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

TL;DR

This paper investigates the projective dimension and resolutions of modules over diagonal hypersurface rings, revealing explicit presentations, linkage-based generators, and connections to algebraic properties like the Weak Lefschetz Property and Frobenius powers.

Contribution

It provides explicit resolutions for modules over diagonal hypersurface rings, explores conditions for finite/infinite projective dimension, and links these to algebraic properties and Frobenius power behaviors.

Findings

Explicit presentations for first and second syzygy modules.

Criteria for finite vs. infinite projective dimension based on parameters.

Periodic behavior of Frobenius power resolutions in positive characteristic.

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 $Q$?