The Stacks project

Lemma 51.5.5. Let $A$ be a Noetherian ring. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. If $\dim (A) < \infty $, then functor $D(\text{Mod}_{A, T}) \to D_ T(A)$ is an equivalence.

Proof. Say $\dim (A) = d$. Then we see that $H^ i_ Z(M) = 0$ for $i > d$ for every closed subset $Z$ of $\mathop{\mathrm{Spec}}(A)$, see Lemma 51.4.7. By Lemma 51.5.3 we find that $H^0_ T$ has bounded cohomological dimension.

Let $K \in D_ T(A)$. We claim that $RH^0_ T(K) \to K$ is an isomorphism. We know this is true when $K$ is bounded below, see Lemma 51.5.4. However, since $H^0_ T$ has bounded cohomological dimension, we see that the $i$th cohomology of $RH_ T^0(K)$ only depends on $\tau _{\geq -d + i}K$ and we conclude. Thus $D(\text{Mod}_{A, T}) \to D_ T(A)$ is an equivalence with quasi-inverse $RH^0_ T$. $\square$

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