The Stacks project

72.10 Mayer-Vietoris

In this section we prove that an elementary distinguished triangle gives rise to various Mayer-Vietoris sequences.

Let $S$ be a scheme. Let $U \to X$ be an étale morphism of algebraic spaces over $S$. In Properties of Spaces, Section 63.27 it was shown that $U_{spaces, {\acute{e}tale}} = X_{spaces, {\acute{e}tale}}/U$ compatible with structure sheaves. Hence in this situation we often think of the morphism $j_ U : U \to X$ as a localization morphism (see Modules on Sites, Definition 18.19.1). In particular we think of pullback $j_ U^*$ as restriction to $U$ and we often denote it by ${}|_ U$; this is compatible with Properties of Spaces, Equation (63.26.1.1). In particular we see that

72.10.0.1
\begin{equation} \label{spaces-perfect-equation-stalk-restriction} (\mathcal{F}|_ U)_{\overline{u}} = \mathcal{F}_{\overline{x}} \end{equation}

if $\overline{u}$ is a geometric point of $U$ and $\overline{x}$ the image of $\overline{u}$ in $X$. Moreover, restriction has an exact left adjoint $j_{U!}$, see Modules on Sites, Lemmas 18.19.2 and 18.19.3. Finally, recall that if $\mathcal{G}$ is an $\mathcal{O}_ X$-module, then

72.10.0.2
\begin{equation} \label{spaces-perfect-equation-stalk-j-shriek} (j_{U!}\mathcal{G})_{\overline{x}} = \bigoplus \nolimits _{\overline{u}} \mathcal{G}_{\overline{u}} \end{equation}

for any geometric point $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$ where the direct sum is over those morphisms $\overline{u} : \mathop{\mathrm{Spec}}(k) \to U$ such that $j_ U \circ \overline{u} = \overline{x}$, see Modules on Sites, Lemma 18.38.1 and Properties of Spaces, Lemma 63.19.13.

Lemma 72.10.1. Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$.

  1. For a sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ we have a short exact sequence

    \[ 0 \to j_{U \times _ X V!}\mathcal{F}|_{U \times _ X V} \to j_{U!}\mathcal{F}|_ U \oplus j_{V!}\mathcal{F}|_ V \to \mathcal{F} \to 0 \]
  2. For an object $E$ of $D(\mathcal{O}_ X)$ we have a distinguished triangle

    \[ j_{U \times _ X V!}E|_{U \times _ X V} \to j_{U!}E|_ U \oplus j_{V!}E|_ V \to E \to j_{U \times _ X V!}E|_{U \times _ X V}[1] \]

    in $D(\mathcal{O}_ X)$.

Proof. To show the sequence of (1) is exact we may check on stalks at geometric points by Properties of Spaces, Theorem 63.19.12. Let $\overline{x}$ be a geometric point of $X$. By Equations (72.10.0.1) and (72.10.0.2) taking stalks at $\overline{x}$ we obtain the sequence

\[ 0 \to \bigoplus \nolimits _{(\overline{u}, \overline{v})} \mathcal{F}_{\overline{x}} \to \bigoplus \nolimits _{\overline{u}} \mathcal{F}_{\overline{x}} \oplus \bigoplus \nolimits _{\overline{v}} \mathcal{F}_{\overline{x}} \to \mathcal{F}_{\overline{x}} \to 0 \]

This sequence is exact because for every $\overline{x}$ there either is exactly one $\overline{u}$ mapping to $\overline{x}$, or there is no $\overline{u}$ and exactly one $\overline{v}$ mapping to $\overline{x}$.

Proof of (2). We have seen in Cohomology on Sites, Section 21.20 that the restriction functors and the extension by zero functors on derived categories are computed by just applying the functor to any complex. Let $\mathcal{E}^\bullet $ be a complex of $\mathcal{O}_ X$-modules representing $E$. The distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section 13.12 and especially Lemma 13.12.1) to the short exact sequence of complexes of $\mathcal{O}_ X$-modules

\[ 0 \to j_{U \times _ X V!}\mathcal{E}^\bullet |_{U \times _ X V} \to j_{U!}\mathcal{E}^\bullet |_ U \oplus j_{V!}\mathcal{E}^\bullet |_ V \to \mathcal{E}^\bullet \to 0 \]

which is short exact by (1). $\square$

Lemma 72.10.2. Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$.

