Classification of the Tor-algebras of codimension four almost complete intersections

Abstract

Let ( R , m , k ) (R,m,k) be a local ring in which 2 2 is a unit. Assume that every element of k k has a square root in k k . We classify the algebras Tor ∙ R ⁡ ( R / J , k ) \operatorname {Tor}_ \bullet ^R(R/J,k) as J J varies over all grade four almost complete intersection ideals in R R . The analogous classification has already been found when J J varies over all grade four Gorenstein ideals [21], and when J J varies over all ideals of grade at most three [5, 30]. The present paper makes use of the classification, in [21], of the Tor-algebras of codimension four Gorenstein rings, as well as the (usually nonminimal) DG {\text {DG}} -algebra resolution of a codimension four almost complete intersection which is produced in [25 and 26].