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.

It is closely related to the crystalline cohomology of that scheme. It may be regarded as a $p$-adic absolute Hodge cohomology.

Related

• Deligne-Beilinson cohomology
• p-adic Hodge theory

References

The syntomic site was introduced in

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

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.