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 $N=a n+r$ 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 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 socle degrees to predict that the tail of the resolution of $F^t(Q)$ is a shift of the tail of the resolution of $Q$?