On direct summands of modules of finite phantom projective dimension (Q2275730)

Let \(R\) be a Noetherian ring of positive prime characteristic, let \(M\) be a finitely generated \(R\)-module, and denote by \(F^e\) the \(e\)th iteration of the Frobenius functor (\(e\geq 0\)); then \(M\) is said to have finite phantom projective dimension (\(\text{ppd}_R(M)<\infty\)), if there is a bounded left complex \(G_\bullet\) of finitely generated projective \(R\)-modules with \(H_0(G_\bullet)=M\) such that, for all \(e\geq 0\) and \(i\geq 1\), the \(i\)th cycles of the complex \(F^e(G_\bullet)\) are contained in the tight closure of the \(i\)th boundaries (i.e. \(F^e(G_\bullet)\)) has phantom homology for \(i\geq 1\). This notion was introduced by \textit{M. Hochster} and \textit{C. Huneke} [J. Am. Math. Soc. 3, No. 1, 31--116 (1990; Zbl 0701.13002)]. The class of modules of finite phantom projective dimension over a fixed ring \(R\) includes those of finite projective dimension but is usually larger (the two classes coincide for Cohen-Macaulay rings). Nevertheless, the notions of phantom projective dimension do not enjoy all the good properties of the usual ones. One of the most important reasons for this is that the behavior of such modules in short exact sequences is more complicated (e.g., it can happen that for \(R\)-modules \(N\subseteq M\) we have \(\text{ppd}_R(M)<\infty\), \(\text{ppd}_R(M/N)<\infty\) but \(N\) has no finite phantom resolution). The main result of the paper under review is the construction of an example of a local ring \(R\) of finite characteristic and cyclic \(R\)-modules \(M\) and \(N\) such that \(M\oplus N\) has finite phantom projective dimension, however neither \(M\) nor \(N\) has a finite phantom projective resolution.