The Stacks project

64.12 Characterizing perfect objects

For the commutative case see More on Algebra, Sections 15.64, 15.66, and 15.74.

Definition 64.12.1. Let $\Lambda $ be a (possibly noncommutative) ring. An object $K\in D(\Lambda )$ has finite $\text{Tor}$-dimension if there exist $a, b \in \mathbf{Z}$ such that for any right $\Lambda $-module $N$, we have $H^ i(N \otimes _{\Lambda }^\mathbf {L} K) = 0$ for all $i \not\in [a, b]$.

This in particular means that $K \in D^ b(\Lambda )$ as we see by taking $N = \Lambda $.

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. $\square$

The reader is strongly urged to try and prove this. The proof relies on the fact that a finite module on a finitely left-presented ring is flat if and only if it is projective.

Remark 64.12.3. A variant of this lemma is to consider a Noetherian scheme $X$ and the category $D_{perf}(\mathcal{O}_ X)$ of complexes which are locally quasi-isomorphic to a finite complex of finite locally free $\mathcal{O}_ X$-modules. Objects $K$ of $D_{perf}(\mathcal{O}_ X)$ can be characterized by having coherent cohomology sheaves and bounded tor dimension.

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