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

上同调理论在代数几何和算术几何中占有很大的比重。在这个项目中，我们将发展和研究上同调理论，特别是在混合特征方面，分别推广和统一了étale上同调，crystaline上同调和de Rham上同调。非对易条件下的Hochschild同调。一个主要的目标是构建一个上同调理论，可以作为相同的目的，算术计划的l-adic或结晶上同调与他们的Frobenius行动品种在有限领域。从代数几何，代数拓扑，算子代数和分析的想法融合在这些调查。

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.