  1. For every sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ we have a short exact sequence

    \[ 0 \to \mathcal{F} \to j_{U, *}\mathcal{F}|_ U \oplus j_{V, *}\mathcal{F}|_ V \to j_{U \times _ X V, *}\mathcal{F}|_{U \times _ X V} \to 0 \]
  2. For any object $E$ of $D(\mathcal{O}_ X)$ we have a distinguished triangle

    \[ E \to Rj_{U, *}E|_ U \oplus Rj_{V, *}E|_ V \to Rj_{U \times _ X V, *}E|_{U \times _ X V} \to E[1] \]

    in $D(\mathcal{O}_ X)$.

Proof. Let $W$ be an object of $X_{\acute{e}tale}$. We claim the sequence

\[ 0 \to \mathcal{F}(W) \to \mathcal{F}(W \times _ X U) \oplus \mathcal{F}(W \times _ X V) \to \mathcal{F}(W \times _ X U \times _ X V) \]

is exact and that an element of the last group can locally on $W$ be lifted to the middle one. By Lemma 72.9.2 the pair $(W \times _ X U \subset W, V \times _ X W \to W)$ is an elementary distinguished square. Thus we may assume $W = X$ and it suffices to prove the same thing for

\[ 0 \to \mathcal{F}(X) \to \mathcal{F}(U) \oplus \mathcal{F}(V) \to \mathcal{F}(U \times _ X V) \]

We have seen that

\[ 0 \to j_{U \times _ X V!}\mathcal{O}_{U \times _ X V} \to j_{U!}\mathcal{O}_ U \oplus j_{V!}\mathcal{O}_ V \to \mathcal{O}_ X \to 0 \]

is a exact sequence of $\mathcal{O}_ X$-modules in Lemma 72.10.1 and applying the right exact functor $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(- , \mathcal{F})$ gives the sequence above. This also means that the obstruction to lifting $s \in \mathcal{F}(U \times _ X V)$ to an element of $\mathcal{F}(U) \oplus \mathcal{F}(V)$ lies in $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathcal{O}_ X, \mathcal{F}) = H^1(X, \mathcal{F})$. By locality of cohomology (Cohomology on Sites, Lemma 21.7.3) this obstruction vanishes étale locally on $X$ and the proof of (1) is complete.

Proof of (2). 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. Then $\mathcal{I}^\bullet |U$ is a K-injective complex (Cohomology on Sites, Lemma 21.20.1). Hence $Rj_{U, *}E|_ U$ is represented by $j_{U, *}\mathcal{I}^\bullet |_ U$. Similarly for $V$ and $U \times _ X V$. Hence the distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section 13.12 and especially Lemma 13.12.1) to the short exact sequence of complexes

\[ 0 \to \mathcal{I}^\bullet \to j_{U, *}\mathcal{I}^\bullet |_ U \oplus j_{V, *}\mathcal{I}^\bullet |_ V \to j_{U \times _ X V, *}\mathcal{I}^\bullet |_{U \times _ X V} \to 0. \]

