3:00PM: Differential Graded Categories and Milnor's Theorem
Oren Ben-BassetIn his book on Algebraic K-theory, Milnor gave a technique for gluing projective modules. This technique constructs a projective module on a ring A out of a projective modules on rings B and C and a certain isomorphism. In joint work with Jonathan Block, we generalized this construction to a gluing (involving a homotopy fiber product) of certain differential graded categories. In this talk, I will review differential graded categories and introduce the construction of Jonathan Block which assigns a differential graded category to each differential graded algebra. This construction is fundamentally different than a well known construction involving differential graded modules. If time permits I will give some geometric consequences relating to compact complex manifolds. One of these consequences relates to forming vector bundles on a complex manifold X out of vector bundles "near" a submanifold Z and vector bundles on X-Z. I will also mention a related gluing construction involving gluing categories of coherent sheaves over a punctured formal neighborhood of a subvariety Z inside a variety X over any field k. The punctured formal neighborhood is described using Berkovich analytic geometry. That construction is joint work with Michael Temkin.