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The Stacks project

Lemma 38.29.3. In Situation 38.29.1 let K be as in Lemma 38.29.2. For any perfect object E of D(\mathcal{O}_ X) the cohomology

M = R\Gamma (X, K \otimes ^\mathbf {L} E)

is a pseudo-coherent object of D(A) and there is a canonical isomorphism

R\Gamma (X_ n, K_ n \otimes ^\mathbf {L} E|_{X_ n}) = M \otimes _ A^\mathbf {L} A_ n

in D(A_ n). Here E|_{X_ n} denotes the derived pullback of E to X_ n.

Proof. Write E_ n = E|_{X_ n} and M_ n = R\Gamma (X_ n, K_ n \otimes ^\mathbf {L} E|_{X_ n}). By Derived Categories of Schemes, Lemma 36.30.5 we see that M_ n is a pseudo-coherent object of D(A_ n) whose formation commutes with base change. Thus the maps M_ n \otimes _{A_ n}^\mathbf {L} A_{n - 1} \to M_{n - 1} coming from \varphi _ n are isomorphisms. By More on Algebra, Lemma 15.97.1 we find that R\mathop{\mathrm{lim}}\nolimits M_ n is pseudo-coherent and that its base change back to A_ n recovers M_ n. On the other hand, the exact functor R\Gamma (X, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(A) of triangulated categories commutes with products and hence with derived limits, whence

R\Gamma (X, E \otimes ^\mathbf {L} K) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, E \otimes ^\mathbf {L} K_ n) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X_ n, E_ n \otimes ^\mathbf {L} K_ n) = R\mathop{\mathrm{lim}}\nolimits M_ n

as desired. \square


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