On the depth and reflexivity of tensor products
Olgur Celikbas, Uyen Le, Hiroki Matsui
TL;DR
This work studies the depth and reflexivity behavior of tensor products over local hypersurface rings, focusing on derived tensor products of complexes with finite complete intersection dimension. It proves a depth-inequality main theorem for $X\otimes^{\bf L}_R Y$ when $\mathrm{CI-dim}_R(X)<\infty$ and $X\otimes^{\bf L}_R Y$ is bounded and $(S_n)$, deriving a Serre-type conclusion for $Y$ and, in many cases, reflexivity of the factors. The results connect with Huneke–Wiegand and Onex, providing criteria under which $M\otimes_R N$ reflexive enforces reflexivity of both $M$ and $N$, with full-support and rank considerations. An appendix applies these ideas to symbolic powers of primes, showing that over non-domain hypersurfaces, a reflexive tensor product of prime-ideal powers forces minimal primes, offering new constraints on prime-ideal tensor products.
Abstract
In this paper we study the depth of tensor products of homologically finite complexes over commutative Noetherian local rings. As an application of our main result, we determine new conditions under which nonzero tensor products of finitely generated modules over hypersurface rings can be reflexive only if both of their factors are reflexive. A result of Asgharzadeh shows that nonzero symbolic powers of prime ideals in a local ring cannot have finite projective dimension, unless the ring in question is a domain. We make use of this fact in the appendix and consider the reflexivity of tensor products of prime ideals over hypersurface rings.