group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Syntomic cohomology is the abelian sheaf cohomology of the syntomic site of a scheme. It is a $p$-adic analogue of Deligne-Beilinson cohomology.
Syntomic cohomology is closely related to the crystalline cohomology of that scheme and may be regarded as a $p$-adic absolute Hodge cohomology.
The syntomic cohomology may also be obtained from prismatic cohomology (Bhatt22). Let $R$ be a p-adically complete ring and let $\Delta_{R}$ be its absolute prismatic cohomology. It has an action of the Frobenius morphism $\phi$. We also have a “Breuil Kisin twist” $\Delta\lbrace 1\rbrace$ and a filtration $\Fil_{N}^{\bullet}\Delta$ called the Nygaard filtration (see see Bhatt21, section 2). The syntomic cohomology $\mathbb{Z}_{p}(i)$ is then defined to be the fiber
This construction globalizes and may be applied to p-adic formal schemes instead of just p-adically complete rings (see Bhatt21, Remark 2.14).
The syntomic site was introduced in
A construction of syntomic cohomology via prismatic cohomology is briefly discussed in
Bhargav Bhatt, Algebraic Geometry in Mixed Characteristic (arXiv:2112.12010)
Bhargav Bhatt, p-adic Hodge Theory and Applications: Connections to Algebraic Topology (Day 3 of Simons Lectures) (YouTube)
Further developments are in
The following shows that, just as Deligne-Beilinson cohomology may be interpreted as absolute Hodge cohomology, syntomic cohomology may be interpreted as $p$-adic absolute Hodge cohomology.
Last revised on November 28, 2022 at 05:26:52. See the history of this page for a list of all contributions to it.