Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, local cohomology is a chapter of homological algebra and sheaf theory introduced into algebraic geometry by Alexander Grothendieck. He developed it in seminars in 1961 at Harvard University, and 1961-2 at IHES. It was later written up as SGA2. Applications to commutative algebra and hyperfunction theory followed. There is a long exact sequence of sheaf cohomology linking the ordinary sheaf cohomology of X and of the open set U = X Y, with the local cohomology groups. The initial applications were to analogues of the Lefschetz hyperplane theorems. In general such theorems state that homology or cohomology is supported on a hyperplane section of an algebraic variety, except for some 'loss' that can be controlled. These results applied to the algebraic fundamental group and to the Picard group.

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