
TL;DR
This paper establishes a precise criterion for when the mod p^n reductions of first syntomic cohomology groups of reflexive F-gauges are isomorphic, linking it to isomorphisms of associated Breuil--Kisin modules with G_K-actions.
Contribution
It provides a new congruence criterion connecting syntomic cohomology groups and Breuil--Kisin modules in p-adic Hodge theory.
Findings
Isomorphism of mod p^n reductions corresponds to isomorphism of Breuil--Kisin modules mod p^{2n}.
Refines understanding of local Bloch--Kato Selmer groups.
Establishes a congruence relation in p-adic Hodge theory.
Abstract
Let O_K be the ring of integers of a finite extension K of Q_p. Given two reflexive F-gauges on O_K, we show that for large enough n, the mod p^n-reductions of their first syntomic cohomology groups, which might be regarded as a refinement of local Bloch--Kato Selmer groups, are isomorphic if and only if the mod p^{2n}-reductions of their attached Breuil--Kisin modules with G_K-actions and Nygaard filtrations are isomorphic.
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