Complexity and curvature of pairs of Burch modules and ideals
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.