Varieties of Modules for Z/2z × Z/2z

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety C, that is the set of pairs of (n× n)-matrices (A,B) such that A2 = B2 = [A,B] = 0, is equidimensional. C can be identified with the ‘variety of n-dimensional modules’ for Z/2Z×Z/2Z, or equivalently, for k[X,Y ]/(X2, Y 2). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic > 2. We also prove that if e2 = 0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of C as a direct sum of indecomposable components of varieties of Z/2Z × Z/2Z-modules. 0 Introduction Let G = GL(n, k), where k is an algebraically closed field of characteristic p > 0, and let g be the Lie algebra of G. We denote the p-th power of matrices on g by x 7→ x, and its iteration m times by x 7→ x m]. (This is the standard notation in the theory of restricted Lie algebras.) Clearly x is nilpotent if and only if x N ] = 0 for N ≫ 0. Denote by N the set of nilpotent elements of g and by N1 the subset of elements satisfying x = 0 (the restricted nullcone). It was proved in [10] that N1 is irreducible. (An explicit description was given in [1].) In [15], Premet proved that the nilpotent commuting variety C(g) := {(x, y) ∈ g × g | x, y ∈ N , [x, y] = 0} is irreducible and of dimension (n − 1). More specifically, C(g) = G · (e, u) where e is a regular nilpotent element of g and u = ke⊕ke ⊕ . . .⊕ke. (Here and in what follows we use the dot to denote the action of G by conjugation on g or the induced diagonal action on g × g, and the notation V for the Zariski closure of a subset V of an arbitrary affine vector space, where the context is clear.) In fact, Premet proved 1 that the nilpotent commuting variety of Lie(G) is equidimensional for any reductive group G over an algebraically closed field of good characteristic. The nilpotent commuting variety, or more accurately the restricted nilpotent commuting variety C 1 (g) = {(x, y) ∈ N1 × N1 : [x, y] = 0} is related to the cohomology of G by work of Suslin, Friedlander & Bendel. It was proved in [17] that C 1 (g) is homeomorphic to the spectrum of the cohomology ring ⊕i≥0H(G2, k), where G2 is the second Frobenius kernel of G. More generally, the restricted nullcone N1 plays an important role in the representation theory of g due to the theory of support varieties of (reduced enveloping algebras of) restricted Lie algebras (studied for the restricted enveloping algebra in [4, 5, 9] and for general reduced enveloping algebras in [6]; see also [13, 14]). Another perspective is that of varieties of modules (see for example [2]): C 1 (g) can be identified with the variety of n-dimensional modules for the truncated polynomial ring k[X, Y ]/(X, Y ). There is an isomorphism k[X, Y ]/(X, Y ) → kΓ, where Γ = Z/pZ × Z/pZ. Specifically, if σ (resp. τ) is a generator for the first (resp. second) copy of Z/pZ in Γ, then X + 1 7→ σ and Y + 1 7→ τ . This paper began as a preliminary investigation of the restricted nilpotent commuting variety in the simplest possible case: hence we assume p = 2. In Section 1 we show that projection onto the first coordinate maps any irreducible component of C 1 (g) onto N1 (that is, the components are ‘determined’ by the dense orbit in N1). Equidimensionality then follows by equidimensionality of zg(e) ∩ N1, which we prove for any e ∈ N1. Proposition. Let k be of characteristic 2 and let C 1 be the restricted nilpotent commuting variety. (a) If n = 2m, then C 1 has ([m/2] + 1) irreducible components, each of dimension 3m . (b) If n = 2m + 1, then C 1 has (m + 1) irreducible components, each of dimension 3m(m+ 1). This result for C 1 might be expected to indicate that the restricted nilpotent commuting variety is equidimensional for general p. However, we show that this is not the case (Remark 2.4). On the other hand, we observe that zg(e) ∩ N1 is equidimensional for many choices of G and e. We conjecture that this is true for reductive G in good characteristic. We remark that the intersection zg(e) ∩ N1 can be identified with the support variety Vzg(e)(k), where k is the trivial zg(e)-module. In the final section we express each irreducible component of C 1 (g) as a direct sum of indecomposable components of modules. Our method for obtaining the above results is a rather crude direct approach. Such a strategy will clearly be inappropriate in general. Notation. We denote by Matr×s the vector space of all r×s matrices over k. If x ∈ g then the centralizer of x in g (resp. G) will be denoted zg(e) (resp. ZG(e)). We will sometimes abuse notation and use N1 to refer to the set of p-nilpotent elements in an arbitrary Lie algebra. This will cause no confusion. We denote by eij the matrix with 1 in the (i, j)-th position, and zeros everywhere else. (The dimension of eij will always be specified or clear from the context.) Our convention is that all modules are left modules. We denote by [m/r] the integer part of the fraction m/r. 2 1 Centralizers Let G = GL(n, k), let g = Lie(G) and let e0, e1, . . . , em be a set of representatives for the orbits in N1 = N1(g). Clearly C 1 = ⋃m i=0 G · (ei, zg(ei)∩N1). In general the set zg(ei)∩N1 is not irreducible. For each i let V (1) i , V (2) i , . . . , V (ri) i be the irreducible components of zg(ei)∩N1. The following Lemma is adapted from [15, Prop. 2.1]. The argument works for arbitrary G and p. (The only requirement is that the number of orbits in N1 is finite. This is well-known if p is good (see [16]) but is true even if p is bad [7].) Lemma 1.1. Let X be an irreducible component of C 1 . Then there is some i, 0 ≤ i ≤ m, and some j, 1 ≤ j ≤ ri, such that X = G · (ei, V (j) i ). Moreover, V (j) i ⊆ G · ei. Proof. Since there are finitely many of the sets G · (ei, V (j) i ) and they cover C nil 1 , the first statement is obvious. For the second statement, define an action of GL(2) on g × g by the morphism GL(2) × (g × g) → g × g, ( ( a b c d ) , (x, y)) 7→ (ax + by, cx + dy). Clearly any element of GL(2) preserves C 1 . Hence GL(2) preserves each irreducible component of C nil 1 . In particular, τ(X) = X, where τ : (x, y) 7→ (y, x). Suppose therefore that X = G · (ei, V (j) i ) is an irreducible component of C 1 . Let π : g × g → g denote the first projection. Then π(X) = G · ei. But X = τ(X), hence V (j) i ⊆ π(X). Suppose from now on that p = 2. For each i, 0 ≤ i ≤ m = [n/2], let ei =   0 0 Ii 0 0 0 0 0 0   ∈ g, where Ii is the i× i identity matrix. Here the top left, top right, bottom left and bottom right submatrices are i× i, the top middle and bottom middle submatrices are i× (n− 2i), the centre left and centre right submatrices are (n − 2i) × i, and the central submatrix is (n− 2i)× (n− 2i). Then {e0, e1, . . . , em} is a set of representatives for the conjugacy classes in N1. It is easy to see, with the standard description of nilpotent orbits via partitions of n, that ei corresponds to the partition 2 .1. Moreover, we have the following inclusions: {0} = G · e0 ⊂ G · e1 ⊂ . . . ⊂ G · eM = N1. The condition V (j) i ⊆ G · ei is clearly equivalent to the inequality: rk(y) ≤ i for all y ∈ V (j) i . Fix i until further notice and let x be an element of the centralizer zg(ei), which must have the form