Lemma 91.16.5 (0DIX)—The Stacks project

Table of contents
Part 6: Deformation Theory
Chapter 91: Deformation Theory
Section 91.16: Deformations of complexes on ringed topoi
Lemma 91.16.5 (cite)

Lemma 91.16.5. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings. Assume given the following data

a complex of $\mathcal{O}$-modules $\mathcal{F}^\bullet $,
a complex $\mathcal{K}_0^\bullet $ of $\mathcal{O}_0$-modules,
a quasi-isomorphism $\mathcal{K}_0^\bullet \to \mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{O}_0$,

Then there exist a quasi-isomorphism $\mathcal{G}^\bullet \to \mathcal{F}^\bullet $ such that the map of complexes $\mathcal{G}^\bullet \otimes _\mathcal {O} \mathcal{O}_0 \to \mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{O}_0$ factors through $\mathcal{K}_0^\bullet $ in the homotopy category of complexes of $\mathcal{O}_0$-modules.

Proof. Set $\mathcal{F}_0^\bullet = \mathcal{F}^\bullet \otimes _\mathcal {O} \mathcal{O}_0$. By Derived Categories, Lemma 13.9.8 there exists a factorization

\[ \mathcal{K}_0^\bullet \to \mathcal{L}_0^\bullet \to \mathcal{F}_0^\bullet \]

of the given map such that the first arrow has an inverse up to homotopy and the second arrow is termwise split surjective. Hence we may assume that $\mathcal{K}_0^\bullet \to \mathcal{F}_0^\bullet $ is termwise surjective. In that case we take

\[ \mathcal{G}^ n = \mathcal{F}^ n \times _{\mathcal{F}^ n_0} \mathcal{K}_0^ n \]

and everything is clear. $\square$

The Stacks project

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