Lemma 52.6.6. Let \mathcal{C} be a site. Let \mathcal{O} \to \mathcal{O}' be a homomorphism of sheaves of rings. Let \mathcal{I} \subset \mathcal{O} be a sheaf of ideals. The inverse image of D_{comp}(\mathcal{O}, \mathcal{I}) under the restriction functor D(\mathcal{O}') \to D(\mathcal{O}) is D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}').
Proof. Using Lemma 52.6.3 we see that K' \in D(\mathcal{O}') is in D_{comp}(\mathcal{O}', \mathcal{I}\mathcal{O}') if and only if T(K'|_ U, f) is zero for every local section f \in \mathcal{I}(U). Observe that the cohomology sheaves of T(K'|_ U, f) are computed in the category of abelian sheaves, so it doesn't matter whether we think of f as a section of \mathcal{O} or take the image of f as a section of \mathcal{O}'. The lemma follows immediately from this and the definition of derived complete objects. \square
Comments (0)
There are also: