The Stacks project

Remark 20.45.5. With notation and assumptions as in Lemma 20.45.4. Suppose that $U, V \in \mathcal{B}$. Let $\mathcal{B}'$ be the set of elements of $\mathcal{B}$ contained in $U \cap V$. Then

\[ (\{ K_{U'}\} _{U' \in \mathcal{B}'}, \{ \rho _{V'}^{U'}\} _{V' \subset U'\text{ with }U', V' \in \mathcal{B}'}) \]

is a system on the ringed space $U \cap V$ satisfying the assumptions of Lemma 20.45.4. Moreover, both $(K_ U|_{U \cap V}, \rho ^ U_{U'})$ and $(K_ V|_{U \cap V}, \rho ^ V_{U'})$ are solutions to this system. By the lemma we find a unique isomorphism

\[ \rho _{U, V} : K_ U|_{U \cap V} \longrightarrow K_ V|_{U \cap V} \]

such that for every $U' \subset U \cap V$, $U' \in \mathcal{B}$ the diagram

\[ \xymatrix{ K_ U|_{U'} \ar[rr]_{\rho _{U, V}|_{U'}} \ar[rd]_{\rho ^ U_{U'}} & & K_ V|_{U'} \ar[ld]^{\rho ^ V_{U'}} \\ & K_{U'} } \]

commutes. Pick a third element $W \in \mathcal{B}$. We obtain isomorphisms $\rho _{U, W} : K_ U|_{U \cap W} \to K_ W|_{U \cap W}$ and $\rho _{V, W} : K_ U|_{V \cap W} \to K_ W|_{V \cap W}$ satisfying similar properties to those of $\rho _{U, V}$. Finally, we have

\[ \rho _{U, W}|_{U \cap V \cap W} = \rho _{V, W}|_{U \cap V \cap W} \circ \rho _{U, V}|_{U \cap V \cap W} \]

This is true by the uniqueness in the lemma because both sides of the equality are the unique isomorphism compatible with the maps $\rho ^ U_{U''}$ and $\rho ^ W_{U''}$ for $U'' \subset U \cap V \cap W$, $U'' \in \mathcal{B}$. Some minor details omitted. The collection $(K_ U, \rho _{U, V})$ is a descent datum in the derived category for the open covering $\mathcal{U} : X = \bigcup _{U \in \mathcal{B}} U$ of $X$. In this language we are looking for “effectiveness of the descent datum” when we look for the existence of a solution.

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