Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
157 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Generalized depth and associated primes in the perfect closure $R^\infty$ (1810.06028v3)

Published 14 Oct 2018 in math.AC

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 $pe$-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 $Fe(M)/N_e$ as $e \in \mathbb{N}$ varies. Here $F$ refers to the Frobenius functor, and in the paper we define an $F$-sequence of submodules $\lbrace N_e \rbrace \subseteq \lbrace Fe(M) \rbrace$ 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$R(Fe(M))$ 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.

Summary

We haven't generated a summary for this paper yet.