Ph.D. Thesis Defense, Gabriel Valenzuela : 'Homological algebra of complete and torsion modules.'

Tuesday, May 5, 2015
4:15 PM - 5:30 PM (ET)
ESC 618
Event Type
Mark Hovey,

Abstract: Let R be a Noetherian regular local ring with maximal ideal 𝔪. The category of 𝔪-adically complete R-modules is not Abelian, but it can be enlarged to an Abelian category of so-called L-complete modules. This category is an Abelian subcategory of the full category of R-modules, but it is not usually a Grothendieck category. It is well known that a Grothendieck category always has a derived category, however, this is much more delicate for arbitrary Abelian categories.In this talk, we will show that the derived category of the L-complete modules exists, and that it is in fact equivalent to a certain Bousfield localization of the full derived category of R. L-complete modules should be dual to 𝔪-torsion modules, which do form a Grothendieck category. We make this precise by showing that although these two Abelian categories are clearly not equivalent, they are derived equivalent. As an application, we will explain how this result can be used to effortlessly recover well known duality theorems currently found in the literature.

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