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A non-finitely generated algebra of Frobenius maps

Katzman, M. (2009) A non-finitely generated algebra of Frobenius maps. arXiv:0906.1083v1 [math.AC]. (Unpublished)

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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 $e\geq 0$ we let $\mathcal{F}^e(M)$ denote the set of all additive functions $\phi: M \to M$ with the property that $\phi(s m)=s^{p^e} \phi(m)$ for all $s\in S$ and $m\in M$. For all $e_1, e_2 \geq 0$, and $\phi_1\in \mathcal{F}^{e_1}(M)$, $\phi_2\in \mathcal{F}^{e_2}(M)$ the composition $\phi_2 \circ \phi_1$ 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 $\phi_0 \phi=\phi_0 \circ \phi$. We now define $\mathcal{F}(M)=\oplus_{e\geq 0} \mathcal{F}^e(M)$ 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.

Item Type: Article
Copyright, Publisher and Additional Information: © 2009 Katzman. Paper Imported from arXiv.
Institution: The University of Sheffield
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 19 Nov 2009 16:29
Last Modified: 08 Feb 2013 16:59
Status: Unpublished
Related URLs:
URI: http://eprints.whiterose.ac.uk/id/eprint/10169

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