A non-finitely generated algebra of Frobenius maps

Abstract

The purpose of this paper is to answer a question raised by Gennady Lyubeznik and Karen Smith. This question involves the finite generation of the following non-commutative algebra. Let $S$ be any commutative algebra of prime characteristic $p$. For any $S$-module $M$ and all $e\ge 0$ we let ${\mathcal{F}}^e(M)$ denote the set of all additive functions $\varphi :M\to M$ with the property that $\varphi (sm)=s^{p^e}\varphi (m)$ for all $s\in S$ and $m\in M$. For all $e_1,e_2\ge 0$, and ${\varphi }_1\in {\mathcal{F}}^{e_1}(M)$, ${\varphi }_2\in {\mathcal{F}}^{e_2}(M)$ the composition ${\varphi }_2\circ {\varphi }_1$ is in ${\mathcal{F}}^{e_1+e_2}(M)$. Also, each ${\mathcal{F}}^e(M)$ is a module over $\mathcal{F}^{0}(M)=\Hom_{S}(M,M)$ via ${\varphi }_0\varphi ={\varphi }_0\circ \varphi$. We now define $\mathcal{F}(M)={\oplus }_{e\ge 0}{\mathcal{F}}^e(M)$ and endow it with the structure of a $\Hom_{S}(M,M)$-algebra with multiplication given by composition. We construct an example of an Artinian module over a complete local ring $S$ for which $\mathcal{F}(M)$ is not a finitely generated $\Hom_{S}(M,M)$-algebra, thus giving a negative answer to the question raised by Lyubeznik and Smith.

Misprints corrected. To appear in the Proceedings of the AMS

References in corpus (1)

• Parameter test ideals of Cohen Macaulay rings