Homological Algebra
Introduction
Homological algebra is a branch of mathematics that studies homology in a general algebraic setting. It emerged as a distinct field in the mid-20th century, primarily through the work of Henri Cartan and Samuel Eilenberg. Homological algebra provides tools and concepts that are essential in various areas of mathematics, including algebraic topology, algebraic geometry, and representation theory.
Basic Concepts
Homological algebra revolves around the study of chain complexes and their homology. A chain complex is a sequence of abelian groups or modules connected by homomorphisms, with the property that the composition of consecutive homomorphisms is zero. Formally, a chain complex \( C_\bullet \) is a sequence of abelian groups \( \{C_n\} \) and homomorphisms \( \{d_n: C_n \to C_{n-1}\} \) such that \( d_{n-1} \circ d_n = 0 \) for all \( n \).
Chain Complexes
A chain complex can be written as: \[ \cdots \to C_{n+1} \xrightarrow{d_{n+1}} C_n \xrightarrow{d_n} C_{n-1} \to \cdots \] The homology groups of a chain complex are defined as: \[ H_n(C_\bullet) = \ker(d_n) / \operatorname{im}(d_{n+1}) \] where \( \ker(d_n) \) is the kernel of \( d_n \) and \( \operatorname{im}(d_{n+1}) \) is the image of \( d_{n+1} \).
Exact Sequences
An exact sequence is a sequence of abelian groups and homomorphisms: \[ \cdots \to A_{n+1} \xrightarrow{f_{n+1}} A_n \xrightarrow{f_n} A_{n-1} \to \cdots \] such that the image of each homomorphism is equal to the kernel of the next. Exact sequences are fundamental in homological algebra because they describe how different algebraic structures are related.
Derived Functors
Derived functors are a central concept in homological algebra. They generalize the notion of taking homology and provide a systematic way to study the properties of functors between abelian categories.
Ext and Tor Functors
Two of the most important derived functors are the Ext and Tor functors. The Ext functor measures the extent to which a module fails to be projective, while the Tor functor measures the extent to which a module fails to be flat.
For modules \( M \) and \( N \) over a ring \( R \), the Ext functor is defined as: \[ \operatorname{Ext}^n_R(M, N) = H^n(\operatorname{Hom}_R(P_\bullet, N)) \] where \( P_\bullet \) is a projective resolution of \( M \).
The Tor functor is defined as: \[ \operatorname{Tor}_n^R(M, N) = H_n(P_\bullet \otimes_R N) \] where \( P_\bullet \) is a projective resolution of \( M \).
Spectral Sequences
Spectral sequences are a powerful computational tool in homological algebra. They provide a way to compute homology groups by filtering a complex and examining the associated graded pieces.
A spectral sequence is a sequence of pages \( E_r \) that converges to the homology of a complex. Each page \( E_r \) is a bigraded module, and the differentials \( d_r \) map between these modules. The spectral sequence stabilizes at some page \( E_\infty \), which provides information about the homology of the original complex.
Applications
Homological algebra has numerous applications in various fields of mathematics.
Algebraic Topology
In algebraic topology, homological algebra is used to study the properties of topological spaces through their homology and cohomology groups. Chain complexes arise naturally in the study of simplicial complexes and CW complexes.
Algebraic Geometry
In algebraic geometry, homological algebra is used to study sheaves and their cohomology. The derived category of sheaves provides a framework for understanding the relationships between different sheaves on a variety.
Representation Theory
In representation theory, homological algebra is used to study the representations of algebras and groups. The Ext and Tor functors provide information about extensions and relations between different representations.
Advanced Topics
Homological algebra continues to evolve, with new concepts and techniques being developed.
Derived Categories
Derived categories provide a framework for working with chain complexes up to homotopy. They allow for a more flexible approach to homological algebra and have applications in various areas of mathematics.
Triangulated Categories
Triangulated categories are a generalization of derived categories. They provide a setting for studying the relationships between different objects in a homological context.
Homotopical Algebra
Homotopical algebra is a branch of mathematics that combines homological algebra with homotopy theory. It provides tools for studying the homotopy properties of algebraic structures.