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/(xN,yN,zN) as a module over the ring R=k[x,y,z]/(xn+yn+zn). 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 (xN,yN,zN):(xn+yn+zn) 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]/(Xa,Ya,Za) 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 Ft(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 Ft(Q) eventually a periodic function of t (up to shift)? In particular, we exhibit a situation where the tail of the resolution of Ft(Q), after shifting, is periodic as a function of t, with an arbitrarily large period. Can one use socle degrees to predict that the tail of the resolution of Ft(Q) is a shift of the tail of the resolution of Q?