,0 ∀ n ∈ N 0 which we prove by induction on k. This assertion is trivial for k = 0, so assume it holds for some k ≥ 0. It follows from ( * * ) and the induction hypothesis that a n,k+1 = a n,k + a n−1,k = k l=0 k l a n−l,0 + k l=0 k l a n−1−l,0 = a n,0 + k l=1 k l a n−l,0 + k l=1 k l − 1 a n−l,0 + a n−1−k,0 = k+1 l=0 k + 1 l a n−l,0 , whence the assertion holds also for k + 1.
Now we are ready to prove (2). It is a consequence of the slightly more general identity which can be shown by finite induction on l. For l = 0 the assertion is trivial, so assume it holds for some 0 ≤ l ≤ k − 1. It follows from the induction hypothesis and ( * * * ) that whence the assertion holds also for l + 1. ✷ We rewrite the first part of Theorem 2.3 as an isomorphism theorem similar to a factorization theorem of Hochschild and Serre in characteristic zero (see [21,Theorem 13]).
Theorem 2.4 Let L be a finite-dimensional Lie algebra over a field F of prime characteristic p, let M be a finite-dimensional L-module, and let L be a p-envelope of L with dim F L/L < ∞. Then, for every non-negative integer n, there is an isomorphism We conclude this section with some immediate consequences of Theorem 2.3 that will be useful in Section 5. The first result states that the cohomology of any finite-dimensional p-envelope of a finite-dimensional modular Lie algebra L vanishes if and only if the cohomology of L vanishes.
The low degree cohomology of finite-dimensional p-envelopes can be computed as follows (for the second part see also [24,Korollar 3.10]). Corollary 2.6 Let L be a finite-dimensional Lie algebra over a field F of prime characteristic p, let M be a finite-dimensional L-module, and let L be a p-envelope of L with k := dim F L/L < ∞. Then the following statements hold:

Cohomological Vanishing Theorems
In this section we reformulate a cohomological vanishing theorem of Dzhumadil'daev by using the concept of a universal p-envelope. Then we derive some consequences which will become useful in the last section. By using the cohomology theory of associative algebras Farnsteiner showed that Dzhumadil'daev's vanishing theorem holds more generally for arbitrary elements of the center of the augmentation ideal of the universal enveloping algebra (see [13, Theorem 2.1 and Corollary 2.2] as well as [14] for further generalizations).
Let F be a field of prime characteristic p. A polynomial of the form f (t) = n≥0 α n t p n ∈ F[t] is called p-polynomial (cf. [22,Exercise V.14,p. 196]). It is clear from Proof. Consider a cocycle ψ ∈ Z n (L, M ) of degree n. According to Lemma 2.2, θ n (z)(ψ) is a coboundary. But z ∈ C(L) and thus θ n (z) In the remainder of this section we apply Theorem 3.1 to the cohomology of simple or indecomposable modules. We begin by proving another result of Dzhumadil'daev (cf. [6, Theorem 2]).
Corollary 3.2 Let L be a restricted Lie algebra over a field of prime characteristic and let S be a simple L-module. If H n (L, S) = 0 for some non-negative integer, then S is a restricted L-module.
Since S is simple, it follows from Schur's lemma that (z) S is invertible. Hence the assertion is a consequence of Theorem 3.1. ✷ Corollary 3.2 can be generalized in several directions (see [14,Theorem 6.2] or [14, Corollary 6.4]) but we will need instead an application of the following result to Z-graded Lie algebras. If L is finite-dimensional, then every element in L + ∪ L − is ad-nilpotent so that one can apply

Truncated Induced Modules
In this section we introduce truncated induced modules and truncated coinduced modules following [9], [10], and [16]. For applications of truncated induced modules in the structure theory of modular Lie algebras we refer the reader to [23]. Let F be a field of prime characteristic p and let L be a finite-dimensional Lie algebra over F. as unital associative F-algebras.
