Tor as a Module over an Exterior Algebra

David Eisenbud; Irena Peeva; Frank-Olaf Schreyer

paper

Tor as a Module over an Exterior Algebra

arXiv:1810.04999

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 $Tor^{S} (M, k)$ . In the case where $M$ is a high syzygy over the complete intersectionR:=S/(f_{1},\dots,f_{c}) $w e d escr ib e t hi s$ E $- m o d u l es t r u c t u r e in d e t ai l, in c l u d in g i t s minima l f r eer eso l u t i o n o v er$ E $. T u r nin g t o$ {\rm Ext}_{R}(M,\, k) $w es h o w t ha t, w h e n$ M $i s ahi g h sy z y g y o v er$ R $, t h e minima l f r eer eso l u t i o n o f$ {\rm Ext}_{R}(M,\, k) $a s am o d u l eo v er t h er in g o f C I o p er a t or s i s t h e B er n s t e in - G e l^{'} f an d - G e l^{'} f an dd u a l o f t h e$ E $- m o d u l e$ {\rm Tor}^{S}(M,\,k) $. F or t h e p r oo f w e in t r o d u ce higher CI operators, an d g i v e a co n s t r u c t i o n o f a (g e n er a l l y n o n - minima l) r eso l u t i o n o f$ M $o v er$ S $s t a r t in g f r o ma r eso l u t i o n o f$ M $o v er$ R$ and its higher CI operators.

To appear in the Journal of the European Mathematical Society (JEMS)