On the Finite F-representation type and F-signature of hypersurfaces

TL;DR

This paper investigates the structure of hypersurfaces over fields of positive characteristic, providing matrix factorizations for Frobenius pushforwards, characterizations for finite F-representation type, and explicit computations of F-signatures.

Contribution

It introduces a matrix factorization approach to analyze Frobenius pushforwards of hypersurfaces, enabling characterization of finite F-representation type and explicit F-signature calculations.

Findings

Existence of matrix factorizations for Frobenius pushforwards of hypersurfaces.

Characterization of when certain hypersurface rings have finite F-representation type.

Explicit computation of F-signatures for specific hypersurface rings.

Abstract

Let $S=K[x_1,...,x_n]$ or $S=K[[x_1,...,x_n]]$ be either a polynomial or a formal power series ring in a finite number of variables over a field $K$ of characteristic $p > 0$ with $[K:K^p] < \infty$. Let $R$ be the hypersurface $S/fS$ where $f$ is a nonzero nonunit element of $S$. If $e$ is a positive integer, $F_*^e(R)$ denotes the $R$-algebra structure induced on $R$ via the $e$-times iterated Frobenius map ( $r\rightarrow r^{p^e}$ ). We show an existence of a matrix factorization of $f$ whose cokernel is isomorphic to $F_*^e(R)$ as $R$-module. The presentation of $F_*^e(R)$ as the cokernel of a matrix factorization of $f$ enables us to find a characterization from which we can decide when the ring $S[[u,v]]/(f+uv)$ has Finite F-representation type (FFRT) where $S=K[[x_1,...,x_n]]$. This allows us to create a class of rings that have Finite F-representation type but it does not have finite CM type. For $S=K[[x_1,...,x_n]]$, we use this presentation to show that the ring $S[[y]]/(y^{p^d} +f)$ has finite F-representation type for any $f$ in $S$. Furthermore, we proved that $S/I$ has Finite F-representation type when $I$ is a monomial ideal in either $S=K[x_1,...,x_n]$ or $S=K[[x_1,...,x_n]]$. Finally, this presentation enables us to compute the F-signature of the rings $S[[u,v]]/(f+uv)$ and $S[[z]]/(f+z^2)$ where $S=K[[x_1,...,x_n]]$ and $f$ is a monomial in the ring $S$.