The direct sum decomposability of \(^eM\) in dimension 2
Let \((R,m)\) be a local noetherian ring of prime characteristic \(p\) and maximal ideal \(m\), and let \(M\) be a finitely generated \(R\)-module. For \(e \in \mathbb{N}\), the map \(F^e:R \to R\) defined by \(F^e(r)=r^{p^e}\) is a ring homomorphism. We denote by \(^e M\) the \(R\)-module defined by the abelian group \(M\) with the scalar multiplication \( r \cdot m = r^{p^e}m\) (\(r \in R\), \(m \in M\)). The ring \(R\) is called \(F\)-finite if \(^1 R\) is a finitely generated \(R\)-module or, equivalently, if \(^e R\) is a finitely generated \(R\)-module for all \(e \in \mathbb{N}\). If \(R\) is \(F\)-finite and \(M\) is a finitely generated \(R\)-module, then \(^e M\) is a finitely generated \(R\)-module for all \(e \in \mathbb{N}\). This paper studies whether the \(R\)-modules \(^e M\) are indecomposable, i.e., they cannot be split as a direct sum of two non-trivial submodules. In several cases, \textit{M. Hochster} [Conf. Commut. Algebra, Kingston 1975, 106--195 (1975; Zbl 0342.13009)] showed that \(^e M\) splits for \(e \gg 0\). In particular, he proved the following result: Assume that the local ring \((R,m)\) is \(F\)-finite and \(\dim (M)=1\). Let \(P\) be an associated prime of \(M\) with \(\dim (R/P)=1\) and let \(A\) be the integral closure of the domain \(R/P\) in its fraction field. Then, for any \(n \in \mathbb{N}\), there exists \(e_0 \in \mathbb{N}\) such that \(^e M\) has a direct summand isomorphic to \(A^n\) for all \(e \geq e_0\). The main result of the paper generalizes this result to the case when \(\dim (M)=2\). More precisely, the author proves the following: Let \((R,m)\) be an \(F\)-finite local ring and let \(M\) be a finitely generated \(R\)-module with \(\dim(M)=2\). Let \(P\) be an associated prime of \(M\) with \(\dim (R/P)=2\) and let \(A\) be the integral closure of the domain \(R/P\) in its fraction field. If \(A\) is strongly \(F\)-regular, then, for any \(n \in \mathbb{N}\), there exists \(e_0 \in \mathbb{N}\) such that \(^e M\) has a direct summand isomorphic to \(A^n\) for all \(e \geq e_0\). We note here that in the case proved by Hochster the integral closure \(A\) of the one-dimensional domain \(R/P\) is regular and hence strongly \(F\)-regular.
scientific article; zbMATH DE number 5604558
scientific article