Lemma 51.5.7. Let A \to B be a flat homomorphism of Noetherian rings. Let T \subset \mathop{\mathrm{Spec}}(A) be a subset stable under specialization. Let T' \subset \mathop{\mathrm{Spec}}(B) be the inverse image of T. Then the canonical map
is an isomorphism for K \in D^+(A). If A and B have finite dimension, then this is true for K \in D(A).
Proof. From the map R\Gamma _ T(K) \to K we get a map R\Gamma _ T(K) \otimes _ A^\mathbf {L} B \to K \otimes _ A^\mathbf {L} B. The cohomology modules of R\Gamma _ T(K) \otimes _ A^\mathbf {L} B are supported on T' and hence we get the arrow of the lemma. This arrow is an isomorphism if T is a closed subset of \mathop{\mathrm{Spec}}(A) by Dualizing Complexes, Lemma 47.9.3. Recall that H^ i_ T(K) is the colimit of H^ i_ Z(K) where Z runs over the (directed set of) closed subsets of T, see Lemma 51.5.3. Correspondingly H^ i_{T'}(K \otimes _ A^\mathbf {L} B) = \mathop{\mathrm{colim}}\nolimits H^ i_{Z'}(K \otimes _ A^\mathbf {L} B) where Z' is the inverse image of Z. Thus the result because \otimes _ A B commutes with filtered colimits and there are no higher Tors. \square
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