Milnor Descent for Cohesive DG Categories

For any curved differential graded algebra A, Block introduced a dg category
called P_A. When A is the Dolbeaut algebra of a compact complex manifold,
Block proved that the homotopy category of P_A is the bounded derived category
of coherent sheaves on the manifold.  His first paper can be found at
http://arxiv.org/abs/math/0509284 [4]

Because P_A is defined without the use of sheaves, its definition is motivated by
non-commutative geometry. After providing some historical context and defining the main
objects, I will discuss a kind of descent theorem. This theorem shows that a fiber product
of differential graded algebras translates via the assignment sending A to P_A into a 
homotopy fiber product of dg categories as was first defined by Drinfeld.  Our theorem
generalizes Milnor's descent theorem for projective modules over  (possibly
non-commutative rings).  It also generalizes descent for dg categories of perfect
complexes of sheaves in the commutative context as was proven by  Hirschowitz
and Simpson. The main tools used in this theorem can be found in Block's original
paper 'Duality and Equivalence of module categories in noncommutative geometry,'
in 'The homotopy theory of dg-categories and derived Morita theory' by Toen and
in Landsburg's work on K-theory.

This talk will be based on joint work with Jonathan Block which can be found
at http://arxiv.org/abs/1201.6118