Cohomology over complete intersections via exterior algebras

Summary

To Karin Erdmann on her 60th birthday.

Abstract. A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the growth of resolutions of complexes over such local rings.

Introduction

This paper concerns homological invariants of modules and complexes over complete intersection local rings. The goal is to explain a method by which one can establish in a uniform way results over such rings by deducing them from results on DG (that is, differential graded) modules over a graded exterior algebra, which are often easier to prove. A secondary purpose is to demonstrate the use of numerical invariants of objects in derived categories, called ‘levels’, introduced in earlier joint work with Buchweitz and Miller [5]; see Section 1. Levels allow one to track homological and structural information under changes of rings or DG algebras, such as those involved when passing from complete intersections to exterior algebras.

We focus on the complexity and the injective complexity of a complex M over a complete intersection ring R. These numbers measure, on a polynomial scale, the rate of growth of the minimal free resolution and the minimal injective resolution of M, respectively. The relevant basic properties are established in Section 2.