Local cohomology

In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by Hartshorne (1967), and in 1961-2 at IHES written up as SGA2 - Grothendieck (1968), republished as Grothendieck (2005). Given a function (more generally, a section of a quasicoherent sheaf) defined on an open subset of an algebraic variety (or scheme), local cohomology measures the obstruction to extending that function to a larger domain. The rational function ${\displaystyle 1/x}$, for example, is defined only on the complement of ${\displaystyle 0}$ on the affine line ${\displaystyle \mathbb {A} _{K}^{1}}$ over a field ${\displaystyle K}$, and cannot be extended to a function on the entire space. The local cohomology module ${\displaystyle H_{(x)}^{1}(K[x])}$ (where ${\displaystyle K[x]}$ is the coordinate ring of ${\displaystyle \mathbb {A} _{K}^{1}}$) detects this in the nonvanishing of a cohomology class ${\displaystyle [1/x]}$. In a similar manner, ${\displaystyle 1/xy}$ is defined away from the ${\displaystyle x}$ and ${\displaystyle y}$ axes in the affine plane, but cannot be extended to either the complement of the ${\displaystyle x}$-axis or the complement of the ${\displaystyle y}$-axis alone (nor can it be expressed as a sum of such functions); this obstruction corresponds precisely to a nonzero class ${\displaystyle [1/xy]}$ in the local cohomology module ${\displaystyle H_{(x,y)}^{2}(K[x,y])}$.^[1]

Outside of algebraic geometry, local cohomology has found applications in commutative algebra,^[2][3][4] combinatorics,^[5][6][7] and certain kinds of partial differential equations.^[8]