The Stacks project

Proof. Let $L \to M \to N \to L[1]$ be a distinguished triangle in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. Then we obtain a commutative diagram

whose rows are exact by Derived Categories, Lemma 13.4.2 and Algebra, Lemma 10.8.8. Hence if the statement of the lemma holds for $N[-1]$, $L$, $N$, and $L[1]$ then it holds for $M$ by the 5-lemma. Thus, using the distinguished triangles for the canonical truncations of $L$ (see Derived Categories, Remark 13.12.4) we reduce to the case that $L$ has only one nonzero cohomology sheaf.

Choose a bounded complex $\mathcal{F}^\bullet $ of coherent $\mathcal{O}_ X$-modules and a quasi-coherent ideal $\mathcal{I} \subset \mathcal{O}_ X$ cutting out $X \setminus U$ such that $K_ n$ is represented by $\mathcal{I}^ n\mathcal{F}^\bullet $. Using “stupid” truncations we obtain compatible termwise split short exact sequences of complexes

which in turn correspond to compatible systems of distinguished triangles in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. Arguing as above we reduce to the case where $\mathcal{F}^\bullet $ has only one nonzero term. This reduces us to the case discussed in the next paragraph.

Given a coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and a coherent $\mathcal{O}_ X$-module $\mathcal{G}$ we have to show that the canonical map

is an isomorphism for all $i \geq 0$. For $i = 0$ this is Cohomology of Schemes, Lemma 30.10.5. Assume $i > 0$.

Injectivity. Let $\xi \in \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{I}^ n\mathcal{F}, \mathcal{G})$ be an element whose restriction to $U$ is zero. We have to show there exists an $m \geq n$ such that the restriction of $\xi $ to $\mathcal{I}^ m\mathcal{F} = \mathcal{I}^{m - n}\mathcal{I}^ n\mathcal{F}$ is zero. After replacing $\mathcal{F}$ by $\mathcal{I}^ n\mathcal{F}$ we may assume $n = 0$, i.e., we have $\xi \in \mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{F}, \mathcal{G})$ whose restriction to $U$ is zero. By Derived Categories of Schemes, Proposition 36.11.2 we have $D^ b_{\textit{Coh}}(\mathcal{O}_ X) = D^ b(\textit{Coh}(\mathcal{O}_ X))$. Hence we can compute the $\mathop{\mathrm{Ext}}\nolimits $ group in the abelian category of coherent $\mathcal{O}_ X$-modules. This implies there exists an surjection $\alpha : \mathcal{F}'' \to \mathcal{F}$ such that $\xi \circ \alpha = 0$ (this is where we use that $i > 0$). Set $\mathcal{F}' = \mathop{\mathrm{Ker}}(\alpha )$ so that we have a short exact sequence

It follows that $\xi $ is the image of an element $\xi ' \in \mathop{\mathrm{Ext}}\nolimits ^{i - 1}_ X(\mathcal{F}', \mathcal{G})$ whose restriction to $U$ is in the image of $\mathop{\mathrm{Ext}}\nolimits ^{i - 1}_ U(\mathcal{F}''|_ U, \mathcal{G}|_ U) \to \mathop{\mathrm{Ext}}\nolimits ^{i - 1}_ U(\mathcal{F}'|_ U, \mathcal{G}|_ U)$. By Artin-Rees the inverse systems $(\mathcal{I}^ n\mathcal{F}')$ and $(\mathcal{I}^ n \mathcal{F}'' \cap \mathcal{F}')$ are pro-isomorphic, see Cohomology of Schemes, Lemma 30.10.3. Since we have the compatible system of short exact sequences

we obtain a commutativew diagram

with exact rows. By induction on $i$ and the comment on inverse systems above we find that the left two vertical arrows are isomorphisms. Now $\xi $ gives an element in the top right group which is the image of $\xi '$ in the middle top group, which in turn maps to an element of the bottom middle group coming from some element in the left bottom group. We conclude that $\xi $ maps to zero in $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{I}^ n\mathcal{F}, \mathcal{G})$ for some $n$ as desired.

Surjectivity. Let $\xi \in \mathop{\mathrm{Ext}}\nolimits ^ i_ U(\mathcal{F}|_ U, \mathcal{G}|_ U)$. Arguing as above using that $i > 0$ we can find an surjection $\mathcal{H} \to \mathcal{F}|_ U$ of coherent $\mathcal{O}_ U$-modules such that $\xi $ maps to zero in $\mathop{\mathrm{Ext}}\nolimits ^ i_ U(\mathcal{H}, \mathcal{G}|_ U)$. Then we can find a map $\varphi : \mathcal{F}'' \to \mathcal{F}$ of coherent $\mathcal{O}_ X$-modules whose restriction to $U$ is $\mathcal{H} \to \mathcal{F}|_ U$, see Properties, Lemma 28.22.4. Observe that the lemma doesn't guarantee $\varphi $ is surjective but this won't matter (it is possible to pick a surjective $\varphi $ with a little bit of additional work). Denote $\mathcal{F}' = \mathop{\mathrm{Ker}}(\varphi )$. The short exact sequence

shows that $\xi $ is the image of $\xi '$ in $\mathop{\mathrm{Ext}}\nolimits ^{i - 1}_ U(\mathcal{F}'|_ U, \mathcal{G}|_ U)$. By induction on $i$ we can find an $n$ such that $\xi '$ is the image of some $\xi '_ n$ in $\mathop{\mathrm{Ext}}\nolimits ^{i - 1}_ X(\mathcal{I}^ n\mathcal{F}', \mathcal{G})$. By Artin-Rees we can find an $m \geq n$ such that $\mathcal{F}' \cap \mathcal{I}^ m\mathcal{F}'' \subset \mathcal{I}^ n\mathcal{F}'$. Using the short exact sequence

