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

上同调理论渗透到代数几何和算术几何的很大一部分中。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，分别概括和统一了故事上同调、晶体上同调和德罗姆上同调。非对易环境下的Hochschild同调。一个主要目标是构造一个上同调理论，它可以用于算术方案，其目的与L上的上同调或晶体上同调及其对有限域上的簇的Frobenius作用相同。这些研究融合了代数几何、代数拓扑学、算子代数和分析的思想。

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.