Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mapping free resolutions of length three II -- Module formats

Sara Angela Filippini, Lorenzo Guerrieri

Abstract

Let $M$ be a perfect module of projective dimension 3 in a Gorenstein, local or graded ring $R$. We denote by $\FF$ the minimal free resolution of $M$. Using the generic ring associated to the format of $\FF$ 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.

Mapping free resolutions of length three II -- Module formats

Abstract

Let be a perfect module of projective dimension 3 in a Gorenstein, local or graded ring . We denote by the minimal free resolution of . Using the generic ring associated to the format of 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 -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 -modules to lie in the Buchsbaum--Rim linkage class of a Buchsbaum--Rim complex of length 3.
Paper Structure (10 sections, 12 theorems, 133 equations)

This paper contains 10 sections, 12 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Critical representations and higher structure maps for module formats
  3. Definition of certain higher structure maps
  4. Perfect modules with resolution of format $(2,4,4,2)$
  5. Perfect modules with resolution of format $(2,5,4,1)$
  6. Buchsbaum--Rim linkage
  7. The free resolution of a linked module
  8. Linking higher structure maps for formats $(2,6,5,1)$ and $(2,5,5,2)$
  9. Examples
  10. Computation over a split exact complex

Key Result

Theorem 2.1

Let $R$ be a local Gorenstein ring and let $M$ be an $R$-module having minimal free resolution of format $(2,4,4,2)$. The following conditions are equivalent:

Theorems & Definitions (28)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 18 more