Fix a prime $p$ and let $X$ be a $\mathbb{Z}_{p}$-scheme. Write $X_{n}:=X\otimes\mathbb{Z}/p^{n}$ and $\phi:X_{1}\rightarrow X_{1}$ for the absolute Frobenius. Let $X\hookrightarrow Z$ be a (suitable) closed immersion. Let $D_{n}$ be the PD-envelope of $X_{n}$ in $Z_{n}$ and set $J_{D_{n}}=\ker(\mathcal{O}_{D_{n}}\rightarrow\mathcal{O}_{X_{n}})$. Then for $0\leq r\leq p-1$ the syntomic complex $\mathscr{S}_{n}(r)_{X,Z}$ is the complex of étale sheaves on $X_{1}$ given by the mapping fibre of \begin{equation*} 1-\frac{\phi}{p^{r}}:\mathbb{J}^{[r]}_{n,X,Z}\rightarrow\mathbb{J}^{[0]}_{n,X,Z} \end{equation*} where $\mathbb{J}^{[r]}_{n,X,Z}$ is is the complex \begin{equation*} J_{D_{n}}^{[r]}\xrightarrow{d}J_{D_{n}}^{[r-1]}\otimes_{\mathcal{O}_{Z_{n}}}\Omega_{Z_{n}}^{1}\xrightarrow{d}\cdots \end{equation*} As Kato explains in [1], the image of $\mathscr{S}_{n}(r)_{X,Z}$ in the derived category is independent of $Z$. Kato then defines \begin{equation*} H_{\text{syn}}^{i}(X,\mathscr{S}_{n}(r)):=\mathbb{H}_{\text{ét}}^{i}(X,\mathscr{S}_{n}(r)_{X,Z}) \end{equation*} In the same paper, Kato says that this computes the syntomic cohomology defined by Fontaine-Messing using the syntomic site, but says that the details are given in [2]. But here I can only find Remark (1.1) which is the same statement, again without proof.

Does anybody have a reference that explains how these two cohomologies are the same? Or perhaps somebody can sketch how this works?

[1] The Explicit Reciprocity Law and the Cohomology of Fontaine-Messing, Bull. Soc. math. France, 119, 1991, p. 397–441.

[2] On $p$-adic Vanishing Cycles (Application of ideas of Fontaine-Messing), Adv. Studies in Pure Math. 10, 1987, pp207-251.

Further remark:- Niziol has many (wonderful) papers studying generalisations of Kato's mapping fibre approach. I haven't been able to find an answer in any of these articles either, but in Niziol's 2006 ICM article $p$-adic motivic cohomology in arithmetic, there is a remark on page 7 claiming that the Fontaine-Messing construction is "philosophically" the same as the Kato construction. This only confuses me further...


Ok, maybe I've figured this out. Hopefully somebody can correct me if this is wrong. Also, I'd still like to know a reference that writes this out in detail, if anybody has one.

I'll change the notation from the question slightly: $k$ is a perfect field of characteristic $p>0$, and $W=W(k)$. First let's recall the definition of Fontaine-Messing's syntomic sheaf:- we are working on the site $(\text{Spf }W)_{\text{NILSYN}}$. This is the site whose objects are $W$-schemes on which $p$ is locally nilpotent, and the topology is the syntomic topology. For each $n\in\mathbb{N}$, we have some sheaves on this site:- \begin{equation*} \mathcal{O}_{n}:X\mapsto\mathcal{O}_{X_{n}}(X_{n}) \\ \mathcal{O}_{n}^{\text{cris}}:X\mapsto H_{\text{cris}}^{0}(X_{1}/W_{n}) \\ J_{n}:=\ker\left(\mathcal{O}_{n}^{\text{cris}}\rightarrow\mathcal{O}_{n}\right) \end{equation*} Define $\tilde{J}_{n}^{[r]}:=\text{im}\left(J_{n+r}^{[r]}\rightarrow J_{n}^{[r]}\right)$. For $r<p$, Fontaine and Messing (2.3 on page 191) show that $\phi(\tilde{J}_{n}^{[r]})\subset p^{r}\mathcal{O}_{n}^{\text{cris}}$ where $\phi:\mathcal{O}_{n}^{\text{cris}}\rightarrow \mathcal{O}_{n}^{\text{cris}}$ is the Frobenius. So we have a map $\frac{\phi}{p^{r}}:\tilde{J}_{n}^{[r]}\rightarrow\mathcal{O}_{n}^{\text{cris}}$. The syntomic sheaf is then defined to be \begin{equation*} s_{n}(r)_{X}:=\ker\left(1-\frac{\phi}{p^{r}}:\tilde{J}_{n}^{[r]}\rightarrow\mathcal{O}_{n}^{\text{cris}}\right). \end{equation*} Now, for $X$ a $W$-scheme, let $u:X_{n,\text{SYN}}\rightarrow X_{n,\text{ET}}$ be the morphism of sites. The claim is that we have an isomorphism \begin{equation*} Ru_{\ast}s_{n}(r)_{X}\simeq\mathscr{S}_{n}(r)_{X}[-1] \end{equation*} By Remark 1.8 on page 213 of Kato's ``On p-adic Vanishing Cycles (Application of ideas of Fontaine-Messing)'', it suffices to prove the claim when $X$ is quasi-projective. But then we can find a closed immersion $X\hookrightarrow Z$ into a smooth $W$-scheme endowed with a Frobenius (e.g. $Z$ could be projective space). In the notation of the question, we are writing $D_{n}=D_{X_{n}}(Z_{n})$ for the PD-envelope of $X_{n}$ in $Z_{n}$ (wrt the PD-structure on $pW_{n}$). In this notation then, Theorem 7.2 of Berthelot-Ogus gives an isomorphism \begin{equation*} \mathbb{R}u_{\ast}J_{n}^{[r]}\simeq\mathbb{J}_{X_{n},Z_{n}}^{[r]} \end{equation*} (I think we also need the result of this question

Crystalline cohomology via the syntomic site

to make this part legit).

Now we simply use the definition of $s_{n}(r)_{X}$ as $\ker\left(1-\frac{\phi}{p^{r}}\right)$ and the triangle associated to mapping fibres to see the claim. (Note, this is where the shift by $-1$ comes in).

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