The resolution of the bracket powers of the maximal ideal in a diagonal hypersurface ring

Abstract

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 ?

The Weak Lefschetz Property for monomial complete intersection in positive characteristic

“…In particular, item (2a) is one of the main concerns in [12]. Early in the investigation that led to [12], we found a relationship between (2a) and (1b). Eventually, we found the equivalence of (1a) and (1b) in [4].…”

“…The paper [12] is about the resolution of R by free R-modules. In particular, item (2a) is one of the main concerns in [12].…”

“…(3)] knew some version of the equivalence of (1a) and (1b) and that this idea is also used in [10].) In [12] we used the numerical values given in [4] to prove (3). Lucho Avramov drew our attention to the equivalence of (2a), (2b), and (2c) in a recent conversation.…”

“…In [12] we used the numerical values given in [4] to prove (3). When we wrote [12] we were surprised by conclusion (3); that is, we were surprised that the homological questions considered in [12] were related to the WLP. When we wrote [12] we were surprised by conclusion (3); that is, we were surprised that the homological questions considered in [12] were related to the WLP.…”

Some Algebras with the Weak Lefschetz Property

“…Recently, there have been many investigations of the presence of the weak Lefschetz property (see, e.g., [1,2,4,9,13,14,15,16,17,18]). A standard graded Artinian algebra A over a field K is said to have the weak Lefschetz property if there is a linear form ℓ ∈ A such that the multiplication map ×ℓ : [A] i → [A] i+1 has maximal rank for all i (i.e., it is injective or surjective).…”