DG Structure on the Length 4 Big From Small Construction

The big from small construction was introduced by Kustin and Miller and can be used to construct resolutions of tightly double linked Gorenstein ideals. In this paper, we expand on the DG-algebra techniques introduced Kustin for building matrix factorizations and construct a DG-algebra structure on the length $4$ big from small construction. The techniques employed involve the construction of a morphism from a Tate-like complex to an acyclic DG-algebra exhibiting Poincar\'e duality. This induces homomorphisms which, after suitable modifications, satisfy a list of identities that end up perfectly encapsulating the required associativity and DG axioms of the desired product structure for the big from small construction.