Tor as a Module over an Exterior Algebra

TL;DR

This paper explores the structure of Tor and Ext modules over complete intersections, revealing a duality mediated by an exterior algebra and introducing higher CI operators for resolutions.

Contribution

It establishes a detailed description of the E-module structure on Tor and its duality with Ext over complete intersections, introducing higher CI operators for resolutions.

Findings

E-module structure on Tor is explicitly described for high syzygies.

Duality between Ext and Tor modules is established via BGG duality.

Higher CI operators are introduced to construct resolutions over S.

Abstract

Let $S$ be a regular local ring with residue field $k$ and let $M$ be a finitely generated $S$-module. Suppose that $f_1,\dots ,f_c\in S$ is a regular sequence that annihilates $M$, and let $E$ be an exterior algebra over $k$ generated by $c$ elements. The homotopies for the $f_{i}$ on a free resolution of $M$ induce a natural structure of graded $E$-module on ${\rm Tor}^{S}(M,k)$. In the case where $M$ is a high syzygy over the complete intersectionR:=S/(f_{1},\dots,f_{c})$ we describe this $E$-module structure in detail, including its minimal free resolution over $E$. Turning to ${\rm Ext}_{R}(M,\, k)$ we show that, when $M$ is a high syzygy over $R$, the minimal free resolution of ${\rm Ext}_{R}(M,\, k)$ as a module over the ring of CI operators is the Bernstein-Gel'fand-Gel'fand dual of the $E$-module ${\rm Tor}^{S}(M,\,k)$. For the proof we introduce \emph{higher CI operators}, and give a construction of a (generally non-minimal) resolution of $M$ over $S$ starting from a resolution of $M$ over $R$ and its higher CI operators.