Let σ : K → F be the Lie algebra homomorphism given by σ(x) := tr(ad L/K x) for every x ∈ K. Since the mapping σ defined by x → x + σ(x)1 is a Lie algebra homomorphism from K into U (K), there exists a unique algebra homomorphism U (σ) from U (K) into U (K) that extends σ.
Let V be a K-module. Then the action of U (K) on V can be extended to O(L, K) by letting the polynomial algebra F[z 1 , . . . , z k ] act via its canonical augmentation mapping. Let V σ be the twisted module with K-action given by The following result is a generalization of [18, Corollary 1.2] to truncated induced and coinduced modules.
Theorem 4.1 Let L be a finite-dimensional Lie algebra over a field of prime characteristic and let K be a subalgebra of L. Then, for every K-module V , there are the following L-module isomorphisms: i − χ(e i ) p · 1 for every 1 ≤ i ≤ l, it can be seen that U (L) and the χ-reduced universal enveloping algebra u(L, χ) are isomorphic as unital associative F-algebras (cf. [16, p. 157]).
Let V be a K-module. Then the action of U (K) on V can be extended to U (K) via z i · v := 0 for every k + 1 ≤ i ≤ l and every v ∈ V . Since I(K) = I(L) ∩ K, one can consider U (K) as a subalgebra of U (L) and then form induced and coinduced modules called truncated induced modules and truncated coinduced modules, respectively. In the special case of restricted Lie algebras, the truncated (co)induced modules are just (co)induced modules over reduced universal enveloping algebras. The following result (cf.   In the last section we will need Shapiro's lemma for truncated induced modules (see [16,Theorem 2.1] or [9, Theorem in §5 and Corollary 1 in §3]): Theorem 4.3 Let L be a finite-dimensional Lie algebra over a field of prime characteristic, let K be a subalgebra of L, and let V be a K-module. Then, for every non-negative integer n, there is an isomorphism Proof. Let n be a non-negative integer. Then the following isomorphisms can be obtained from  The next result will be useful in Section 5 and is included here for the convenience of the reader. Recall that the divided power algebra A(m) is a natural W (m)-module via (f ∂) · g := f ∂(g) for all f, g ∈ A(m).
for every 0 ≤ a ≤ p m − 1, Ψ is surjective. But dim F V (p − 1) = p m = dim F A and thus Ψ is bijective.
(2) & (4): Finally, consider the Lie algebra homomorphism σ : b → F defined by σ(x) := tr(ad W/b x) for every x ∈ b. We have [e i , e −1 ] = −e i−1 and therefore σ(e i ) = −δ i0 for every 0 ≤ i ≤ p m − 2. Hence [F −λ ] σ = F p−1−λ and thus Theorem 4.1(2) yields V (λ) * ∼ = V (p − 1 − λ) for every λ ∈ F p . Then (2) and (4)  It is well-known that the generalized baby Verma modules of a Zassenhaus algebra W (m) can be realized on the divided power algebra A(m) (see [6, p. 134]). Let λ ∈ F and let A λ (m) denote the vector space A(m) with W (m)-action defined by (f ∂) · g := f ∂(g) + λ∂(f )g for all f, g ∈ A(m). As in the proof of Proposition 4.4(3), one obtains the following isomorphisms: In general, there is a similar relationship between truncated (co)induced modules of modular Lie algebras L ⊇ K and (co)induced modules over restricted universal enveloping algebras of certain p-envelopes L ⊇ K of L and K, respectively. Namely, Farnsteiner [15, Theorem 2.3] proved that for every finite-dimensional K-module V there exist finite-dimensional p-envelopes K ⊆ L of K and L, respectively, such that V is a restricted K-module and U (L) ⊗ O(L,K) V ∼ = u(L) ⊗ u(K) V as L-modules.

Cohomology of Zassenhaus Algebras
In this section we compute the cohomology of Zassenhaus algebras with coefficients in generalized baby Verma modules and in simple modules.
The first result reduces the computation of the cohomology of the Zassenhaus algebra W (m) with coefficients in a generalized baby Verma module to the t-invariants of the cohomology of the maximal nilpotent subalgebra u(m) with coefficients in the one-dimensional trivial module (cf. [6,Theorem 5]).
