Homological algebra

Homological algebra is a part of mathematics that studies a special idea called "homology" using algebra. It began in the late 1800s when mathematicians like Henri Poincaré and David Hilbert were exploring shapes and structures in combinatorial topology and abstract algebra. Over time, homological algebra became its own field and is closely linked to category theory.

One important tool in homological algebra is called a chain complex. These help mathematicians study features of different objects, such as rings, modules, and topological spaces, by turning information into homological invariants.

Homological algebra is very useful in many areas of math and science. It helps in algebraic topology, commutative algebra, algebraic geometry, and mathematical physics. Tools like spectral sequences help experts solve difficult problems in these fields. Today, homological algebra is used in many areas, from studying numbers to understanding space and complex equations.

History

Homological algebra began in the late 19th century as a part of topology. It became its own subject in the 1940s when mathematicians started studying special tools like the ext functor and the tor functor.

Chain complexes and homology

Main article: Chain complex

Homological algebra uses something called a chain complex to study shapes and structures in mathematics. A chain complex is a sequence of groups or spaces connected by special maps. These maps have an important rule: when you use two maps one after the other, you return to where you started.

In a chain complex, we look at two key groups: cycles, which are elements that map to zero, and boundaries, which are elements that come from the next group. The homology group helps us understand the cycles that are not boundaries, giving us useful information about the object being studied.

Chain complexes are used in many areas, such as studying shapes in topology or structures in algebra, helping us learn about these objects through their homology.

Foundational aspects

Cohomology theories help us understand many mathematical objects such as topological spaces, sheaves, groups, rings, Lie algebras, and C*-algebras. In modern algebraic geometry, sheaf cohomology is very important.

A key idea in homological algebra is the exact sequence, which helps us do calculations. Important tools include derived functors like Ext and Tor. Over time, new methods were created to organize these ideas, leading to useful tools for solving hard problems.

Standard tools

Main article: Exact sequence

Homological algebra uses special sequences of algebraic structures called exact sequences. These sequences show how different structures link together. For example, in group theory, an exact sequence tells us that the output of one mapping is exactly the starting point for the next mapping.

One common type is the short exact sequence, which looks like this: A → B → C. This shows us that A is a part of B, and B can be "split" by A to get C. These sequences help mathematicians understand how different algebraic objects relate to each other.

Main article: Five lemma

Main article: Snake lemma

Main article: Abelian category

Main article: Derived functor

Main article: Ext functor

Main article: Tor functor

Main article: Spectral sequence

Functoriality

A continuous map between spaces creates links between their homology groups for all levels. This idea comes from algebraic topology and helps explain how many spaces work together.

In homological algebra, we study maps between chain complexes. These maps follow certain rules and create matching maps between homology groups. When objects are linked by a map, their chain complexes are also linked, and this linking still works when we combine maps. This linking also applies to homology groups. Maps between objects create matching maps between their homology groups.

One important pattern in algebra and topology involves three chain complexes and two maps between them. This pattern leads to a special sequence in homology. The homology groups appear in a cycle, connected by special maps. This idea is used in topics like the Mayer–Vietoris sequence and sequences for relative homology.