Lemma 51.5.9. Let $A$ be a Noetherian ring. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. Assume $A$ has finite dimension. Then
\[ R\Gamma _ T(K) = R\Gamma _ T(A) \otimes _ A^\mathbf {L} K \]
for $K \in D(A)$. For $K, L \in D(A)$ we have
\[ R\Gamma _ T(K \otimes _ A^\mathbf {L} L) = K \otimes _ A^\mathbf {L} R\Gamma _ T(L) = R\Gamma _ T(K) \otimes _ A^\mathbf {L} L = R\Gamma _ T(K) \otimes _ A^\mathbf {L} R\Gamma _ T(L) \]
If $K$ or $L$ is in $D_ T(A)$ then so is $K \otimes _ A^\mathbf {L} L$.
Proof.
By construction we may represent $R\Gamma _ T(A)$ by a complex $J^\bullet $ in $\text{Mod}_{A, T}$. Thus if we represent $K$ by a K-flat complex $K^\bullet $ then we see that $R\Gamma _ T(A) \otimes _ A^\mathbf {L} K$ is represented by the complex $\text{Tot}(J^\bullet \otimes _ A K^\bullet )$ in $\text{Mod}_{A, T}$. Using the map $R\Gamma _ T(A) \to A$ we obtain a map $R\Gamma _ T(A) \otimes _ A^\mathbf {L} K\to K$. Thus by the adjointness property of $R\Gamma _ T$ we obtain a canonical map
\[ R\Gamma _ T(A) \otimes _ A^\mathbf {L} K \longrightarrow R\Gamma _ T(K) \]
factoring the just constructed map. Observe that $R\Gamma _ T$ commutes with direct sums in $D(A)$ for example by Lemma 51.5.3, the fact that directed colimits commute with direct sums, and the fact that usual local cohomology commutes with direct sums (for example by Dualizing Complexes, Lemma 47.9.1). Thus by More on Algebra, Remark 15.59.11 it suffices to check the map is an isomorphism for $K = A[k]$ where $k \in \mathbf{Z}$. This is clear.
The final statements follow from the result we've just shown by transitivity of derived tensor products.
$\square$
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