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.
Categories: math.AC