2023/10/19 by Benjamin Antieau, Achim Krause, Antieau, Benjamin +3
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2310.12770
We develop prismatic and syntomic cohomology relative to a δ-ring. This simultaneously generalizes Bhatt and Scholze's absolute and relative prismatic cohomology and shows that the latter, which was defined relative to a prism, is in fact independent of the prism structure and only depends on the underlying δ-ring. We give several possible definitions of our new version of prismatic cohomology: a site theoretic definition, one using prismatic crystals, and a stack theoretic definition. These are equivalent under mild syntomicity hypotheses. As an application, we note how the theory of prismatic cohomology of filtered rings arises naturally in this context.