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Topology/Geometry Seminar
Thursday, October 27, 2005
5:30pm,
260 Hawthorn (Altoona)
Jim Gillespie
Penn State McKeesport
The Flat Model Structure on Complexes of Sheaves
Abstract: Let $\mathcal{A}$ be an abelian category. A Quillen model structure on the associated chain complex category is a "way to do homological algebra" in $\mathcal{A}$. We will see that a nice enough class of objects in $\mathcal{A}$, called Kaplansky classes, give rise to model structures on the associated chain complex category. Each model structure essentially translates into a new way to define derived functors such as Ext and Tor via resolutions by objects in $\mathcal{A}$. All of the usual examples of model structures on chain complex categories can be recovered from an appropriate Kaplansky class. More importantly, the method can be used to get new "flat model structures" when $\mathcal{A}$ is 1) the category of modules over a ring, 2) the category of modules on a ringed space or even 3) the category of quasi-coherent sheaves on a nice enough scheme. The last two categories do not have enough projectives, so the resulting flat model structure is useful for definining and studying the derived tensor product.
We meet weekly on Thursday afternoons, with the third meeting of each month in Altoona. If you are interested in giving a talk in our seminar please contact one of the coordinators for the Fall 2005 semester: Aissa Wade and Wojciech Dorabiala.
