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 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 the classical work of Burch, and subsequently extended to modules by Dey-Kobayashi. This class includes several large and well-studied families of modules and ideals over a Noetherian local ring $(R,\mathfrak{m},k)$. For example, these include the residue field $k$ as an $R$-module, every nonzero module of the form $\mathfrak{m}M$ (e.g., ${\mathfrak{m}}^n$ for $n\ge 1$), and under mild conditions every integrally closed ideal $I$ with $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}(R\mathrm{/}I)=0$. Suppose $I$ and $J$ are Burch ideals such that $I$ is $\mathfrak{m}$-primary. Motivated by Avramov’s result that Burch modules exhibit extremal complexity and curvature, we establish in this article that ${\mathrm{c}\mathrm{x}}_R(I,J)={\mathrm{t}\mathrm{c}\mathrm{x}}_R(I,J)={\mathrm{c}\mathrm{x}}_R(k)$. Moreover, we show that $R$ is complete intersection if and only if ${\mathrm{c}\mathrm{x}}_R(I,J)$ or ${\mathrm{t}\mathrm{c}\mathrm{x}}_R(I,J)$ is finite if and only if ${\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}}_R(I,J)$ or ${\mathrm{t}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}}_R(I,J)$ is at most $1$. We deduce these results from the corresponding more general results on Burch modules.