# Generic structure of free resolutions, and homological properties of symmetric and exterior powers of modules

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- Purdue University
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## Abstract

In Chapter I we study the generic properties of free resolutions. Given a commutative ring k and positive integers b0,b1,b2 for which b1-b2≥1 and b0-b1+b2≥0, results of Hochster and Bruns show the existence of a universal commutative k-algebra R and a universal free resolution over R with these integers as Betti numbers, from which any other free resolution with the same Betti numbers is obtained by a unique base change. We show that the universal ring R is a finitely presented positively graded k-algebra which is a free k-module, and list explicitly its generators, relations, and basis over k. We prove that R is reduced (resp. a domain, Krull domain, factorial, Cohen-Macaulay, Gorenstein) if and only if k is, and provide necessary and sufficient conditions for the regularity of R. These results are special cases of theorems that completely describe the generic structure of finite free complexes acyclic in depth 1. In Chapter II we construct, starting from a free resolution of a module M, canonical complexes that approximate free resolutions of the symmetric and exterior powers of the module M. We provide criteria for the acyclicity of these approximate resolutions, and use them to study the torsion freeness of modules of projective dimension 1. ^

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