[Paper Review] Log Rigid Syntomic Cohomology for Strictly Semistable Schemes

Kazuki Yamada|arXiv (Cornell University)|May 14, 2015

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper introduces log rigid syntomic cohomology for strictly semistable schemes over the ring of integers of a $p$-adic field, defining it as the extension group of admissible filtered $(\phi,N)$-modules. It establishes that this cohomology realizes the absolute $p$-adic Hodge cohomology, providing a framework for $p$-adic Beilinson conjectures and enabling explicit calculations via $p$-adic Hodge theory.

ABSTRACT

We construct log rigid syntomic cohomology for strictly semistable schemes over the ring of integers of a p-adic field, and prove that it is interpreted as the extension group of the complex of admissible filtered $(ϕ,N)$-modules.

Motivation & Objective

• To define a cohomology theory for strictly semistable schemes over $V$, the ring of integers of a $p$-adic field, that realizes absolute $p$-adic Hodge cohomology.
• To extend the theory of rigid syntomic cohomology beyond smooth schemes to include strictly semistable ones with potentially good reduction.
• To provide a cohomological framework compatible with the $p$-adic Beilinson conjectures, especially for schemes with bad reduction.
• To establish a Chern class map from $K$-theory to syntomic cohomology using the derived category of admissible filtered $(\phi,N)$-modules.

Proposed method

• Constructs log rigid syntomic cohomology as $ H^i_{\mathrm{syn}}(\mathcal{X},n) = \operatorname{Ext}^i_{p\mathrm{HD}_K}(K_0, \mathbb{R}\Gamma_{\mathrm{Hdg}}(\mathcal{X})(n)) $, where $ \mathbb{R}\Gamma_{\mathrm{Hdg}}(\mathcal{X}) $ is the $p$-adic Hodge complex.
• Uses generalized Godement resolution to define log rigid cohomology and constructs Frobenius and monodromy operators on the complex.
• Establishes an equivalence $ \Theta: D^b(\mathrm{MF}^{\mathrm{ad}}_K(\phi,N)) \to \widetilde{p\mathrm{HD}_K} $, linking syntomic cohomology to extension groups of admissible filtered $(\phi,N)$-modules.
• Defines the syntomic Chern class map $ c_{\mathrm{syn}}: K_i(\mathcal{X}) \to H^{2j-i}_{\mathrm{syn}}(\mathcal{X},j) $ via the universal Chern class in the derived category.
• Applies the theory to the case of strictly semistable schemes satisfying the Hyodo-Kato condition (HK), ensuring compatibility with known $p$-adic Hodge theory.

Experimental results

Research questions

• RQ1 How can rigid syntomic cohomology be extended to strictly semistable schemes with potentially bad reduction?
• RQ2 Can log rigid syntomic cohomology be interpreted as an extension group of admissible filtered $(\phi,N)$-modules?
• RQ3 What is the role of the $p$-adic Hodge complex in unifying de Rham and log rigid cohomology for strictly semistable schemes?
• RQ4 How does the Chern class map from $K$-theory to syntomic cohomology behave in the context of $p$-adic Beilinson conjectures?
• RQ5 What is the relationship between log rigid syntomic cohomology and the derived category of $p$-adic Hodge complexes?

Key findings

• Log rigid syntomic cohomology is defined as the extension group of the $p$-adic Hodge complex $ \mathbb{R}\Gamma_{\mathrm{Hdg}}(\mathcal{X}) $ in the derived category $ p\mathrm{HD}_K $, providing a cohomological realization of absolute $p$-adic Hodge theory.
• The cohomology $ H^i_{\mathrm{syn}}(\mathcal{X},n) $ is isomorphic to the extension group of admissible filtered $(\phi,N)$-modules when $ \mathcal{X} $ satisfies the Hyodo-Kato condition.
• The Chern class map $ c_{\mathrm{syn}}: K_i(\mathcal{X}) \to H^{2j-i}_{\mathrm{syn}}(\mathcal{X},j) $ is constructed via the universal Chern class in the derived category of $(\phi,N)$-modules.
• The construction is compatible with the motivic framework, where $ D\mathcal{R}_{\mathrm{Hdg}} $ factors through the derived category of mixed motives.
• The theory provides a $p$-adic Hodge-theoretic interpretation of syntomic cohomology that is amenable to explicit $p$-adic analysis and $L$-function computations.