Algebra structures on resolutions of rings defined by grade four almost complete intersections
Let (R, m, k) be a local ring in which 2 is a unit and let J be an indeal in R. The minimal R-free resolution of R/J is called a DG-algebra if it has the structure of an associative, differential, graded commutative algebra over R. If J is a grade four almost complete intersection ideal in R, then J is necessarily linked to a grade four Gorenstein ideal I by a grade four complete intersection K. It is well known that the minimal R-free resolutions of R/I and R/K admit DG-algebra structures. In this paper, we show that if W denotes the image of the map TorR1(R/K, k) → TorR1(R/I, k) and W2 = 0 (for example, if K ⊆ mI), then the minimal R-free resolution of S = R/J has a DG-algebra structure. In the case where the ring R is regular local and K ⊆ mI we prove that the Poincaré series PNS(t) of every finitely generated S-module N is rational and the ring S satisfies the Eisenbud Conjecture.
Slattery, S. Palmer. "Algebra structures on resolutions of rings defined by grade four almost complete intersections." Journal of Algebra 159, no. 1 (1993): 1-46.
Journal of Algebra