Rigidity and Poincaré Duality for algebraic varieties.

The history of topological methods in algebraic geometry started almost 60 years ago when André Weil formulated his famous conjectures. A lot of scheme invariants of cohomological nature were introduced since that time and many classical problems were solved in that way. However, until mid-90s, it was unclear whether cohomology theories on schemes can be represented by objects of some category playing the same role as the category of spectra does in topology. The situation changed revolutionary after works of Voevodsky and collaborators. Their approach gave us a lot of new cohomology theories on schemes and also raised many interesting questions.

We are going to look at one old result from algebraic K-theory (Suslin’s rigidity theorem) from this new point of view. We’ll consider it as the particular case of a more general statement holding for a wide class of cohomology theories. In order to describe this class, we introduce a notion of an orientable theory in the algebraic context. Orientable theories inherit some nice properties from topology. For example, such a classical topological result as Poincaré duality also holds for orientable theories on algebraic varieties. Besides algebraic K-theory, important examples of orientable theories include motivic cohomology and algebraic cobordism.