Homological algebra

Homological algebra is a part of mathematics that looks at a special idea called "homology" in a more general, algebraic way. It started in the late 1800s when mathematicians like Henri Poincaré and David Hilbert were studying shapes and structures in two different areas: combinatorial topology and abstract algebra. Over time, homological algebra grew into its own field, closely connected to another big idea in math called category theory.

One of the main tools in homological algebra is something called a chain complex. These complexes help mathematicians understand and describe important features of many different objects, like rings, modules, and even topological spaces. They do this by turning information into something called homological invariants.

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

History

Homological algebra started in the late 19th century as a part of topology. It became its own subject in the 1940s when mathematicians began 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 like a sequence of groups or spaces connected by special maps. These maps have a key property: when you follow two maps in a row, you end up back at the starting point.

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

Chain complexes appear in many areas, like studying shapes in topology or structures in algebra, helping us understand the properties of these objects through their homology.

Foundational aspects

Cohomology theories help us understand many different 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, different methods were developed to organize these ideas, leading to powerful tools like spectral sequences for solving complex problems.

Standard tools

Main article: Exact sequence

Homological algebra uses special sequences of algebraic structures called exact sequences. These sequences show how different structures connect to each other. For example, in group theory, an exact sequence tells us that the result of one mapping is exactly the input for the next mapping.

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

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 keep certain rules, and they create matching maps between homology groups. When objects are linked by a map, their chain complexes are also linked, and this linking keeps working even when we combine maps. This linking also applies to homology groups, meaning 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 where the homology groups appear in a cycle, connected by special maps. This idea shows up in topics like the Mayer–Vietoris sequence and sequences for relative homology.