This sequence is exact by (1). $\square$

Lemma 72.10.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $(U \subset X, V \to X)$ be an elementary distinguished square. Denote $a = f|_ U : U \to Y$, $b = f|_ V : V \to Y$, and $c = f|_{U \times _ X V} : U \times _ X V \to Y$ the restrictions. For every object $E$ of $D(\mathcal{O}_ X)$ there exists a distinguished triangle

\[ Rf_*E \to Ra_*(E|_ U) \oplus Rb_*(E|_ V) \to Rc_*(E|_{U \times _ X V}) \to Rf_*E[1] \]

in $D(\mathcal{O}_ Y)$. This triangle is functorial in $E$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $E$. We may assume $\mathcal{I}^ n$ is an injective object of $\textit{Mod}(\mathcal{O}_ X)$ for all $n$, see Injectives, Theorem 19.12.6. Then $Rf_*E$ is computed by $f_*\mathcal{I}^\bullet $. Similarly for $U$, $V$, and $U \cap V$ by Cohomology on Sites, Lemma 21.20.1. Hence the distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section 13.12 and especially Lemma 13.12.1) to the short exact sequence of complexes

\[ 0 \to f_*\mathcal{I}^\bullet \to a_*\mathcal{I}^\bullet |_ U \oplus b_*\mathcal{I}^\bullet |_ V \to c_*\mathcal{I}^\bullet |_{U \times _ X V} \to 0. \]

To see this is a short exact sequence of complexes we argue as follows. Pick an injective object $\mathcal{I}$ of $\textit{Mod}(\mathcal{O}_ X)$. Apply $f_*$ to the short exact sequence

\[ 0 \to \mathcal{I} \to j_{U, *}\mathcal{I}|_ U \oplus j_{V, *}\mathcal{I}|_ V \to j_{U \times _ X V, *}\mathcal{I}|_{U \times _ X V} \to 0 \]

of Lemma 72.10.2 and use that $R^1f_*\mathcal{I} = 0$ to get a short exact sequence

\[ 0 \to f_*\mathcal{I} \to f_*j_{U, *}\mathcal{I}|_ U \oplus f_*j_{V, *}\mathcal{I}|_ V \to f_*j_{U \times _ X V, *}\mathcal{I}|_{U \times _ X V} \to 0 \]

The proof is finished by observing that $a_* = f_*j_{U, *}$ and similarly for $b_*$ and $c_*$. $\square$

Lemma 72.10.4. Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. For objects $E$, $F$ of $D(\mathcal{O}_ X)$ we have a Mayer-Vietoris sequence

\[ \xymatrix{ & \ldots \ar[r] & \mathop{\mathrm{Ext}}\nolimits ^{-1}(E_{U \times _ X V}, F_{U \times _ X V}) \ar[lld] \\ \mathop{\mathrm{Hom}}\nolimits (E, F) \ar[r] & \mathop{\mathrm{Hom}}\nolimits (E_ U, F_ U) \oplus \mathop{\mathrm{Hom}}\nolimits (E_ V, F_ V) \ar[r] & \mathop{\mathrm{Hom}}\nolimits (E_{U \times _ X V}, F_{U \times _ X V}) } \]

where the subscripts denote restrictions to the relevant opens and the $\mathop{\mathrm{Hom}}\nolimits $'s are taken in the relevant derived categories.

Proof. Use the distinguished triangle of Lemma 72.10.1 to obtain a long exact sequence of $\mathop{\mathrm{Hom}}\nolimits $'s (from Derived Categories, Lemma 13.4.2) and use that $\mathop{\mathrm{Hom}}\nolimits (j_{U!}E|_ U, F) = \mathop{\mathrm{Hom}}\nolimits (E|_ U, F|_ U)$ by Cohomology on Sites, Lemma 21.20.8. $\square$

Lemma 72.10.5. 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 72.10.2 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. $\square$

Lemma 72.10.6. Let $S$ be a scheme. Let $j : U \to X$ be a étale morphism of algebraic spaces over $S$. Given an étale morphism $V \to Y$, set $W = V \times _ X U$ and denote $j_ W : W \to V$ the projection morphism. Then $(j_!E)|_ V = j_{W!}(E|_ W)$ for $E$ in $D(\mathcal{O}_ U)$.

Proof. This is true because $(j_!\mathcal{F})|_ V = j_{W!}(\mathcal{F}|_ W)$ for an $\mathcal{O}_ X$-module $\mathcal{F}$ as follows immediately from the construction of the functors $j_!$ and $j_{W!}$, see Modules on Sites, Lemma 18.19.2. $\square$

Lemma 72.10.7. Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Set $T = |X| \setminus |U|$.

