The Stacks project

Lemma 77.13.3. In Situation 77.13.1 let $K$ be as in Lemma 77.13.2. 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 Spaces, Lemma 75.25.5 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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