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 $i \not\in [a, b]$.

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