Lemma 15.91.12. Let A be a ring and let I \subset A be a finitely generated ideal. Let K^\bullet be a complex of A-modules such that f : K^\bullet \to K^\bullet is an isomorphism for some f \in I, i.e., K^\bullet is a complex of A_ f-modules. Then the derived completion of K^\bullet is zero.
Proof. Indeed, in this case the R\mathop{\mathrm{Hom}}\nolimits _ A(K, L) is zero for any derived complete complex L, see Lemma 15.91.1. Hence K^\wedge is zero by the universal property in Lemma 15.91.10. \square
Comments (0)
There are also: