nLab syntomic cohomology

Idea

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.

Construction via Prismatic 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

$$\mathbb{Z}_{p}(i)(R)=\fib(\Fil_{N}^{i}\Delta_{R}\lbrace i\rbrace\xrightarrow{\phi_{i}-1}\Delta_{R}\lbrace i\rbrace)$$

This construction globalizes and may be applied to p-adic formal schemes instead of just p-adically complete rings (see Bhatt21, Remark 2.14).

Related

• Deligne-Beilinson cohomology

• p-adic Hodge theory

• prismatic cohomology

References

The syntomic site was introduced in

• Jean-Marc Fontaine and William Messing, $p$-Adic periods and $p$-adic etale cohomology (pdf)

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

• Amnon Besser, Syntomic regulators and $p$-adic integration I: rigid syntomic regulators (pdf)

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.

• Frédéric Déglise, Wiesława Nizioł?, On $p$-adic absolute Hodge cohomology and syntomic coefficients, I, arXiv:1508.02567.