# Rigid versus log-rigid cohomology for semistable varieties

If $$K$$ is a p-adic field, with maximal unramified subfield $$K_0$$, and $$X$$ is a proper semi-stable $$O_K$$-scheme, then there's a canonical way to make the special fibre $$X_k$$ into a log-scheme; and there's a theory of log-rigid cohomology adapted to such things, which gives $$K_0$$-vector spaces $$H^i_{log-rig}(X_k / K_0)$$ equipped with a Frobenius $$\varphi$$ and a monodromy operator $$N$$.

However, if we forget the log-structure, we can also make sense of the rigid cohomology of $$X_k$$: rigid cohomology is defined for any $$k$$-variety (not necessarily smooth), and this gives spaces $$H^i_{rig}(X_k / K_0)$$ which have Frobenius actions (but not monodromy).

How are the groups $$H^i_{rig}(X_k / K_0)$$ and $$H^i_{log-rig}(X_k / K_0)$$ related?

I tried writing everything down for a semistable elliptic curve and convinced myself that in this case, the rigid cohomology was isomorphic to the kernel of the monodromy operator on the log-rigid cohomology. Is this true more generally?

• I think the general relationship is described in Chiarellotto's Duke 1999 paper "Rigid cohomology and invariant cycles for a semistable log scheme". He shows that the weight-monodromy conjecture implies that there is an exact sequence $H_{rig}^{i}(X_{k}/K_{0})\rightarrow H_{log-cris}^{i}(X_{k}/K_{0})\xrightarrow{N}H_{log-cris}^{i}(X_{k}/K_{0})$. This is unconditional if, for example, $X$ is a semistable family of curves or surfaces. In these cases the first map is an injection for $i=1$. For searching purposes, the elements of $\ker N$ are often called "$p$-adic invariant cycles". – Oli Gregory Apr 10 at 13:28
• Aha! "Invariant cycles" was the key googlable phrase I was missing. Thanks for this! Do you know if the map from $H^i_{rig}$ to $H^i_{log-cris}$ is expected to be injective? (Chiarellotto shows this for $i = 1$ only.) – David Loeffler Apr 10 at 15:54

$$\require{AMScd}$$I'll expand a little on my comment to give an answer to David's follow up question:

Firstly, the general relationship is described in Chiarellotto's Duke 1999 paper "Rigid cohomology and invariant cycles for a semistable log scheme". He shows that the weight-monodromy conjecture implies that there is an exact sequence $$\begin{equation*} H_{rig}^{n}(X_{k}/K_{0})\rightarrow H_{log-cris}^{n}(X_{k}/K_{0})\xrightarrow{N}H_{log-cris}^{n}(X_{k}/K_{0}) \end{equation*}$$ This is unconditional if, for example, $$X$$ is a semistable family of curves or surfaces. In these cases the first map is an injection for $$n=1$$. For searching purposes, the elements of $$\ker N$$ are often called "$$p$$-adic invariant cycles".

As for your second question, Chiarellotto states in the same paper that one cannot expect the first arrow to be an injection for $$n\geq 2$$ in general. Here is my guess as to why:

For simplicity, let us suppose that there exists a proper scheme $$P$$ over $$W(k)$$ such that $$X\hookrightarrow P$$ and $$P$$ is smooth around $$X_{k}$$. Let $$X_{k}=\bigcup_{i\in I}X_{k,i}$$ be the decomposition of $$X_{k}$$ into irreducible components. Then $$\left\{]X_{k,i}[_{X_{K_{0}}^{rig}}\right\}_{i\in I}$$ is an admissible covering of the rigid analytic space $$X_{K_{0}}^{rig}$$ associated to $$X_{K_{0}}$$, and similarly $$\left\{]X_{k,i}[_{P_{K_{0}}^{rig}}\right\}_{i\in I}$$ is an admissible cover of $$]X_{k}[_{P_{K_{0}}^{rig}}$$. The Čech cohomology of the first cover computes $$H_{log-cris}^{\ast}(X_{k}/K_{0})$$ and the Čech cohomology of the second cover computes $$H_{rig}^{\ast}(X_{k}/K_{0})$$.

Let us call the first map in above sequence $$\psi^{\ast}$$; it is the map on cohomology induced by $$\psi:X_{K_{0}}^{rig}\rightarrow ]X_{k}[_{P_{K_{0}}^{rig}}$$. It induces the following commutative diagram where the rows are the Mayer-Vietoris sequences for the two coverings:

$$\begin{CD} \displaystyle\bigoplus_{i\in I}H_{dR}^{1}(]X_{k,i}[_{X_{K_{0}}^{rig}}) @>\alpha>> \displaystyle\bigoplus_{i>> H_{log-cris}^{2}(X_{k}/K_{0}) @>>> \displaystyle\bigoplus_{i\in I}H_{dR}^{2}(]X_{k,i}[_{X_{K_{0}}^{rig}}) @>\beta>> \displaystyle\bigoplus_{i\alpha'>> \displaystyle\bigoplus_{i>> H_{rig}^{2}(X_{k}/K_{0}) @>>> \displaystyle\bigoplus_{i\in I}H_{dR}^{1}(]X_{k,i}[_{P_{K_{0}}^{rig}}) @>\beta'>> \displaystyle\bigoplus_{i

