The Stacks project

Proof. By Cohomology, Lemma 20.55.7 the complex $\mathcal{Q}^\bullet = \eta _\mathcal {I}\mathcal{M}^\bullet $ represents $L\eta _\mathcal {I}M$. To check that this complex is locally bounded and consists of finite locally free, we may work affine locally. Then the boundedness is clear. Choose an affine chart $(U, A, f, M^\bullet )$ for $(X, D, M)$ such that the ideals $I_ i(M^\bullet , f)$ are principal and such that $\mathcal{M}^ i|_ U$ is finite free for each $i$. By our assumption that $(X, D, M)$ is a good triple we can do this. Writing $N^ i = \Gamma (U, \mathcal{M}^ i|_ U)$ we get a bounded complex $N^\bullet $ of finite free $A$-modules representing the same object in $D(A)$ as the complex $M^\bullet $ (by Derived Categories of Schemes, Lemma 36.3.5). Then $I_ i(N^\bullet , f)$ is a principal ideal for all $i$ by More on Algebra, Lemma 15.96.1. Hence the complex $\eta _ fN^\bullet $ is a bounded complex of finite locally free $A$-modules. Since $\mathcal{Q}^ i|_ U$ is the quasi-coherent $\mathcal{O}_ U$-module corresponding to $\eta _ fN^ i$ by Lemma 38.42.1 we conclude. $\square$