The Stacks project

Lemma 20.25.4. In the situation above, assume Čech cohomology agrees with cohomology for the sheaves $\mathcal{F}_ i^ p$ and $\mathcal{G}_ j^ q$. Let $a_3 \in H^ n(X, \mathcal{F}_3^\bullet )$ and $b_1 \in H^ m(X, \mathcal{G}_1^\bullet )$. Then we have

\[ \gamma _1( \partial a_3 \cup b_1) = (-1)^{n + 1} \gamma _3( a_3 \cup \partial b_1) \]

in $H^{n + m}(X, \mathcal{H}^\bullet )$ where $\partial $ indicates the boundary map on cohomology associated to the short exact sequences of complexes above.

Proof. We will use the following conventions and notation. We think of ${\mathcal F}_1^ p$ as a subsheaf of ${\mathcal F}_2^ p$ and we think of ${\mathcal G}_3^ q$ as a subsheaf of ${\mathcal G}_2^ q$. Hence if $s$ is a local section of ${\mathcal F}_1^ p$ we use $s$ to denote the corresponding section of ${\mathcal F}_2^ p$ as well. Similarly for local sections of ${\mathcal G}_3^ q$. Furthermore, if $s$ is a local section of ${\mathcal F}_2^ p$ then we denote $\bar s$ its image in ${\mathcal F}_3^ p$. Similarly for the map ${\mathcal G}_2^ q \to {\mathcal G}^ q_1$. In particular if $s$ is a local section of ${\mathcal F}_2^ p$ and $\bar s = 0$ then $s$ is a local section of ${\mathcal F}_1^ p$. The commutativity of the diagrams above implies, for local sections $s$ of ${\mathcal F}_2^ p$ and $t$ of ${\mathcal G}_3^ q$ that $\gamma _2(s \otimes t) = \gamma _3(\bar s \otimes t)$ as sections of ${\mathcal H}^{p + q}$.

Let ${\mathcal U} : X = \bigcup _{i \in I} U_ i$ be an open covering of $X$. Suppose that $\alpha _3$, resp. $\beta _1$ is a degree $n$, resp. $m$ cocycle of $\text{Tot}( \check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal F}_3^\bullet ))$, resp. $\text{Tot}( \check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal G}_1^\bullet ))$ representing $a_3$, resp. $b_1$. After refining $\mathcal{U}$ if necessary, we can find cochains $\alpha _2$, resp. $\beta _2$ of degree $n$, resp. $m$ in $\text{Tot}( \check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal F}_2^\bullet ))$, resp. $\text{Tot}( \check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal G}_2^\bullet ))$ mapping to $\alpha _3$, resp. $\beta _1$. Then we see that

\[ \overline{d(\alpha _2)} = d(\bar\alpha _2) = 0 \quad \text{and}\quad \overline{d(\beta _2)} = d(\bar\beta _2) = 0. \]

This means that $\alpha _1 = d(\alpha _2)$ is a degree $n + 1$ cocycle in $\text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal F}_1^\bullet ))$ representing $\partial a_3$. Similarly, $\beta _3 = d(\beta _2)$ is a degree $m + 1$ cocycle in $\text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal G}_3^\bullet ))$ representing $\partial b_1$. Thus we may compute

\begin{align*} d(\gamma _2(\alpha _2 \cup \beta _2)) & = \gamma _2(d(\alpha _2 \cup \beta _2)) \\ & = \gamma _2(d(\alpha _2) \cup \beta _2 + (-1)^ n \alpha _2 \cup d(\beta _2) ) \\ & = \gamma _2( \alpha _1 \cup \beta _2) + (-1)^ n \gamma _2( \alpha _2 \cup \beta _3) \\ & = \gamma _1(\alpha _1 \cup \beta _1) + (-1)^ n \gamma _3(\alpha _3 \cup \beta _3) \end{align*}

So this even tells us that the sign is $(-1)^{n + 1}$ as indicated in the lemma1. $\square$

[1] The sign depends on the convention for the signs in the long exact sequence in cohomology associated to a triangle in $D(X)$. The conventions in the Stacks project are (a) distinguished triangles correspond to termwise split exact sequences and (b) the boundary maps in the long exact sequence are given by the maps in the snake lemma without the intervention of signs. See Derived Categories, Section 13.10.

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