Local cohomology

Remarks

One has $H^0_I(M) = \Gamma_I(M)$, and a short exact sequence of modules gives a long exact sequence of local cohomology. Equivalently, $H^i_I(M) \cong \varinjlim_n \operatorname{Ext}^i_R(R/I^n, M)$, a direct limit of Ext modules. If $I = (x_1, \ldots, x_n)$, then $H^i_I(M)$ is also the cohomology of the Čech complex $0 \to M \to \bigoplus_i M_{x_i} \to \bigoplus_{i<j} M_{x_i x_j} \to \cdots \to M_{x_1 \cdots x_n} \to 0$. Example: for $R = K[x]$ and $I = (x)$, the Čech complex of $R$ is $0 \to R \to R_x \to 0$, so $H^0_I(R) = 0$ and $H^1_I(R) = K[x, x^{-1}]/K[x]$. The functor $\Gamma_I$ is not right exact: for $0 \to K[x] \xrightarrow{x} K[x] \to K \to 0$ with $I = (x)$ one gets $0 \to 0 \to K$, and the last map is not onto. If $(R, \mathfrak{m})$ is local and $M \ne 0$ is finitely generated, then $H^i_{\mathfrak{m}}(M) = 0$ for $i < \operatorname{depth} M$ and $i > \dim M$, and it is nonzero for $i = \operatorname{depth} M$ and $i = \dim M$.