Lemma 51.2.4. Let S \subset A be a multiplicative set of a ring A. Let M be an A-module with S^{-1}M = 0. Then \mathop{\mathrm{colim}}\nolimits _{f \in S} H^0_{V(f)}(M) = M and \mathop{\mathrm{colim}}\nolimits _{f \in S} H^1_{V(f)}(M) = 0.
Proof. The statement on H^0 follows directly from the definitions. To see the statement on H^1 observe that R\Gamma _{V(f)} and H^1_{V(f)} commute with colimits. Hence we may assume M is annihilated by some f \in S. Then H^1_{V(ff')}(M) = 0 for all f' \in S (for example by Lemma 51.2.3). \square
Comments (0)