Koszul homology and blowing-up rings J. Herzog 1 Aron Simis 2 W. Vasconcelos 3 Contents 1 Introduction 2 2 Ideals of linear type 5 3 Ascent and descent 9 4 Approximation complexes 12 5 Depth and acyclicity 18 6 Sequences 21 6.1 Generalizations of regular sequences ...................... 21 6.2 Syzygetic ideals .................................. 24 6.3 Coarse Cohen-Macaulayfication ......................... 25 7 Depth of the Koszul homology modules 25 7.1 Ideals generated by proper sequences ...................... 25 7.2 Koszul homology and linkage .......................... 26 8 Dimension of symmetric algebras 29 9 Cohen–Macaulay blowing-up rings 32 10 Cohen–Macaulay symmetric algebras 34 11 The syzygetic reduction groups 39 12 Sequences and the approximation complexes 42 13 Linear resolutions 49 These notes are a combined and expanded account of the individual talks given by the authors at the Trento conference. They aim at providing a framework for studying arithmetical properties of Rees algebras, associated graded rings, symmetric algebras and the theory of linear resolutions of ideals and modules. 0 1 2 3 1