A formalism of $F$-modules for rings with complete local finite $F$-representation type

Abstract

We develop a formalism of unit F -modules in the style of Lyubeznik and Emerton-Kisin for rings which have finite F -representation type after localization and completion at every prime ideal. As applications, we show that if R is such a ring then the iterated local cohomology modulesIs (R) have finitely many associated primes, and that all local cohomology modules H n I (R/gR) have closed support when g is a nonzerodivisor on R. Background The Frobenius endomorphismLet p be a prime number and R be a commutative ring of characteristic p > 0. Then the function F : R → R given by F (x) = x p for all x ∈ R is additive, i.e. satisfies F (x + y) = F (x) + F (y) for all x, y ∈ R. Since it is clearly multiplicative and maps 1 to 1, F is a ring homomorphism. This ring homomorphism F is called the Frobenius endomorphism of R. Given an integer e ≥ 0 we denote by F e the e-th iterate of Frobenius, which is given by F e (x) = x p e for all x ∈ R; in particular, F 0 is the identity map on R. Since F e : R → R is a ring homomorphism, its image is a subring of R, which we denote by R p e := {x p e | x ∈ R}.Associated to any ring homomorphism one always has a restriction of scalars functor, and the same is true for F . However, the fact that F has the same target and source can make this very confusing, and special notation is introduced to deal with this situation. Here we will follow the notation from algebraic geometry, but other notations are also common in the literature; most notably some authors write M 1/p e for what we denote F e * M . Given an R-module M and an integer e ≥ 0 we denote by F e * M the R-module obtained via restriction of scalars via F e : R → R. In particular, F e * M is equal to M as an abelian group. Given an element u ∈ M we will usually write F e * u when we want to emphasize that we view u as an element of F e * M and not as an element of M . With this notation the R-module structure of F e * M is given byfor all x ∈ R and u ∈ M . Remark 2.1. Here we are making a slight departure from standard terminology in that, for us, F e * M = M as sets, whereas most authors would define F e * M as the set {F e * u | u ∈ M }. Our definition has the pleasant consequence that, given integers i, j ≥ 0 the set Hom R (F i * M, F j * M ) is actually the subset of Hom Z (M, M ) given by Homwhereas in the other terminology Hom R (F i * M, F j * M ) is only naturally identified with this set. This means, for example, that given α ∈ Hom R (F i * M, F j * M ) and β ∈ Hom R (F a * M, F b * M ) the composition β • α is well defined as an element of End Z (M, M ); one can then think about what the linearity of β • α is if necessary. Definition 2.2. Let R be a noetherian ring of characteristic p > 0. We say that R is F -finite whenever the Frobenius endomorphism F : R → R is module-finite, i.e. whenever F * R is a finitely generated R-module.Recall that a composition of module-finite ring homomorphisms is again module-finite. In particular, if R is F -finite then F e * R is a finitely generated R-module for every e ≥ 0. Every...