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?



Rigid versus log-rigid cohomology for semistable varieties
$\require{AMScd}$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<j\in I}H\_{dR}^{1}(]X\_{k,ij}[\_{X\_{K\_{0}}^{rig}}) @>>> 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<j\in I}H\_{dR}^{2}(]X\_{k,ij}[\_{X\_{K\_{0}}^{rig}}) \\
@A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA\\
\displaystyle\bigoplus\_{i\in I}H\_{dR}^{1}(]X\_{k,i}[\_{P\_{K\_{0}}^{rig}}) @>\alpha'>> \displaystyle\bigoplus\_{i<j\in I}H\_{dR}^{1}(]X\_{k,ij}[\_{P\_{K\_{0}}^{rig}}) @>>> 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<j\in I}H\_{dR}^{1}(]X\_{k,ij}[\_{P\_{K\_{0}}^{rig}}) \\ \end{CD}


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<j\in I}H\_{rig}^{1}(U\_{ij}/K\_{0}) @>>> H\_{log-cris}^{2}(X\_{k}/K\_{0}) @>>> \displaystyle\bigoplus\_{i\in I}H\_{rig}^{2}(U\_{i}/K\_{0}) @>\beta>> \displaystyle\bigoplus\_{i<j\in I}H\_{rig}^{2}(U\_{ij}/K\_{0}) \\
@A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA @A\psi^{\ast}AA\\
\displaystyle\bigoplus\_{i\in I}H\_{cris}^{1}(X\_{k,i}/K\_{0}) @>\alpha'>> \displaystyle\bigoplus\_{i<j\in I}H\_{cris}^{1}(X\_{k,ij}/K\_{0}) @>>> H\_{rig}^{2}(X\_{k}/K\_{0}) @>>> \displaystyle\bigoplus\_{i\in I}H\_{cris}^{2}(X\_{k,i}/K\_{0}) @>\beta'>> \displaystyle\bigoplus\_{i<j\in I}H\_{cris}^{2}(X\_{k,ij}/K\_{0}) \\ \end{CD}


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<j\in I}\rho\_{ij}(H\_{rig}^{0}(D\_{ij}/K\_{0}))\right).
\end{equation\*}
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 $n\geq 2$.


