Definition 15.68.1. Let $R$ be a ring. Let $K$ be an object of $D(R)$. We say $K$ has finite projective dimension if $K$ can be represented by a bounded complex of projective modules. We say $K$ as projective-amplitude in $[a, b]$ if $K$ is quasi-isomorphic to a complex

$\ldots \to 0 \to P^ a \to P^{a + 1} \to \ldots \to P^{b - 1} \to P^ b \to 0 \to \ldots$

where $P^ i$ is a projective $R$-module for all $i \in \mathbf{Z}$.

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).