Theorem 5.1 Let F be a field of prime characteristic p, let m be a positive integer, and let λ ∈ F. Then, for every non-negative integer n, there is an isomorphism of F-vector spaces.
Proof. It is a consequence of Corollary 2.6(1) and Theorem 5.1 that (1) If p = 3, then (2) If p > 3, then Proof. Set W := W (m) and u := u(m). Since u acts trivially on F λ+1 , it follows from Theorem 5.1 that For the last two summands we have that Since u acts trivially on F λ+1 , we obtain for the first summand that where t acts via (e 0 · η)(x) = e 0 · η(x) − η([e 0 , x]) for every x ∈ u. (Here x denotes the coset of x in u/ [u, u].) By a straightforward computation (see [24,Lemma 5.14]) one can show that Suppose now that η ∈ Hom F (u/[u, u], F λ+1 ) t , i.e., e 0 · η = 0. Then Moreover, we obtain the following results which, in conjunction with (+), finish the proof of the theorem.
(1) If p = 3, then Remark. It is an immediate consequence of Theorem 5.3 and Proposition 4.4(1) that In fact, the outer derivations of W (m) are induced by (ad e −1 ) p r for any 1 ≤ r ≤ m − 1 (see also the remarks at the end of Section 1).
In view of the discussion before Lemma 2.2, every W (m)-module is a W(m)-module for the minimal p-envelope W(m) of W (m). In particular, V (λ) is a W(m)-module for every λ ∈ F. The next result is a consequence of Corollary 2.6(2), Theorem 5.3, and Proposition 5.2.
Theorem 5.4 Let F be a field of prime characteristic p > 2, let m be a positive integer, and let λ ∈ F. Then the following statements hold: (2) If p > 3, then Remark. According to Proposition 4.6, V (λ) ∼ = u(W(m)) ⊗ u(b(m)) F λ is a restricted baby Verma module for W(m) if λ ∈ F p . This isomorphism, in conjunction with [16, Proposition 1.5] and Theorem 4.3, can be employed to give a proof of Theorem 5.4 for λ ∈ F p which does neither use Theorem 5.3 nor the results in Section 2.
Following Hochschild [20] we define the restricted cohomology of a restricted Lie algebra L with coefficients in a restricted L-module M by means of Corollary 5.5 Let F be a field of prime characteristic p > 2, let m be a positive integer, and let λ ∈ F p . Then the following statements hold: (2) If p > 3, then Proof. Set W := W(m), b := b(m), and u := u(m). According to Proposition 4.6, Now the assertions follow from the dimensions formulas at the end of the proof of Theorem 5.3. ✷ Remark. For λ = p − 1, Corollary 5.5 is an immediate consequence of Theorem 5.4 and Proposition 5.2 in conjunction with the six-term exact sequence of Hochschild [20, p. 575] or the Appendix of [19].
Let S be a simple W (m)-module. According to Corollary 3.4, H • (W (m), S) = 0 unless e −1 and u(m) act nilpotently on S. In the following we determine the simple modules of the Zassenhaus algebras on which e −1 and u(m) act nilpotently (cf. [24,Lemma 5.2]).
Lemma 5.6 Let F be an algebraically closed field of prime characteristic p and let S be a simple W (m)-module on which e −1 and u(m) act nilpotently. Then there is a non-zero element v ∈ S and there is an element λ ∈ F such that e 0 · v = λv and e i · v = 0 for every 1 ≤ i ≤ p m − 2.