Since the $$X_{k,i}$$ and $$X_{k,ij}:=X_{k,i}\cap X_{k,j}$$ are smooth and proper (by the semi-stability assumption), we have $$H_{dR}^{\ast}(]X_{k,i}[_{P_{K_{0}}^{rig}})\simeq H_{cris}^{\ast}(X_{k,i}/K_{0})$$ and $$H_{dR}^{\ast}(]X_{k,ij}[_{P_{K_{0}}^{rig}})\simeq H_{cris}^{\ast}(X_{k,ij}/K_{0})$$. If we give each $$X_{k,i}$$ the log-structure induced by the divisor $$D_{i}$$ given by intersecting $$X_{k,i}$$ with the other irreducible components, and similarly for the $$X_{k,ij}$$, we have $$H_{dR}^{\ast}(]X_{k,i}[_{X_{K_{0}}^{rig}})\simeq H_{log-cris}^{\ast}(X_{k,i}/K_{0})$$ and $$H_{dR}^{\ast}(]X_{k,ij}[_{X_{K_{0}}^{rig}})\simeq H_{log-cris}^{\ast}(X_{k,ij}/K_{0})$$. If we let $$U_{i}$$ be the complement of $$D_{i}$$ in $$X_{k,i}$$, and similarly for $$U_{ij}$$, we have $$H_{log-cris}^{\ast}(X_{k,i}/K_{0})\simeq H_{rig}^{\ast}(U_{i}/K_{0})$$ and $$H_{log-cris}^{\ast}(X_{k,ij}/K_{0})\simeq H_{rig}^{\ast}(U_{ij}/K_{0})$$. The above diagram is then

$$\begin{CD} \displaystyle\bigoplus_{i\in I}H_{rig}^{1}(U_{i}/K_{0}) @>\alpha>> \displaystyle\bigoplus_{i>> H_{log-cris}^{2}(X_{k}/K_{0}) @>>> \displaystyle\bigoplus_{i\in I}H_{rig}^{2}(U_{i}/K_{0}) @>\beta>> \displaystyle\bigoplus_{i\alpha'>> \displaystyle\bigoplus_{i>> H_{rig}^{2}(X_{k}/K_{0}) @>>> \displaystyle\bigoplus_{i\in I}H_{cris}^{2}(X_{k,i}/K_{0}) @>\beta'>> \displaystyle\bigoplus_{i

The first, second, fourth and fifth vertical arrows are the direct sums of the arrows appearing in the Gysin sequences: $$\begin{equation*} 0\rightarrow H_{cris}^{1}(X_{k,i}/K_{0})\xrightarrow{\psi^{\ast}} H_{rig}^{1}(U_{i}/K_{0})\rightarrow H_{rig}^{0}(D_{i}/K_{0})\xrightarrow{\rho_{i}} H_{cris}^{2}(X_{k,i}/K_{0})\xrightarrow{\psi^{\ast}} H_{rig}^{2}(U_{i}/K_{0}) \end{equation*}$$ and $$\begin{equation*} 0\rightarrow H_{cris}^{1}(X_{k,ij}/K_{0})\xrightarrow{\psi^{\ast}} H_{rig}^{1}(U_{ij}/K_{0})\rightarrow H_{rig}^{0}(D_{ij}/K_{0})\xrightarrow{\rho_{ij}} H_{cris}^{2}(X_{k,ij}/K_{0})\xrightarrow{\psi^{\ast}} H_{rig}^{2}(U_{ij}/K_{0}) \end{equation*}$$ We get a diagram

$$\begin{CD} 0 @>>> \mathrm{coker}(\alpha) @>>> H_{log-cris}^{2}(X_{k}/K_{0}) @>>> \ker(\beta) @>>> 0\\ @. @AAA @A\psi^{\ast}AA @AAA @.\\ 0 @>>> \mathrm{coker}(\alpha') @>>> H_{rig}^{2}(X_{k}/K_{0}) @>>> \ker(\beta') @>>> 0 \\ \end{CD}$$

and from the Gysin sequences we can see that the left vertical arrow is an injection, but the kernel of the right vertical arrow will be $$\begin{equation*} \ker\left(\bigoplus_{i\in I}\rho_{i}(H_{rig}^{0}(D_{i}/K_{0})\rightarrow\bigoplus_{i Probably this kernel is non-trivial in general, and hence $$\psi^{\ast}:H_{rig}^{2}(X_{k}/K_{0})\rightarrow H_{log-cris}^{2}(X_{k}/K_{0})$$ won't be injective in general. The same argument should work for all $$n\geq 2$$.