FFRT properties of hypersurfaces and their F-signatures

Abstract

This paper studies properties of certain hypersurfaces in prime characteristic: we give a sufficient and necessary conditions for some classes of such hypersurfaces to have Finite Frepresentation Type (FFRT) and we compute the F -signatures of these hypersurfaces. The main method used in this paper is based on finding explicit matrix factorizations. Recall that a non-zero finitely generated module M over a local ring R is Cohen-Macaulay if depth R M = dim M and R is a Cohen-Macaulay ring if R itself is a Cohen-Macaulay module. However, if depth R M = dim R, M is called maximal Cohen-Macaulay module (or MCM module). When M is a finitely generated module over non-local ring R, M is Cohen-Macaualy if M m is a Cohen-Macaulay module for all maximal ideals m ∈ Supp M.Later in the paper we will look at the modules F e * (R) when R is a Cohen-Macaulay ring. Note that a regular sequence on an R-module M is also a regular sequence on F e * M, and in particular, if R is Cohen-Macaulay, F e * (R) are MCM modules for all e ≥ 0. Definition 2.1. A finitely generated R-module M is said to have finite F-representation type (henceforth abbreviated FFRT) by finitely generated R-modules M 1 , . . . , M s if for every positive integer e, the R-module F e * (M) can be written as a finite direct sum in which each direct summand is isomorphic to some module in {M 1 , . . . , M s } , that is, there exist non-negative integers t (e,1) , . . . , t (e,s) such that F e * (M) = s j=1 M ⊕t (e,j)