Local cohomology of graded rings and projective schemes

Abstract

0. Introduction Local cohomology of graded rings has been studied in the classical situation (locally closed support) by Goto and Watanabe [9], essentially in order to study pro- jective varieties through their associated graded algebras. On the other hand, a lot of information is known about local cohomology of ungraded rings, with arbitrary supports, cf. [3,4,7, 16, . ..I. Here local cohomology may be expressed as the derived functors of idempotent kernel functors associated to the supports one considers. The purpose of this note is to derive a similar set-up in the graded case. Our main result expresses the local cohomology groups of quasi coherent sheaves on Proj(R) with support in an arbitrary subset as the local cohomology groups of graded R- modules with respect to a suitable chosen idempotent kernel functor in R-gr, the category of graded R-modules. The main difficulties we will encounter are (i) that we do not necessarily assume R to be noetherian (hence certain ‘classical’ vanishing theorems do not apply) and (ii) that quasi coherent sheaves on Proj(R) and graded R-modules do not uniquely determine each other, essentially due to the fact that ‘the top’ R, = a,,>,, R, is missing in Proj(R). 1. Background on graded localization 1.1. Throughout, we denote by R a commutative, positively graded ring and by R-gr the category of graded R-modules. For generalities about graded rings and modules we refer to [12]. Let us, however, recall some of the details we used. If MER-~~ and if n is an integer, then we denote by M(n) the n th shifted version of M, i.e. M(n) coincides with M as an ungraded R-module and has gradation given by M(n), = M n+p for any integer