Mapping free resolutions of length three II -- Module formats

Let $M$ be a perfect module of projective dimension 3 in a Gorenstein, local or graded ring $R$. We denote by $\text{FF}$ the minimal free resolution of $M$. Using the generic ring associated to the format of $\text{FF}$ we define higher structure maps, according to the theory developed by Weyman in “Generic free resolutions and root systems” (Annales de l’Institut Fourier} 68.3 (2018), pp. 1241–1296). We introduce a generalization of classical linkage for $R$-module using the Buchsbaum–Rim complex, and study the behaviour of structure maps under this Buchsbaum–Rim linkage. In particular, for certain formats we obtain criteria for these $R$-modules to lie in the Buchsbaum–Rim linkage class of a Buchsbaum–Rim complex of length 3.