Complexity and curvature of pairs of Burch modules and ideals
Souvik Dey, Dipankar Ghosh, Mouma Samanta
TL;DR
This work investigates the Ext- and Tor- complexity and curvature for pairs of Burch modules/ideals over a Noetherian local ring, revealing deep links to complete intersection structures. The authors prove that, under suitable Tor/Ext vanishing and depth hypotheses, the (Tor/Ext)cx and curvature of a module N are controlled by the pair invariants tcx(M,N) and tcurv(M,N); in CI rings these inequalities become equalities. A key contribution is showing that, for Burch ideals I,J with I m-primary, tcx(I,J) = cx(I,J) = cx(k) and that finiteness of these invariants (or their ≤1 curvature counterparts) characterizes complete intersections, including hypersurfaces, under broad conditions. The paper also provides injective-analogues, CM-context refinements, and concrete examples to illustrate and sharpen these characterizations.
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 that of pairs of modules, which measures the polynomial growth rate of the minimal number of generators of their Ext modules. Dao studied a similar notion of Tor-complexity. Recently, Dey-Ghosh-Saha initiated the study of Ext and Tor curvature of a pair of modules, which measure the exponential growth rates of the corresponding Ext and Tor, respectively. On the other hand, the concept of Burch ideals was introduced by Dao-Kobayashi-Takahashi, motivated by classical work of Burch. This class includes several large and well-studied families of ideals in a Noetherian local ring $(R,\mathfrak{m},k)$. For example, every nonzero ideal of the form $\mathfrak{a}\mathfrak{m}$ (e.g., $\mathfrak{m}^n$ for $n\ge 1$), and under mild conditions every integrally closed ideal $I$ with $\mathrm{depth}(R/I)=0$, is a Burch ideal. Suppose $I$ and $J$ are Burch ideals such that $I$ is $\mathfrak{m}$-primary. One of the main results of this article is $\mathrm{cx}_R(I,J)=\mathrm{tcx}_R(I,J)=\mathrm{cx}_R(k)$. Moreover, we show that $R$ is complete intersection if and only if $\mathrm{cx}_R(I,J)$ or $\mathrm{tcx}_R(I,J)$ is finite if and only if $\mathrm{curv}_R(I,J)$ or $\mathrm{tcurv}_R(I,J)$ is at most $1$. We deduce these results from the corresponding more general results on Burch modules (in the sense of Dey-Kobayashi).