Generalized depth and associated primes in the perfect closure $R^\infty$

Abstract

For a reduced Noetherian ring $R$ of characteristic $p > 0$, in this paper we discuss an extension of $R$ called its perfect closure $R^\infty$. This extension contains all $p^e$-th roots of elements of $R$, and is usually non-Noetherian. We first define the generalized notions of associated primes of a module over a non-Noetherian ring. Then for any $R$-module $M$, we state a correspondence between certain generalized prime ideals of $(R^\infty \otimes_R M)/N$ over $R^\infty$, and the union of associated prime ideals of $F^e(M)/N_e$ as $e \in \mathbb{N}$ varies. Here $F$ refers to the Frobenius functor, and in the paper we define an $F$-sequence of submodules $\lbrace N_e \rbrace \subseteq \lbrace F^e(M) \rbrace$ as $e$ varies, while $\underrightarrow{\lim} \ N_e = N$. Under the further assumptions that $M$ is finitely generated and $(R,\mathfrak{m})$ is an $F$-pure local ring, we then show that depth$_R(F^e(M))$ is constant for $e \gg 0$, and we call this value the stabilizing depth, or s depth$_R(M)$. Lastly, we turn to non-Noetherian measures of the depth of $R^\infty \otimes_R M$ over $R^\infty$, which generalize as well. Two of these values are the k depth and the c depth, and we show k depth$_{R^\infty} (R^\infty \otimes_R M) =$ s depth$_R (M) \geq$ c depth$_{R^\infty} (R^\infty \otimes_R M)$, while all three values are equal under certain assumptions.