The Stacks project

Lemma 64.12.2. Let $\Lambda $ be a left Noetherian ring and $K\in D(\Lambda )$. Then $K$ is perfect if and only if the two following conditions hold:

  1. $K$ has finite $\text{Tor}$-dimension, and

  2. for all $i \in \mathbf{Z}$, $H^ i(K)$ is a finite $\Lambda $-module.

Proof. See More on Algebra, Lemma 15.74.2 for the proof in the commutative case. $\square$

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