A homotopy-theoretic approach to derived categories of toric varieties

Thomas Huettemann (Queen's University, Belfast)

Monday 23rd March, 2009 16:00-17:00 Mathematics Building, room 204


A (complete, regular) toric variety X, defined over an arbitrary commutative ring, comes equipped with a preferred covering by open affines. A quasi-coherent sheaf on X gives rise, by evaluation on the affine sets, to a diagram of modules over various rings, satisfying a strong compatibility conditions: sections over a large affine piece agree, after restriction, with sections over a smaller affine piece ("gluing"). The derived category D(X) is obtained from the category of (unbounded) chain complexes of quasi-coherent sheaves on X by inverting the quasi-isomorphisms. In the talk I will explain that as far as the derived category is concerned, one can dispose of the compatibility condition mentioned above. The price is moderate: to obtain D(X) one has to invert a larger class C of morphisms in the category of chain complexes of diagrams. The class C admits an explicit combinatorial description; it consists of those morphisms that induce hyper-cohomology isomorphisms for sufficiently many twists of the sheaves involved. The proof uses the machinery of model categories. As an interesting by-product it shows that D(X) can also be obtained from diagrams of modules satisfying a weak compatibility condition ("gluing up to homotopy").

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