  1. If $E$ is an object of $D(\mathcal{O}_ X)$ supported on $T$, then (a) $E \to Rj_*(E|_ V)$ and (b) $j_!(E|_ V) \to E$ are isomorphisms.

  2. If $F$ is an object of $D(\mathcal{O}_ V)$ supported on $j^{-1}T$, then (a) $F \to (j_!F)|_ V$, (b) $(Rj_*F)|_ V \to F$, and (c) $j_!F \to Rj_*F$ are isomorphisms.

Proof. Let $E$ be an object of $D(\mathcal{O}_ X)$ whose cohomology sheaves are supported on $T$. Then we see that $E|_ U = 0$ and $E|_{U \times _ X V} = 0$ as $T$ doesn't meet $U$ and $j^{-1}T$ doesn't meet $U \times _ X V$. Thus (1)(a) follows from Lemma 72.10.2. In exactly the same way (1)(b) follows from Lemma 72.10.1.

Let $F$ be an object of $D(\mathcal{O}_ V)$ whose cohomology sheaves are supported on $j^{-1}T$. By Lemma 72.3.1 we have $(Rj_*F)|_ U = Rj_{W, *}(F|_ W) = 0$ because $F|_ W = 0$ by our assumption. Similarly $(j_!F)|_ U = j_{W!}(F|_ W) = 0$ by Lemma 72.10.6. Thus $j_!F$ and $Rj_*F$ are supported on $T$ and $(j_!F)|_ V$ and $(Rj_*F)|_ V$ are supported on $j^{-1}(T)$. To check that the maps (2)(a), (b), (c) are isomorphisms in the derived category, it suffices to check that these map induce isomorphisms on stalks of cohomology sheaves at geometric points of $T$ and $j^{-1}(T)$ by Properties of Spaces, Theorem 63.19.12. This we may do after replacing $X$ by $V$, $U$ by $U \times _ X V$, $V$ by $V \times _ X V$ and $F$ by $F|_{V \times _ X V}$ (restriction via first projection), see Lemmas 72.3.1, 72.10.6, and 72.9.2. Since $V \times _ X V \to V$ has a section this reduces (2) to the case that $j : V \to X$ has a section.

Assume $j$ has a section $\sigma : X \to V$. Set $V' = \sigma (X)$. This is an open subspace of $V$. Set $U' = j^{-1}(U)$. This is another open subspace of $V$. Then $(U' \subset V, V' \to V)$ is an elementary distinguished square. Observe that $F|_{U'} = 0$ and $F|_{V' \cap U'} = 0$ because $F$ is supported on $j^{-1}(T)$. Denote $j' : V' \to V$ the open immersion and $j_{V'} : V' \to X$ the composition $V' \to V \to X$ which is the inverse of $\sigma $. Set $F' = \sigma ^*F$. The distinguished triangles of Lemmas 72.10.1 and 72.10.2 show that $F = j'_!(F|_{V'})$ and $F = Rj'_*(F|_{V'})$. It follows that $j_!F = j_!j'_!(F|_{V'}) = j_{V'!}F = F'$ because $j_{V'} : V' \to X$ is an isomorphism and the inverse of $\sigma $. Similarly, $Rj_*F = Rj_*Rj'_*F = Rj_{V', *}F = F'$. This proves (2)(c). To prove (2)(a) and (2)(b) it suffices to show that $F = F'|_ V$. This is clear because both $F$ and $F'|_ V$ restrict to zero on $U'$ and $U' \cap V'$ and the same object on $V'$. $\square$

We can glue complexes!

Lemma 72.10.8. Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$. Suppose given

  1. an object $A$ of $D(\mathcal{O}_ U)$,

  2. an object $B$ of $D(\mathcal{O}_ V)$, and

  3. an isomorphism $c : A|_{U \times _ X V} \to B|_{U \times _ X V}$.

