Lemma 63.12.2. 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).