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arXiv · 2310.12770

Prismatic cohomology relative to $\delta$-rings

Benjamin Antieau·Achim Krause·Thomas Nikolaus
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Abstract

We develop prismatic and syntomic cohomology relative to a $\delta$-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 $\delta$-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.

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Benjamin Antieau, Achim Krause, Thomas Nikolaus. 2023-10-19. Prismatic cohomology relative to $\delta$-rings. https://arxiv.org/abs/2310.12770

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