The big from small construction was introduced by Kustin and Miller in [A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra85 (1983) 303–322] and can be used to construct resolutions of tightly double linked Gorenstein ideals. In this paper, we expand on the DG-algebra techniques introduced in [A. Kustin, Use DG methods to build a matrix factorization, preprint (2019), arXiv:1905.11435] and construct a DG $R$-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 $R$-algebra exhibiting Poincaré 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.

Kustin 和 Miller 在 [A. Kustin 和 M. Miller，从小的 Gorenstein 理想构建大 Gorenstein 理想，J. Algebra 85 (1983) 303-322]，可用于构建紧密双联的 Gorenstein 理想的解决方案。在本文中，我们扩展了 [A. Kustin，使用 DG 方法构建矩阵分解，预印本 (2019)，arXiv:1905.11435] 并构建 DG$R$- 长度上的代数结构$4$大从小建设。所采用的技术涉及从 Tate 样复合物到无环 DG 的态射的构建$R$-表现庞加莱对偶的代数。这导致了同态，经过适当的修改，这些同态最终完美地封装了所需的关联性和 DG 公理，用于从大到小构建所需的产品结构。