Lemma 48.30.7 (0G4Q)—The Stacks project

Lemma 48.30.7. Let $j : U \to X$ be an open immersion of Noetherian schemes. Let

$K_ n \to L_ n \to M_ n \to K_ n[1]$

be an inverse system of distinguished triangles in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ . If $(K_ n)$ and $(M_ n)$ are pro-isomorphic to Deligne systems, then so is $(L_ n)$ .

Proof. Observe that the systems $(K_ n|_ U)$ and $(M_ n|_ U)$ are essentially constant as they are pro-isomorphic to constant systems. Denote $K$ and $M$ their values. By Derived Categories, Lemma 13.42.2 we see that the inverse system $L_ n|_ U$ is essentially constant as well. Denote $L$ its value. Let $N \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ . Consider the commutative diagram

$\xymatrix{ \ldots \ar[r] & \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ X(M_ n, N) \ar[r] \ar[d] & \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ X(L_ n, N) \ar[r] \ar[d] & \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ X(K_ n, N) \ar[r] \ar[d] & \ldots \\ \ldots \ar[r] & \mathop{\mathrm{Hom}}\nolimits _ U(M, N|_ U) \ar[r] & \mathop{\mathrm{Hom}}\nolimits _ U(L, N|_ U) \ar[r] & \mathop{\mathrm{Hom}}\nolimits _ U(K, N|_ U) \ar[r] & \ldots }$

By Lemma 48.30.1 and the fact that isomorphic ind-systems have the same colimit, we see that the vertical arrows two to the right and two to the left of the middle one are isomorphisms. By the 5-lemma we conclude that the middle vertical arrow is an isomorphism. Now, if $(L'_ n)$ is a Deligne system whose restriction to $U$ has constant value $L$ (which exists by Lemma 48.30.3), then we have $\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ X(L'_ n, N) = \mathop{\mathrm{Hom}}\nolimits _ U(L, N|_ U)$ as well. Hence the pro-systems $(L_ n)$ and $(L'_ n)$ are pro-isomorphic by Categories, Remark 4.22.7. $\square$

The Stacks project

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