Then there exists an object $F$ of $D(\mathcal{O}_ X)$ and isomorphisms $f : F|_ U \to A$, $g : F|_ V \to B$ such that $c = g|_{U \times _ X V} \circ f^{-1}|_{U \times _ X V}$. Moreover, given

  1. an object $E$ of $D(\mathcal{O}_ X)$,

  2. a morphism $a : A \to E|_ U$ of $D(\mathcal{O}_ U)$,

  3. a morphism $b : B \to E|_ V$ of $D(\mathcal{O}_ V)$,

such that

\[ a|_{U \times _ X V} = b|_{U \times _ X V} \circ c. \]

Then there exists a morphism $F \to E$ in $D(\mathcal{O}_ X)$ whose restriction to $U$ is $a \circ f$ and whose restriction to $V$ is $b \circ g$.

Proof. Denote $j_ U$, $j_ V$, $j_{U \times _ X V}$ the corresponding morphisms towards $X$. Choose a distinguished triangle

\[ F \to Rj_{U, *}A \oplus Rj_{V, *}B \to Rj_{U \times _ X V, *}(B|_{U \times _ X V}) \to F[1] \]

Here the map $Rj_{V, *}B \to Rj_{U \times _ X V, *}(B|_{U \times _ X V})$ is the obvious one. The map $Rj_{U, *}A \to Rj_{U \times _ X V, *}(B|_{U \times _ X V})$ is the composition of $Rj_{U, *}A \to Rj_{U \times _ X V, *}(A|_{U \times _ X V})$ with $Rj_{U \times _ X V, *}c$. Restricting to $U$ we obtain

\[ F|_ U \to A \oplus (Rj_{V, *}B)|_ U \to (Rj_{U \times _ X V, *}(B|_{U \times _ X V}))|_ U \to F|_ U[1] \]

Denote $j : U \times _ X V \to U$. Compatibility of restriction and total direct image (Lemma 72.3.1) shows that both $(Rj_{V, *}B)|_ U$ and $(Rj_{U \times _ X V, *}(B|_{U \times _ X V}))|_ U$ are canonically isomorphic to $Rj_*(B|_{U \times _ X V})$. Hence the second arrow of the last displayed equation has a section, and we conclude that the morphism $F|_ U \to A$ is an isomorphism.

To see that the morphism $F|_ V \to B$ is an isomorphism we will use a trick. Namely, choose a distinguished triangle

\[ F|_ V \to B \to B' \to F[1]|_ V \]

in $D(\mathcal{O}_ V)$. Since $F|_ U \to A$ is an isomorphism, and since we have the isomorphism $c : A|_{U \times _ X V} \to B|_{U \times _ X V}$ the restriction of $F|_ V \to B$ is an isomorphism over $U \times _ X V$. Thus $B'$ is supported on $j_ V^{-1}(T)$ where $T = |X| \setminus |U|$. On the other hand, there is a morphism of distinguished triangles

\[ \xymatrix{ F \ar[r] \ar[d] & Rj_{U, *}F|_ U \oplus Rj_{V, *}F|_ V \ar[r] \ar[d] & Rj_{U \times _ X V, *}F|_{U \times _ X V} \ar[r] \ar[d] & F[1] \ar[d] \\ F \ar[r] & Rj_{U, *}A \oplus Rj_{V, *}B \ar[r] & Rj_{U \times _ X V, *}(B|_{U \times _ X V}) \ar[r] & F[1] } \]

The all of the vertical maps in this diagram are isomorphisms, except for the map $Rj_{V, *}F|_ V \to Rj_{V, *}B$, hence that is an isomorphism too (Derived Categories, Lemma 13.4.3). This implies that $Rj_{V, *}B' = 0$. Hence $B' = 0$ by Lemma 72.10.7.

The existence of the morphism $F \to E$ follows from the Mayer-Vietoris sequence for $\mathop{\mathrm{Hom}}\nolimits $, see Lemma 72.10.4. $\square$


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