Homological Algebra

Homological algebra is a branch of algebra which studies homology and cohomology groups of objects in algebraic structures. Homology and cohomology are tools to measure the "holes" or "obstructions" in objects in a given algebraic structure, which reveal important information about the structure itself.

Homological algebra was developed in the 1940s by Samuel Eilenberg and Saunders Mac Lane as an extension of algebraic topology. The theory has since found applications in various areas of mathematics, including algebraic geometry, representation theory, and number theory.

Homology and Cohomology

Homology and cohomology are algebraic tools used to measure the "holes" or "obstructions" in objects in a given algebraic structure. For instance, in algebraic topology, homology groups measure the number of "holes" or "obstructions" in a topological space. Similarly, in algebraic geometry, cohomology groups measure the "holes" or "obstructions" in a variety.

Homology groups are defined as the quotient groups of cycles modulo boundaries. A cycle is a chain of elements in the algebraic structure that is closed, i.e. its boundary is zero. A boundary is a chain that is the boundary of some other chain.

$Z_n = \{ x_n \in A_n \ | \ \partial_n(x_n) = 0 \}$

$B_n = \{ \partial_{n+1}(y_{n+1}) \in A_n \ | \ y_{n+1} \in A_{n+1} \}$

$H_n(A) = \frac{Z_n}{B_n}$

Cohomology groups are defined as the quotient groups of cocycles modulo coboundaries. A cocycle is a cochain that is closed, i.e. its coboundary is zero. A coboundary is a cochain that is the coboundary of some other cochain.

$Z^n = \{ x^n \in A^n \ | \ \delta^n(x^n) = 0 \}$

$B^n = \{ \delta^{n-1}(y^{n-1}) \in A^n \ | \ y^{n-1} \in A^{n-1} \}$

$H^n(A) = \frac{Z^n}{B^n}$

Applications

Homological algebra has found applications in various areas of mathematics, including algebraic geometry, representation theory, and number theory. One important application is the study of sheaf cohomology in algebraic geometry. Sheaf cohomology is used to study the behavior of algebraic varieties under deformations and to classify them up to isomorphism.

Another application of homological algebra is in the study of representation theory. The homology and cohomology of Lie algebras play an important role in the classification of Lie groups and their representations.

Finally, homological algebra has found applications in number theory, particularly in the study of Galois cohomology. Galois cohomology is used to study the properties of fields and their extensions, and to classify them.

Conclusion

Homological algebra is a powerful tool in algebraic structures, providing a way to measure the "holes" or "obstructions" in objects in a given algebraic structure. The theory has found applications in various areas of mathematics, including algebraic geometry, representation theory, and number theory. Homological algebra has proven to be a valuable tool for studying the structure of algebraic objects, with wide-ranging applications.

Links

ホモロジー代数学[JA]