[Paper Review] Comparison between rigid syntomic and crystalline syntomic cohomology for strictly semistable log schemes with boundary

Veronika Ertl, Kazuki Yamada|arXiv (Cornell University)|May 14, 2018

Algebraic Geometry and Number Theory23 references5 citations

TL;DR

This paper introduces rigid syntomic cohomology for strictly semistable log schemes over a complete discrete valuation ring of mixed characteristic (0,p), and establishes a comparison isomorphism with Nekovář–Nizioł's crystalline syntomic cohomology when a good compactification exists. The key technical advance is a modification of Große-Klönne's rigid Hyodo–Kato theory and its generalization to log schemes with boundary, enabling canonical complexes and compatibility of Frobenius and Hyodo–Kato maps on Frobenius eigenspaces for eigenvalues $p^r$, $r \geq -1$. The result confirms the compatibility of rigid and crystalline syntomic cohomology in the compactifiable case.

ABSTRACT

We introduce rigid syntomic cohomology for strictly semistable log schemes over a complete discrete valuation ring of mixed characteristic (0,p). In case a good compactification exists, we compare this cohomology theory to Nekovář-Nizioł's crystalline syntomic cohomology of the generic fibre. The main ingredients are a modification of Große-Klönne's rigid Hyodo-Kato theory and a generalisation of it for strictly semistable log schemes with boundary.

Motivation & Objective

• To develop a p-adic cohomology theory for strictly semistable log schemes in mixed characteristic (0,p) that is purely analytic and suitable for computing p-adic regulators.
• To overcome technical difficulties in log rigid cohomology arising from dependence on local liftings by constructing canonical rigid complexes.
• To generalize Große-Klönne's rigid Hyodo–Kato theory to log schemes with boundary, enabling a comparison with crystalline syntomic cohomology.
• To establish a comparison isomorphism between rigid and crystalline syntomic cohomology in the case of good compactification.
• To show compatibility of the rigid and crystalline Hyodo–Kato maps on Frobenius eigenspaces for eigenvalues $p^r$, $r \geq -1$.

Proposed method

• Construct canonical rigid complexes for fine log schemes and log schemes with boundary using a modification of Große-Klönne's rigid Hyodo–Kato theory.
• Define rigid Hyodo–Kato cohomology for strictly semistable log schemes with boundary via simplicial resolutions and T-log schemes with boundary.
• Use log convergent cohomology as a bridge to compare rigid and crystalline cohomology, leveraging the crystalline and convergent Poincaré lemmas.
• Establish compatibility of base change and Frobenius structures on the rigid Hyodo–Kato complex, ensuring compatibility with Frobenius eigenspaces.
• Construct a commutative diagram involving rigid, convergent, and crystalline cohomology theories, with comparison maps and quasi-isomorphisms.
• Prove that the canonical morphism $\mathrm{sp}: R\Gamma_{\mathrm{dR}}(X_K) \to R\Gamma_{\mathrm{rig}}(X_0/O_\pi^K)$ is a quasi-isomorphism, implying compatibility of Hyodo–Kato maps on Frobenius eigenspaces.

Experimental results

Research questions

• RQ1 How can rigid syntomic cohomology be defined for strictly semistable log schemes with boundary in mixed characteristic (0,p)?
• RQ2 What modifications to Große-Klönne's rigid Hyodo–Kato theory are required to achieve functoriality and compatibility with Frobenius?
• RQ3 Is there a canonical construction of rigid cohomology complexes that avoids dependence on local liftings?
• RQ4 Can rigid syntomic cohomology be compared to crystalline syntomic cohomology in the presence of a good compactification?
• RQ5 Are the rigid and crystalline Hyodo–Kato maps compatible on Frobenius eigenspaces for eigenvalues $p^r$, $r \geq -1$?

Key findings

• Rigid syntomic cohomology is constructed for strictly semistable log schemes with boundary over a complete discrete valuation ring of mixed characteristic (0,p).
• The rigid Hyodo–Kato map is shown to be compatible with the crystalline Hyodo–Kato map on Frobenius eigenspaces for eigenvalues $p^r$, $r \geq -1$, via a diagram involving log convergent and crystalline cohomology.
• The canonical morphism $\mathrm{sp}: R\Gamma_{\mathrm{dR}}(X_K) \to R\Gamma_{\mathrm{rig}}(X_0/O_\pi^K)$ is a quasi-isomorphism, confirming compatibility of the Hyodo–Kato maps.
• The construction of rigid syntomic cohomology is compatible with cup products, as it follows the same method as in [8, §2.4].
• The Frobenius endomorphism on $R\Gamma_{\mathrm{rig}}((V^\bullet, P^\bullet)/T)$ is well-defined and compatible with Frobenius on the rigid cohomology of the special fiber.
• A quasi-isomorphism is established between the diagrams defining $R\Gamma_{\mathrm{rig}}^{\mathrm{syn}}((X_0,X_0), r, \pi)$ and $R\Gamma_{\mathrm{cr}}^{\mathrm{syn}}(X, r, \pi)$ for $r \geq 0$, confirming the comparison in the compactifiable case.