# The Stacks Project

## Tag 0CRS

Lemma 66.10.5. Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. For an object $E$ of $D(\mathcal{O}_X)$ we have a distinguished triangle $$R\Gamma(X, E) \to R\Gamma(U, E) \oplus R\Gamma(V, E) \to R\Gamma(U \times_X V, E) \to R\Gamma(X, E)[1]$$ and in particular a long exact cohomology sequence $$\ldots \to H^n(X, E) \to H^n(U, E) \oplus H^n(V, E) \to H^n(U \times_X V, E) \to H^{n + 1}(X, E) \to \ldots$$ The construction of the distinguished triangle and the long exact sequence is functorial in $E$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet$ representing $E$ whose terms $\mathcal{I}^n$ are injective objects of $\textit{Mod}(\mathcal{O}_X)$, see Injectives, Theorem 19.12.6. In the proof of Lemma 66.10.2 we found a short exact sequence of complexes $$0 \to \mathcal{I}^\bullet \to j_{U, *}\mathcal{I}^\bullet|_U \oplus j_{V, *}\mathcal{I}^\bullet|_V \to j_{U \times_X V, *}\mathcal{I}^\bullet|_{U \times_X V} \to 0$$ Since $H^1(X, \mathcal{I}^n) = 0$, we see that taking global sections gives an exact sequence of complexes $$0 \to \Gamma(X, \mathcal{I}^\bullet) \to \Gamma(U, \mathcal{I}^\bullet) \oplus \Gamma(V, \mathcal{I}^\bullet) \to \Gamma(U \times_X V, \mathcal{I}^\bullet) \to 0$$ Since these complexes represent $R\Gamma(X, E)$, $R\Gamma(U, E)$, $R\Gamma(V, E)$, and $R\Gamma(U \times_X V, E)$ we get a distinguished triangle by Derived Categories, Section 13.12 and especially Lemma 13.12.1. $\square$

The code snippet corresponding to this tag is a part of the file spaces-perfect.tex and is located in lines 1817–1836 (see updates for more information).

\begin{lemma}
\label{lemma-unbounded-mayer-vietoris}
Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary
distinguished square of algebraic spaces over $S$. For an object $E$
of $D(\mathcal{O}_X)$ we have a distinguished triangle
$$R\Gamma(X, E) \to R\Gamma(U, E) \oplus R\Gamma(V, E) \to R\Gamma(U \times_X V, E) \to R\Gamma(X, E)[1]$$
and in particular a long exact cohomology sequence
$$\ldots \to H^n(X, E) \to H^n(U, E) \oplus H^n(V, E) \to H^n(U \times_X V, E) \to H^{n + 1}(X, E) \to \ldots$$
The construction of the distinguished triangle and the
long exact sequence is functorial in $E$.
\end{lemma}

\begin{proof}
Choose a K-injective complex $\mathcal{I}^\bullet$ representing $E$
whose terms $\mathcal{I}^n$ are injective objects of
$\textit{Mod}(\mathcal{O}_X)$, see Injectives, Theorem
\ref{injectives-theorem-K-injective-embedding-grothendieck}.
In the proof of Lemma \ref{lemma-exact-sequence-j-star}
we found a short exact sequence
of complexes
$$0 \to \mathcal{I}^\bullet \to j_{U, *}\mathcal{I}^\bullet|_U \oplus j_{V, *}\mathcal{I}^\bullet|_V \to j_{U \times_X V, *}\mathcal{I}^\bullet|_{U \times_X V} \to 0$$
Since $H^1(X, \mathcal{I}^n) = 0$, we see that
taking global sections gives an exact sequence of complexes
$$0 \to \Gamma(X, \mathcal{I}^\bullet) \to \Gamma(U, \mathcal{I}^\bullet) \oplus \Gamma(V, \mathcal{I}^\bullet) \to \Gamma(U \times_X V, \mathcal{I}^\bullet) \to 0$$
Since these complexes represent
$R\Gamma(X, E)$, $R\Gamma(U, E)$, $R\Gamma(V, E)$, and
$R\Gamma(U \times_X V, E)$ we
get a distinguished triangle by
Derived Categories, Section
\ref{derived-section-canonical-delta-functor} and especially
Lemma \ref{derived-lemma-derived-canonical-delta-functor}.
\end{proof}

Comment #2792 by Pieter Belmans (site) on August 31, 2017 a 9:28 am UTC

There shouldn't be a 0 in the long exact cohomology sequence, rather an $n$.

Comment #2897 by Johan (site) on October 7, 2017 a 3:19 pm UTC

THanks. Fixed here.

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