
21.26 Mayer-Vietoris

For the usual statement and proof of Mayer-Vietoris, please see Cohomology, Section 20.9.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Consider a commutative diagram

$\xymatrix{ E \ar[d] \ar[r] & Y \ar[d] \\ Z \ar[r] & X }$

in the category $\mathcal{C}$. In this situation, given an object $K$ of $D(\mathcal{O})$ we get what looks like the beginning of a distinguished triangle

$R\Gamma (X, K) \to R\Gamma (Z, K) \oplus R\Gamma (Y, K) \to R\Gamma (E, K)$

In the following lemma we make this more precise.

Lemma 21.26.1. In the situation above, choose a K-injective complex $\mathcal{I}^\bullet$ of $\mathcal{O}$-modules representing $K$. Using $-1$ times the canonical map for one of the four arrows we get maps of complexes

$\mathcal{I}^\bullet (X) \xrightarrow {\alpha } \mathcal{I}^\bullet (Z) \oplus \mathcal{I}^\bullet (Y) \xrightarrow {\beta } \mathcal{I}^\bullet (E)$

with $\beta \circ \alpha = 0$. Thus a canonical map

$c^ K_{X, Z, Y, E} : \mathcal{I}^\bullet (X) \longrightarrow C(\beta )^\bullet [-1]$

This map is canonical in the sense that a different choice of K-injective complex representing $K$ determines an isomorphic arrow in the derived category of abelian groups. If $c^ K_{X, Z, Y, E}$ is an isomorphism, then using its inverse we obtain a canonical distinguished triangle

$R\Gamma (X, K) \to R\Gamma (Z, K) \oplus R\Gamma (Y, K) \to R\Gamma (E, K) \to R\Gamma (X, K)[1]$

All of these constructions are functorial in $K$.

Proof. This lemma proves itself. For example, if $\mathcal{J}^\bullet$ is a second K-injective complex representing $K$, then we can choose a quasi-isomorphism $\mathcal{I}^\bullet \to \mathcal{J}^\bullet$ which determines quasi-isomorphisms between all the complexes in sight. Details omitted. For the construction of cones and the relationship with distinguished triangles see Derived Categories, Sections 13.9 and 13.10. $\square$

Lemma 21.26.2. In the situation above, let $K_1 \to K_2 \to K_3 \to K_1[1]$ be a distinguished triangle in $D(\mathcal{O})$. If $c^{K_ i}_{X, Z, Y, E}$ is a quasi-isomorphism for two $i$ out of $\{ 1, 2, 3\}$, then it is a quasi-isomorphism for the third $i$.

Proof. By rotating the triangle we may assume $c^{K_1}_{X, Z, Y, E}$ and $c^{K_2}_{X, Z, Y, E}$ are quasi-isomorphisms. Choose a map $f : \mathcal{I}^\bullet _1 \to \mathcal{I}^\bullet _2$ of K-injective complexes of $\mathcal{O}$-modules representing $K_1 \to K_2$. Then $K_3$ is represented by the K-injective complex $C(f)^\bullet$, see Derived Categories, Lemma 13.29.3. Then the morphism $c^{K_3}_{X, Z, Y, E}$ is an isomorphism as it is the third leg in a map of distinguished triangles in $K(\textit{Ab})$ whose other two legs are quasi-isomorphisms. Some details omitted; use Derived Categories, Lemma 13.4.3. $\square$

Let us give a criterion for when this does produce a distinguished triangle.

Lemma 21.26.3. In the situation above assume

1. $h_ X^\# = h_ Y^\# \amalg _{h_ E^\# } h_ Z^\#$, and

2. $h_ E^\# \to h_ Y^\#$ is injective.

Then the construction of Lemma 21.26.1 produces a distinguished triangle

$R\Gamma (X, K) \to R\Gamma (Z, K) \oplus R\Gamma (Y, K) \to R\Gamma (E, K) \to R\Gamma (X, K)[1]$

functorial for $K$ in $D(\mathcal{C})$.

Proof. We can represent $K$ by a K-injective complex whose terms are injective abelian sheaves, see Section 21.20. Thus it suffices to show: if $\mathcal{I}$ is an injective abelian sheaf, then

$0 \to \mathcal{I}(X) \to \mathcal{I}(Z) \oplus \mathcal{I}(Y) \to \mathcal{I}(E) \to 0$

is a short exact sequence. The first arrow is injective because by condition (1) the map $h_ Y \amalg h_ Z \to h_ X$ becomes surjective after sheafification, which means that $\{ Y \to X, Z \to X\}$ can be refined by a covering of $X$. The last arrow is surjective because $\mathcal{I}(Y) \to \mathcal{I}(E)$ is surjective. Namely, we have $\mathcal{I}(E) = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}_ E^\# , \mathcal{I})$, $\mathcal{I}(Y) = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}_ Y^\# , \mathcal{I})$, the map $\mathbf{Z}_ E^\# \to \mathbf{Z}_ Y^\#$ is injective by (2), and $\mathcal{I}$ is an injective abelian sheaf. Please compare with Modules on Sites, Section 18.5. Finally, suppose we have $s \in \mathcal{I}(Y)$ and $t \in \mathcal{F}(Z)$ mapping to the same element of $\mathcal{I}(E)$. Then $s$ and $t$ define a map

$s \amalg t : h_ Y^\# \amalg h_ Z^\# \longrightarrow \mathcal{I}$

which by assumption factors through $h_ Y^\# \amalg _{h_ E^\# } h_ Z^\#$. Thus by assumption (1) we obtain a unique map $h_ X^\# \to \mathcal{I}$ which corresponds to an element of $\mathcal{I}(X)$ restricting to $s$ on $Y$ and $t$ on $Z$. $\square$

Lemma 21.26.4. Let $\mathcal{C}$ be a site. Consider a commutative diagram

$\xymatrix{ \mathcal{D} \ar[r] \ar[d] & \mathcal{F} \ar[d] \\ \mathcal{E} \ar[r] & \mathcal{G} }$

of presheaves of sets on $\mathcal{C}$ and assume that

1. $\mathcal{G}^\# = \mathcal{E}^\# \amalg _{\mathcal{D}^\# } \mathcal{F}^\#$, and

2. $\mathcal{D}^\# \to \mathcal{F}^\#$ is injective.

Then there is a canonical distinguished triangle

$R\Gamma (\mathcal{G}, K) \to R\Gamma (\mathcal{E}, K) \oplus R\Gamma (\mathcal{F}, K) \to R\Gamma (\mathcal{D}, K) \to R\Gamma (\mathcal{G}, K)[1]$

functorial in $K \in D(\mathcal{C})$ where $R\Gamma (\mathcal{G}, -)$ is the cohomology discussed in Section 21.14.

Proof. Since sheafification is exact and since $R\Gamma (\mathcal{G}, -) = R\Gamma (\mathcal{G}^\# , -)$ we may assume $\mathcal{D}, \mathcal{E}, \mathcal{F}, \mathcal{G}$ are sheaves of sets. Moreover, the cohomology $R\Gamma (\mathcal{G}, -)$ only depends on the topos, not on the underlying site. Hence by Sites, Lemma 7.29.5 we may replace $\mathcal{C}$ by a “larger” site with a subcanonical topology such that $\mathcal{G} = h_ X$, $\mathcal{F} = h_ Y$, $\mathcal{E} = h_ Z$, and $\mathcal{D} = h_ E$ for some objects $X, Y, Z, E$ of $\mathcal{C}$.. In this case the result follows from Lemma 21.26.3. $\square$

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