Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun mathematics A theory associating a system of
quotient groups to eachtopological space .  noun mathematics A system of
quotient groups associated to atopological space .
Etymologies
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Examples

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."

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

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.

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."
qms commented on the word cohomology
Damn! I’ve been using this to mean the study of salmon.
February 6, 2019