the image of $\xi '_ n$ in $\mathop{\mathrm{Ext}}\nolimits ^{i - 1}_ X(\mathcal{F}' \cap \mathcal{I}^ m\mathcal{F}'', \mathcal{G})$ maps by the boundary map to an element $\xi _ m$ of $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{I}^ m\mathop{\mathrm{Im}}(\varphi ), \mathcal{G})$ which maps to $\xi $. Since $\mathop{\mathrm{Im}}(\varphi )$ and $\mathcal{F}$ agree over $U$ we see that $\mathcal{F}/\mathcal{I}^ m\mathop{\mathrm{Im}}(\varphi )$ is supported on $X \setminus U$. Hence there exists an $l \geq m$ such that $\mathcal{I}^ l\mathcal{F} \subset \mathcal{I}^ m\mathop{\mathrm{Im}}(\varphi )$, see Cohomology of Schemes, Lemma 30.10.2. Taking the image of $\xi _ m$ in $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(\mathcal{I}^ l\mathcal{F}, \mathcal{G})$ we win. $\square$

Proof. Namely, if $(K_ n)$ is a Deligne system then there exists a $b \in \mathbf{Z}$ such that $H^ i(K_ n) = 0$ for $i > b$. Then $\mathop{\mathrm{Hom}}\nolimits (K_ n, L) = \mathop{\mathrm{Hom}}\nolimits (K_ n, \tau _{\leq b}L)$ and $\mathop{\mathrm{Hom}}\nolimits (K, L) = \mathop{\mathrm{Hom}}\nolimits (K, \tau _{\leq b}L)$. Hence using the result of the lemma for $\tau _{\leq b}L$ we win. $\square$

Proof. Part (1) is an immediate consequence of Lemma 48.30.1 and the fact that morphisms between pro-systems are the same as morphisms between the functors they corepresent, see Categories, Remark 4.22.7.

Let $K$ be as in (2). We can choose $K' \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ whose restriction to $U$ is isomorphic to $K$, see Derived Categories of Schemes, Lemma 36.13.2. By Derived Categories of Schemes, Proposition 36.11.2 we can represent $K'$ by a bounded complex $\mathcal{F}^\bullet $ of coherent $\mathcal{O}_ X$-modules. Choose a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ whose vanishing locus is $X \setminus U$ (for example choose $\mathcal{I}$ to correspond to the reduced induced subscheme structure on $X \setminus U$). Then we can set $K_ n$ equal to the object represented by the complex $\mathcal{I}^ n\mathcal{F}^\bullet $ as in the introduction to this section.

Part (3) is immediate from parts (1) and (2). $\square$

Proof. Let $(K_ n)$ be as in Lemma 48.30.3 part (2). Choose an object $L'$ of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ whose restriction to $U$ is $L$ (we can do this as the lemma shows). By Lemma 48.30.1 we can find an $n$ and a morphism $K_ n \to L'$ on $X$ whose restriction to $U$ is the given arrow $K \to L$. We conclude there is a morphism $K' \to L'$ of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ whose restriction to $U$ is the given arrow $K \to L$.

By Derived Categories of Schemes, Proposition 36.11.2 we can find a morphism $\alpha ^\bullet : \mathcal{F}^\bullet \to \mathcal{G}^\bullet $ of bounded complexes of coherent $\mathcal{O}_ X$-modules representing $K' \to L'$. Choose a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ whose vanishing locus is $X \setminus U$. Then we let $K_ n = \mathcal{I}^ n\mathcal{F}^\bullet $ and $L_ n = \mathcal{I}^ n\mathcal{G}^\bullet $. Observe that $\alpha ^\bullet $ induces a morphism of complexes $\alpha _ n^\bullet : \mathcal{I}^ n\mathcal{F}^\bullet \to \mathcal{I}^ n\mathcal{G}^\bullet $. From the construction of cones in Derived Categories, Section 13.9 it is clear that

and hence we can set $M_ n = C(\alpha _ n)^\bullet $. Namely, we have a compatible system of distinguished triangles (see discussion in Derived Categories, Section 13.12)

whose restriction to $U$ is isomorphic to the distinguished triangle we started out with by axiom TR3 and Derived Categories, Lemma 13.4.3. $\square$

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

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. $\square$

Proof. If $X$ is affine, translated into algebra this is More on Algebra, Lemma 15.101.1. In the general case, argue exactly as in the proof of that lemma replacing the reference to Artin-Rees in algebra with a reference to Cohomology of Schemes, Lemma 30.10.3. Details omitted. $\square$

Proof. Assume (1). To prove (2) holds we may assume $(K_ n)$ is a Deligne system. By definition we may choose a bounded complex $\mathcal{F}^\bullet $ of coherent $\mathcal{O}_ X$-modules and a quasi-coherent sheaf of ideals cutting out $X \setminus U$ such that $K_ n$ is represented by $\mathcal{I}^ n\mathcal{F}^\bullet $. Thus the result follows from Lemma 48.30.8.

Assume (2). We will prove that $(K_ n)$ is as in (1) by induction on $b - a$. If $a = b$ then (1) holds essentially by assumption. If $a < b$ then we consider the compatible system of distinguished triangles

See Derived Categories, Remark 13.12.4. By induction on $b - a$ we know that $\tau _{\leq a}K_ n$ and $\tau _{\geq a + 1}K_ n$ are pro-isomorphic to Deligne systems. We conclude by Lemma 48.30.7. $\square$

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

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. $\square$

Proof. By Lemma 48.30.10 the question is local on $X$. Thus we may assume $X$ is the spectrum of a Noetherian ring. In this case the statement follows from the algebra version which is More on Algebra, Lemma 15.101.6. $\square$