Complexity and curvature of (pairs of) Cohen-Macaulay modules, and their applications
Souvik Dey, Dipankar Ghosh, Aniruddha Saha
TL;DR
The paper generalizes Avramov’s complexity and curvature to pairs of modules by introducing Ext-curvature and Tor-curvature for pairs and establishing dualities with classical invariants. It provides sharp lower bounds for the growth of Ext and Tor for pairs, and derives upper bounds for the curvature of the residue field in terms of a Cohen–Macaulay module M via annihilation hypotheses on $ ext{Tor}$ and $ ext{Ext}$. These bounds are then leveraged to characterize complete intersections, hypersurfaces, and regular local rings through the complexity and curvature of CM pairs, including applications to modules of Kähler differentials and Berger’s conjecture. The work also develops a detailed theory around CM modules of minimal multiplicity, Ulrich modules, and their impact on CI-type criteria, with several sharp examples illustrating the necessity of the conditions. Overall, the results provide a cohesive framework linking homological growth invariants of module pairs to fundamental ring-theoretic properties and differential-module behavior, advancing partial progress on several open questions in the area.
Abstract
The complexity and curvature of a module, introduced by Avramov, measure the growth of Betti and Bass numbers of a module, and distinguish the modules of infinite homological dimension. The notion of complexity was extended by Avramov-Buchweitz to pairs of modules that measure the growth of Ext modules. The related notion of Tor complexity was first studied by Dao. Inspired by these notions, we define Ext and Tor curvature of pairs of modules. The aim of this article is to study (Ext and Tor) complexity and curvature of pairs of certain CM (Cohen-Macaulay) modules, and establish lower bounds of complexity and curvature of pairs of modules in terms of that of a single module. It is known that among all modules, the residue field has maximal complexity and curvature, moreover they characterize complete intersection local rings. As applications of our results, we provide some upper bounds of the curvature of the residue field in terms of curvature and multiplicity of any nonzero CM module. As a final upshot, these allow us to characterize complete intersection local rings (including hypersurfaces and regular rings) in terms of complexity and curvature of pairs of certain CM modules. In particular, under some additional hypotheses, we characterize complete intersection and regular local rings via injective curvature of the ring and that of the module of Kähler differentials respectively. Thus, we make partial progress towards a question of Christensen-Striuli-Veliche, as well as another by Vasconcelos.