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A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, lie-algebra-cohomology, motivic-cohomology, equivariant-cohomology, ...
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English reference for the Grauert–Riemenschneider vanishing theorem
nowadays it also holds in positive characteristic:
Higher direct images of the structure sheaf in positive characteristic. Algebra & Number Theory, vol 5, No. 6 (2011), 693-775