On the projective dimension of tensor products of modules
Olgur Celikbas, Souvik Dey, Toshinori Kobayashi
TL;DR
The paper investigates when finite projective dimension of a tensor product $M\otimes_R N$ forces finiteness of the projective dimensions of $M$ and $N$. It provides counterexamples showing that $\mathrm{pd}_R(M\otimes_R N)<\infty$ does not in general imply $\mathrm{pd}_R(M)<\infty$ or $\mathrm{pd}_R(N)<\infty$, nor finite G-dimension or controlled Betti growth for the factors. It contrasts these with nontrivial positive results: if $M$ is totally reflexive and $\mathrm{pd}_R(M\otimes_R N)<\infty$, then $M$ is projective and $\mathrm{pd}_R(N)<\infty$, and a Cohen–Macaulay, syzygy-based rigidity result shows that, under certain conditions, $M=0$ or $N=0$ when $M=\Omega_R L$ with $L$ maximal CM. The work blends explicit counterexamples (including non-Cohen–Macaulay, Golod, and other pathological rings) with structural results via Tor/Ext vanishing, syzygies, transposes, semidualizing modules, and Sharp-type theorems, yielding both negative answers to the questions and meaningful positive criteria with applications to Ulrich and CM contexts.
Abstract
In this paper, we consider finitely generated modules over commutative Noetherian rings whose tensor products have finite projective dimension. We construct examples of modules of infinite projective dimension (and also of infinite Gorenstein dimension) whose tensor products nonetheless have finite projective dimension. Furthermore, we establish nontrivial conditions under which such examples cannot arise. For example, we prove that if the tensor product of two nonzero modules--at least one of which is totally reflexive--has finite projective dimension, then both modules in question must have finite projective dimension.