Cohomological dimension of complexes
arXiv:math/0305119
Abstract
In the derived category of the category of modules over a commutative Noetherian ring , we define, for an ideal $\fa$ of , two different types of cohomological dimensions of a complex in a certain subcategory of the derived category, namely $\cd(\fa, X)=\sup\{\cd(\fa, \H_{\ell}(X))-\ell|\ell\in\Bbb Z\}$ and $-\inf{\mathbf R}\G_{\fa}(X)$, where $\cd(\fa, M)=\sup\{\ell\in\Bbb Z|\H^{\ell}_{\fa}(M)\neq 0\}$ for an --module . In this paper, it is shown, among other things, that, for any complex bounded to the left, $-\inf {\mathbf R}\G_{\fa}(X)\le\cd(\fa, X)$ and equality holds if indeed is finitely generated.
13 pages