In particular, for n = p m − 1 one has that is a non-zero proper W -submodule of S(p − 1), a contradiction since by definition S(p − 1) is simple. ✷ Remark. It follows from Lemma 5.13 below that dim F S(p − 1) = p m − 1 and therefore The generalized baby Verma module V (p − 1) has a one-dimensional maximal submodule (cf. [ Proof. Set W := W (m). As in the proof of Lemma 5.11, one can show that F(e p m −1 −1 Since F ⊆ M and α s = 0, it follows that e p m −2  Proof. Set W := W (m). It follows from the above computations, using the Cartan-Weyl identity, that the linear transformation Φ p−1 : V (p − 1) → S(p − 1) given by e n −1 ⊗ 1 p−1 → e n −1 · v, for every 0 ≤ n ≤ p m − 1, is a non-zero W -module epimorphism. By virtue of Lemma 5.11, i.e., F ⊆ Ker(Φ p−1 ). Hence Φ p−1 induces a non-zero W -module epimorphism Φ p−1 from V (p − 1)/F onto S(p − 1). According to Lemma 5.12, V (p − 1)/F is simple and thus Φ p−1 is a W -module isomorphism. ✷ As a consequence of Lemma 5.12 and Lemma 5.13, we obtain the short exact sequence of W (m)-modules. By virtue of Theorem 4.1(2), the second short exact sequence is the dual of the first short exact sequence. But the second short exact sequence can also be obtained directly by showing that the (p m − 1)-dimensional W (m)-submodule p m −1 n=1 F(e n −1 ⊗ 1 0 ) of V (0) is isomorphic to S(p − 1) (see the remark after Lemma 5.11). Now we apply Theorem 5.3 in order to determine the 1-cohomology of the Zassenhaus algebras with coefficients in simple modules.
Theorem 5.14 Let F be a field of prime characteristic p > 2, let m be a positive integer, and let λ ∈ F p . Then the following statements hold: Proof. Set W := W (m) and u := u(m). Then the assertions for λ = 0, p − 1 are immediate consequences of Lemma 5.8 and Theorem 5.3. According to [29,Theorem 4.2.4(1)], W is simple and therefore perfect which yields the assertions for λ = 0. Since H 1 (W, S(0)) = 0, we obtain the exact sequence (++) by applying the long exact cohomology sequence to the short exact sequence Furthermore, it follows from Proposition 5.2 that V (0) W = 0. Hence we obtain from (++) that and therefore we conclude from Lemma 5.9 and Theorem 5.3 that which finishes the proof of the theorem. ✷ Remark. If the ground field F is algebraically closed, then Theorem 5.14 completely describes the 1-cohomology of Zassenhaus algebras with coefficients in simple modules.
In view of the discussion before Lemma 2.2, every simple W (m)-module is also simple as a W(m)-module. In particular, S(λ) is a simple W(m)-module for every λ ∈ F p . The following result is an immediate consequence of Corollary 2.6(2) and Theorem 5.14: Theorem 5.15 Let F be a field of prime characteristic p > 2, let m be a positive integer, and let λ ∈ F p . Then the following statements hold: (2) If p > 3, then (W(m), S) = 0, then it follows from Corollary 3.2 that S is a restricted W(m)-module. One can show that S is isomorphic to S(λ) for some λ ∈ F p by employing arguments similar to those used above in the case of W (m) and by replacing the generalized baby Verma module V (λ) by the restricted baby Verma module u(W(m)) ⊗ u(b(m)) F λ . As a consequence, Theorem 5.15 completely describes the 1-cohomology of W(m) with coefficients in simple modules if the ground field F is algebraically closed. Corollary 5.16 Let F be a field of prime characteristic p > 2, let m be a positive integer, and let λ ∈ F p . Then the following statements hold: (1) If p = 3, then (2) If p > 3, then Proof. The vanishing of H 1 * (W(m), S(0)) follows from [18,Proposition 2.7], the perfectness of W (m), and the definition of the p-mapping of W(m) (see Section 1). The remaining statements can be obtained from Theorem 5.15 and Proposition 5.2 in conjunction with the six-term exact sequence of Hochschild [20, p. 575] or the Appendix of [19]. ✷   We leave it to the interested reader to apply Theorem 5.18 in conjunction with the six-term exact sequence of Hochschild [20, p. 575] to obtain upper bounds for the dimensions of the second restricted cohomology of W(m) with coefficients in the one-dimensional trivial module. It would be interesting to know the exact dimensions of H 2 * (W(m), F).