The Stacks project

Lemma 48.30.10. Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $(K_ n)$ be an inverse system in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. Let $X = W_1 \cup \ldots \cup W_ r$ be an open covering. The following are equivalent

  1. $(K_ n)$ is pro-isomorphic to a Deligne system,

  2. for each $i$ the restriction $(K_ n|_{W_ i})$ is pro-isomorphic to a Deligne system with respect to the open immersion $U \cap W_ i \to W_ i$.

Proof. By induction on $r$. If $r = 1$ then the result is clear. Assume $r > 1$. Set $V = W_1 \cup \ldots \cup W_{r - 1}$. By induction we see that $(K_ n|_ V)$ is a Deligne system. This reduces us to the discussion in the next paragraph.

Assume $X = V \cup W$ is an open covering and $(K_ n|_ W)$ and $(K_ n|_ V)$ are pro-isomorphic to Deligne systems. We have to show that $(K_ n)$ is pro-isomorphic to a Deligne system. Observe that $(K_ n|_{V \cap W})$ is pro-isomorphic to a Deligne system (it follows immediately from the construction of Deligne systems that restrictions to opens preserves them). In particular the pro-systems $(K_ n|_{U \cap V})$, $(K_ n|_{U \cap W})$, and $(K_ n|_{U \cap V \cap W})$ are essentially constant. It follows from the distinguished triangles in Cohomology, Lemma 20.33.2 and Derived Categories, Lemma 13.42.2 that $(K_ n|_ U)$ is essentially constant. Denote $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ U)$ the value of this system. Let $L$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. Consider the diagram

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Ext}}\nolimits ^{-1}(K_ n|_ V, L|_ V) \oplus \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Ext}}\nolimits ^{-1}(K_ n|_ W, L|_ W) \ar[r] \ar[d] & \mathop{\mathrm{Ext}}\nolimits ^{-1}(K|_{U \cap V}, L|_{U \cap V}) \oplus \mathop{\mathrm{Ext}}\nolimits ^{-1}(K|_{U \cap W}, L|_{U \cap W}) \ar[d] \\ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Ext}}\nolimits ^{-1}(K_ n|_{V \cap W}, L|_{V \cap W}) \ar[r] \ar[d] & \mathop{\mathrm{Ext}}\nolimits ^{-1}(K|_{U \cap V \cap W}, L|_{U \cap V \cap W}) \ar[d] \\ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (K_ n, L) \ar[d] \ar[r] & \mathop{\mathrm{Hom}}\nolimits (K|_ U, L|_ U) \ar[d] \\ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (K_ n|_ V, L|_ V) \oplus \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (K_ n|_ W, L|_ W) \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits (K|_{U \cap V}, L|_{U \cap V}) \oplus \mathop{\mathrm{Hom}}\nolimits (K|_{U \cap W}, L|_{U \cap W}) \ar[d] \\ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits (K_ n|_{V \cap W}, L|_{V \cap W}) \ar[r] & \mathop{\mathrm{Hom}}\nolimits (K|_{U \cap V \cap W}, L|_{U \cap V \cap W}) } \]

The vertical sequences are exact by Cohomology, Lemma 20.33.3 and the fact that filtered colimits are exact. All horizontal arrows except for the middle one are isomorphisms by Lemma 48.30.1 and the fact that pro-isomorphic systems have the same colimits. Hence the middle one is an isomorphism too by the 5-lemma. It follows that $(K_ n)$ is pro-isomorphic to a Deligne system for $K$. Namey, if $(K'_ n)$ is a Deligne system whose restriction to $U$ has constant value $K$ (which exists by Lemma 48.30.3), then we have $\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ X(K'_ n, L) = \mathop{\mathrm{Hom}}\nolimits _ U(K, L|_ U)$ as well. Hence the pro-systems $(K_ n)$ and $(K'_ n)$ are pro-isomorphic by Categories, Remark 4.22.7. $\square$

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