Lemma 21.26.4 (0EVZ)—The Stacks project

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

$\mathcal{G}^\# = \mathcal{E}^\# \amalg _{\mathcal{D}^\# } \mathcal{F}^\# $, and
$\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 21.13.

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. $\square$

The Stacks project

Comments (0)

Post a comment