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

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 describe a matrix factorizations 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 not 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 prove 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 $S$. When $R$ is a Noetherian ring of prime characteristic that has FFRT, we prove that $R[x_1,...,x_n]$ and $R[\![x_1,...,x_n]\!]$ have FFRT. We prove also that over local ring of prime characteristic a module has FFRT if and only it has FFRT by a FFRT system. This enables us to show that if $M$ is a finitely generated module over Noetherian ring $R$ of prime characteristic $p$, then the set of all prime ideals $Q$ such that $M_Q$ has FFRT over $R_Q$ is an open set in the Zariski topology on \Spec(R)

FFRT properties of hypersurfaces and their F-signatures

This paper studies properties of certain hypersurfaces in prime characteristic: we give sufficient and necessary conditions for some classes of such hypersurfaces to have Finite F-representation 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