Generalized cohomology theories and applications to algebraic and arithmetic geometry (A07)

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Cohomology theories pervade large parts of algebraic and arithmetic geometry. In this project we will develop and study cohomology theories, especially in mixed characteristic that generalize and unify étale cohomology, crystalline cohomology and de Rham cohomology resp. Hochschild homology in the non-commutative setting. A main goal is to construct a cohomology theory that can serve the same purposes for arithmetic schemes as the l-adic or crystalline cohomology with their Frobenius actions for varieties over finite fields. Ideas from algebraic geometry, algebraic topology, operator algebras and analysis blend in these investigations.