The Stacks project

Lemma 47.13.4. With notation as above, assume $A \to B$ is a finite ring map of Noetherian rings. Then $R\mathop{\mathrm{Hom}}\nolimits (B, -)$ maps $D^+_{\textit{Coh}}(A)$ into $D^+_{\textit{Coh}}(B)$.

Proof. We have to show: if $K \in D^+(A)$ has finite cohomology modules, then the complex $R\mathop{\mathrm{Hom}}\nolimits (B, K)$ has finite cohomology modules too. This follows for example from Lemma 47.13.3 if we can show the ext modules $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(B, K)$ are finite $A$-modules. Since $K$ is bounded below there is a convergent spectral sequence

\[ \mathop{\mathrm{Ext}}\nolimits ^ p_ A(B, H^ q(K)) \Rightarrow \text{Ext}^{p + q}_ A(B, K) \]

This finishes the proof as the modules $\mathop{\mathrm{Ext}}\nolimits ^ p_ A(B, H^ q(K))$ are finite by Algebra, Lemma 10.70.9. $\square$


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