Remarks on modules of finite projective dimension

TL;DR

This work examines finitely generated modules of finite projective dimension over Noetherian local rings, focusing on how tensor products $M\otimes_R M$ and $M\otimes_R M^*$ encode freeness and reflexivity, thereby extending Auslander-type rigidity beyond regular rings. It develops sharp bounds for the dimensions of Ext-modules $\operatorname{Ext}^i_R(M,R)$ and connects these to the grade conjecture, while also addressing Gorenstein dimension, normality via integral closure, and the behavior of Ext with respect to primes and equidimensional modules. The paper also analyzes the projective dimension of tensor products $M\otimes_R N$, providing both nonexistence results over hypersurfaces and constructive counterexamples in bimodule/Frobenius-twist contexts, and it investigates how chain conditions in the prime spectrum influence $\operatorname{pd}$, including pathological cases in non-catenary rings. Overall, the results offer a comprehensive homological framework to detect freeness, reflexivity, normality, and spectral pathologies from tensorial and Ext-theoretic data, contributing to rigidity theory and homological conjectures in commutative algebra.

Abstract

We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the torsion-freeness or reflexivity of tensor products of the form \( M \otimes_R M \) and \( M \otimes_R M^* \). Under mild homological assumptions, we prove that such properties of these tensor products impose strong structural constraints on \( M \), often forcing it to be free. These results generalize classical theorems of Auslander beyond the regular case and contribute to the broader understanding of rigidity phenomena in commutative algebra. The second part of the paper is devoted to the dimension and support of Ext-modules, particularly \( {Ext}^i_R(M, R) \) for critical values of i, when \( M \) has finite projective dimension. We establish sharp bounds on their Krull dimensions, analyze their behavior for prime and equidimensional modules, and relate these findings to the grade conjecture and other homological conjectures. Applications include new cases of a question of Jorgensen, which asks whether \( {pd}(M) < i \) whenever \({Ext}^i_R(M, M) = 0 \) and M has finite projective dimension over a complete intersection ring. Finally, we examine the projective dimension of prime ideals in rings that fail chain conditions. We show that in non-catenary or pathological saturated chain settings, such prime ideals often have infinite projective dimension.

Theorems & Definitions (120)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.5
• Theorem 1.6
• Conjecture 2.1: Vasconcelos 2010
• Proposition 2.3
• proof
• Corollary 2.4
• proof