Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A theory associating a system of quotient groups to each topological space.
 n. A system of quotient groups associated to a topological space.
Etymologies
Examples

Section 33.9 is six pages of Penrose trying to explain "sheaf cohomology" whose ideas "… are fairly sophisticated mathematically, but actually very natural."

Section 33.10 explains how positive/negative frequency splitting along with "holomorphic first sheaf cohomology" plays "a direct role in generating deformations of twistor space."

This then leads us into some very rich theory of the algebraic topology of the Dirac operator and something called quantum cohomology.

BRST then becomes essentially Lie algebra cohomology, where you construct the invariant, trivial piece of a representation by cancelling a sequence of nontrivial representations against each other.

The action of a real semisimple lie group on a complex flag manifold, II: Unitary representations on partially holomorphic cohomology spaces by Joseph Albert Wolf

On the mixed Hodge structure on the cohomology of the Milnor fibre.

Several mathematical concepts have been named after him including the Tate module, Tate curve, Tate cycle, HodgeTate decompositions, Tate cohomology, SerreTate parameter, and LubinTate group.

Several mathematical concepts have been named after him, including the Tate module, Tate curve, Tate cycle, HodgeTate decompositions, Tate cohomology, SerreTate parameter and

This gives rise to a slogan "when you have partitions of unity, sheaf cohomology vanishes."

So any theorem about sheaf cohomology vanishing on affine spaces needs to be phrased carefully.
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