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This topic has appeared in the English Wikipedia rankings 1 time. It first appeared on 2026-08-22 and was most recently seen on 2026-08-22.
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 defined on an open subset of an algebraic variety, local cohomology measures the obstruction to extending that function to a larger domain. The rational function , for example, is defined only on the complement of on the affine line over a field , and cannot be extended to a function on the entire space. The local cohomology module detects this in the nonvanishing of a cohomology class . In a similar manner, is defined away from the and axes in the affine plane, but cannot be extended to either the complement of the -axis or the complement of the -axis alone ; this obstruction corresponds precisely to a nonzero class in the local cohomology module .
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