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

Abstract:

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