The depth formula for modules with reducible complexity (Q351751)

Let \(A\) be a commutative noetherian local ring and \(M\) and \(N\) two nonzero finitely generated \(A\)-modules such that \(\mathrm{Tor}_i^A(M,N)=0\) for all \(i \geq 1\). In [\textit{C. Huneke} and \textit{R. Wiegand}, Math. Ann. 299, No. 3, 449--476 (1994; Zbl 0803.13008)] it is proved that NEWLINE\[NEWLINE\mathrm{depth}(M)+\mathrm{depth}(N) = \mathrm{depth}(A)+\mathrm{depth}(M\otimes_AN)NEWLINE\]NEWLINE when \(A\) is complete intersection. The same result was proved previously for an arbitrary local ring in [\textit{M. Auslander}, Ill. J. Math. 5, 631--647 (1961; Zbl 0104.26202)] when \(M\) (or \(N\)) is of finite projective dimension. These two results are generalized at once in [\textit{T. Araya} and \textit{Y. Yoshino}, Commun. Algebra 26, No. 11, 3793-3806 (1998; Zbl 0906.13002)]: if \(A\) is a noetherian local ring and \(M\) is of finite complete intersection dimension, then the above equality holds. The class of modules with reducible complexity was introduced in [\textit{P. A. Bergh}, Commun. Algebra 37, No. 6, 1908--1920 (2009; Zbl 1173.13012)], where it is proved that contains strictly the class of modules of finite complete intersection dimension. In the paper under review, the above result of Araya and Yoshino is extended to the case when \(M\) is of reducible complexity and an extra hypothesis holds (like \(A\) Gorenstein, or \(A\) Cohen-Macaulay and \(N\) also has reducible complexity, etc.). In the last section of the paper, some examples are giving showing that there exist modules of any complexity \(\geq 1\), of infinite complete intersection dimension but reducible complexity.