# The Stacks Project

## Tag 07ML

Remark 54.24.2 (Mayer-Vietoris). In the situation of Remark 54.24.1 suppose we have an open covering $X = X' \cup X''$. Denote $X''' = X' \cap X''$. Let $f'$, $f''$, and $f''$ be the restriction of $f$ to $X'$, $X''$, and $X'''$. Moreover, Let $\mathcal{F}'$, $\mathcal{F}''$, and $\mathcal{F}'''$ be the restriction of $\mathcal{F}$ to the crystalline sites of $X'$, $X''$, and $X'''$. Then there exists a distinguished triangle $$Rf_{\text{cris}, *}\mathcal{F} \longrightarrow Rf'_{\text{cris}, *}\mathcal{F}' \oplus Rf''_{\text{cris}, *}\mathcal{F}'' \longrightarrow Rf'''_{\text{cris}, *}\mathcal{F}''' \longrightarrow Rf_{\text{cris}, *}\mathcal{F}[1]$$ in $D(\mathcal{O}_{X'/S'})$.

Hints: This is a formal consequence of the fact that the subcategories $\text{Cris}(X'/S)$, $\text{Cris}(X''/S)$, $\text{Cris}(X'''/S)$ correspond to open subobjects of the final sheaf on $\text{Cris}(X/S)$ and that the last is the intersection of the first two.

The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4446–4470 (see updates for more information).

\begin{remark}[Mayer-Vietoris]
\label{remark-mayer-vietoris}
In the situation of Remark \ref{remark-compute-direct-image}
suppose we have an open covering $X = X' \cup X''$. Denote
$X''' = X' \cap X''$. Let $f'$, $f''$, and $f''$ be the restriction of $f$
to $X'$, $X''$, and $X'''$. Moreover, Let $\mathcal{F}'$, $\mathcal{F}''$,
and $\mathcal{F}'''$ be the restriction of $\mathcal{F}$ to the crystalline
sites of $X'$, $X''$, and $X'''$. Then there exists a distinguished triangle
$$Rf_{\text{cris}, *}\mathcal{F} \longrightarrow Rf'_{\text{cris}, *}\mathcal{F}' \oplus Rf''_{\text{cris}, *}\mathcal{F}'' \longrightarrow Rf'''_{\text{cris}, *}\mathcal{F}''' \longrightarrow Rf_{\text{cris}, *}\mathcal{F}[1]$$
in $D(\mathcal{O}_{X'/S'})$.

\medskip\noindent
Hints: This is a formal consequence of the fact that the subcategories
$\text{Cris}(X'/S)$, $\text{Cris}(X''/S)$, $\text{Cris}(X'''/S)$ correspond
to open subobjects of the final sheaf on $\text{Cris}(X/S)$ and that the
last is the intersection of the first two.
\end{remark}

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