The Stacks project

Lemma 52.7.2. Let $X$ be a locally Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $K$ be an object of $D(\mathcal{O}_ X)$. Then

  1. the derived completion $K^\wedge $ is equal to $R\mathop{\mathrm{lim}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X/\mathcal{I}^ n)$.

Let $K$ is a pseudo-coherent object of $D(\mathcal{O}_ X)$. Then

  1. the cohomology sheaf $H^ q(K^\wedge )$ is equal to $\mathop{\mathrm{lim}}\nolimits H^ q(K)/\mathcal{I}^ nH^ q(K)$.

Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module1. Then

  1. the derived completion $\mathcal{F}^\wedge $ is equal to $\mathop{\mathrm{lim}}\nolimits \mathcal{F}/\mathcal{I}^ n\mathcal{F}$,

  2. $\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n \mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n \mathcal{F}$,

  3. $H^ p(U, \mathcal{F}^\wedge ) = 0$ for $p \not= 0$ for all affine opens $U \subset X$.

Proof. Proof of (1). There is a canonical map

\[ K \longrightarrow R\mathop{\mathrm{lim}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X/\mathcal{I}^ n), \]

see Remark 52.6.13. Derived completion commutes with passing to open subschemes (Remark 52.6.14). Formation of $R\mathop{\mathrm{lim}}\nolimits $ commutes with passing to open subschemes. It follows that to check our map is an isomorphism, we may work locally. Thus we may assume $X = U = \mathop{\mathrm{Spec}}(A)$. Say $I = (f_1, \ldots , f_ r)$. Let $K_ n = K(A, f_1^ n, \ldots , f_ r^ n)$ be the Koszul complex. By More on Algebra, Lemma 15.94.1 we have seen that the pro-systems $\{ K_ n\} $ and $\{ A/I^ n\} $ of $D(A)$ are isomorphic. Using the equivalence $D(A) = D_{\mathit{QCoh}}(\mathcal{O}_ X)$ of Derived Categories of Schemes, Lemma 36.3.5 we see that the pro-systems $\{ K(\mathcal{O}_ X, f_1^ n, \ldots , f_ r^ n)\} $ and $\{ \mathcal{O}_ X/\mathcal{I}^ n\} $ are isomorphic in $D(\mathcal{O}_ X)$. This proves the second equality in

\[ K^\wedge = R\mathop{\mathrm{lim}}\nolimits \left( K \otimes _{\mathcal{O}_ X}^\mathbf {L} K(\mathcal{O}_ X, f_1^ n, \ldots , f_ r^ n) \right) = R\mathop{\mathrm{lim}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X/\mathcal{I}^ n) \]

The first equality is Lemma 52.6.9.

Assume $K$ is pseudo-coherent. For $U \subset X$ affine open we have $H^ q(U, K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ q(U, K)/\mathcal{I}^ n(U)H^ q(U, K)$ by Lemma 52.7.1. As this is true for every $U$ we see that $H^ q(K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ q(K)/\mathcal{I}^ nH^ q(K)$ as sheaves. This proves (2).

Part (3) is a special case of (2). Parts (4) and (5) follow from Derived Categories of Schemes, Lemma 36.3.2. $\square$

[1] For example $H^ q(K)$ for $K$ pseudo-coherent on our locally Noetherian $X$.

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