Linearity defects of modules over commutative rings

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the applications, it is proved that a local ring is Koszul if and only if it admits a Koszul module that is Cohen-Macaulay of minimal degree. An injective analogue of the linearity defect is introduced and studied. The main results express this new invariant in terms of linearity defects of free resolutions, and relate it to other ring theoretic and homological invariants of the module.Comment: 23 pages, minor modification

Homological dimensions and regular rings

A question of Avramov and Foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings. A key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules. Analogous results for flat dimension and projective dimension are also established.Comment: 8 page

The Bousfield lattice of a triangulated category and stratification

For a tensor triangulated category which is well generated in the sense of Neeman, it is shown that the collection of Bousfield classes forms a set. This set has a natural structure of a complete lattice which is then studied, using the notions of stratification and support.Comment: 25 pages; minor changes from earlier version; this will appear in the Math.

Cohomology over complete intersections via exterior algebras

A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the growth of resolutions of complexes over such local rings.Comment: 18 pages; to appear in "Triangulated categories (Leeds, 2006)", LMS lecture notes series