SHEAF COHOMOLOGY Meaning and
Definition
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Sheaf cohomology is a mathematical concept that arises within the field of algebraic topology. It is a mathematical tool used to study the properties and behavior of sheaves over a topological space. In general, a sheaf is a way of assigning data or functions to points of a topological space in a locally consistent manner.
More precisely, sheaf cohomology involves the study of the cohomology groups associated with a sheaf. These cohomology groups capture the information about the "holes" or "twists" in the sheaf that cannot be filled or resolved locally, and give insights into the topological structure of the space. Sheaf cohomology is a powerful tool for understanding the topological properties of a space, providing a way to extract global information from local data.
The cohomology groups of a sheaf are computed by considering certain complexes of abelian groups associated with the sheaf, namely the sheaf cochain complexes. These complexes encode the structure of the sheaf and its relationships with the space. The cohomology groups then reflect the obstruction to finding global sections of the sheaf.
In summary, sheaf cohomology is a technique in algebraic topology that allows us to analyze the topology of a space using the tools of sheaf theory. It involves studying the cohomology groups associated with a sheaf, which provide valuable information about the global structure of the space.
Etymology of SHEAF COHOMOLOGY
The word "sheaf cohomology" has its etymology rooted in the mathematics of algebraic topology and algebraic geometry.
The term "cohomology" comes from the combination of two words: "co" meaning "together" or "jointly" and "homology" which refers to a mathematical concept capturing the notion of "sameness" or "similarity". In algebraic topology, cohomology is a tool used to understand the topological properties of spaces. It provides a way to measure holes or obstructions to the existence of certain structures in a space.
The word "sheaf" in "sheaf cohomology" comes from the field of algebraic geometry. A sheaf is a generalization of functions on a space that takes into account their local behavior. It is a mathematical construct that encapsulates information about local data and how it "glues" together to form a global object.
