, where k is an infinite field. If k has characteristic zero, then Stanley proved that A has the Weak Lefschetz Property (WLP). Henceforth, k has positive characteristic p. If n = 3, then Brenner and Kaid have identified all d, as a function of p, for which A has the WLP. In the present paper, the analogous project is carried out for 4 ≤ n. If 4 ≤ n and p = 2, then A has the WLP if and only if d = 1. If n = 4 and p is odd, then we prove that A has the WLP if and only if d = kq + r for integers k, q, r with 1 ≤ k ≤ p−1 2 , r ∈ q−1 2 , q+1 2 , and q = p e for some non-negative integer e. If 5 ≤ n, then we prove that A has the WLP if and only if n(d−1)+3 2

Let k be a field. For each pair of positive integers (n, N), we resolve Q = R/(x N , y N , z N ) as a module over the ring R = k[x, y, z]/(x n + y n + z n ). Write N in the form N = an + r for integers a and r, with r between 0 and n − 1. If n does not divide N and the characteristic of k is fixed, then the value of a determines whether Q has finite or infinite projective dimension. If Q has infinite projective dimension, then value of r, together with the parity of a, determines the periodic part of the infinite resolution. When Q has infinite projective dimension we give an explicit presentation for the module of first syzygies of Q . This presentation is quite complicated. We also give an explicit presentation for the module of second syzygies for Q . This presentation is remarkably uncomplicated. We use linkage to find an explicit generating set for the grade three Gorenstein ideal (x N , y N , z N ) : (x n + y n + z n ) in the polynomial ring k[x, y, z].The question "Does Q have finite projective dimension?" is intimately connected to the question "Does k[X, Y , Z ]/(X a , Y a , Z a ) have the Weak Lefschetz Property?". The second question is connected to the enumeration of plane partitions. When the field k has positive characteristic, we investigate three questions about the Frobenius powers F t (Q ) of Q . When does there exist a pair (n, N) so that Q has infinite projective dimension and F (Q ) has finite projective dimension? Is the tail of the resolution of the Frobenius power F t (Q ) eventually a periodic function of t (up to shift)? In particular, we exhibit a situation where the tail of the resolution of F t (Q ), after shifting, is periodic as a function of t, with an arbitrarily large period. Can one use 257 socle degrees to predict that the tail of the resolution of F t (Q ) is a shift of the tail of the resolution of Q ?

We introduce classes of rings which are close to being Gorenstein. These rings arise naturally as specializations of rings of countable CM type. We study these rings in detail, and along the way generalize an old result of Teter which characterized Artinian rings which are Gorenstein rings modulo their socle.

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In this paper we introduce two new notions and show how they contribute to a better understanding of tight closure.  2002 Elsevier Science (USA)