Complexity and curvature of pairs of Burch modules and ideals

Summary · qwen2.5:32b

The paper explores the complexity and curvature of pairs involving Burch modules and ideals, unifying and extending previous findings on their extremal properties and Ext/Tor vanishing characteristics. A key new element in the proofs involves demonstrating that the Burch property remains independent under embedding, particularly for Burch modules of depth zero.

Suggested post angle

Exploring the complexities of Burch modules and ideals in mathematics

Excerpt

arXiv:2511.13258v3 Announce Type: replace Abstract: The complexity and curvature of a module were first introduced by Avramov to distinguish modules of infinite homological dimension. Later, Avramov-Buchweitz extended the notion of complexity from a single module to pairs of modules, measuring the polynomial growth rate of the minimal number of generators of their Ext-modules. By taking one of the modules in the pair to be the residue field, one recovers the standard projective and injective complexity of modules, whereas the vanishing of the complexity of a pair is equivalent to the eventual vanishing of Ext-modules, giving rise to the study of what are popularly known as Ext-pd and Ext-id test modules. Dao studied a similar notion of Tor-complexity. In the same vein, the vanishing of the Tor-complexity of pairs gives rise to Tor-pd test modules. On the other hand, the concept of Burch ideals was introduced by Dao-Kobayashi-Takahashi, motivated by the classical work of Burch, and subsequently extended to modules by Dey-Kobayashi. It follows from a result of Avramov that Burch modules exhibit extremal complexity and curvature. Moreover, Dey-Kobayashi and Ghosh-Saha showed, respectively, that Burch modules are Tor-pd and Ext-pd test, and that Burch ideals are Ext-id test. In this paper, we unify and significantly extend these two themes of extremal complexity and curvature, and Ext/Tor vanishing results of Burch modules. A key new ingredient in our proofs, particularly in dealing with Burch modules of depth zero, is the independence of the Burch property under embedding.