Local cohomology is a powerful tool in algebraic geometry and commutative algebra that provides a way to study the behavior of sheaves and their sections in relation to a specific closed subset of a space. It helps capture local properties of sheaves by measuring the sections that vanish on the closed subset, and it plays an essential role in understanding depth, support, and dimensions of varieties. This concept is crucial in linking various deep results, such as duality theories and Riemann-Roch theorems, allowing for a better grasp of geometric and topological properties.
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