Chain complex

A ( co) chain complex in mathematics is a sequence of -modules or abelian groups, or generally objects in abelian categories, which are linked like a chain of images.

Definition

Chain complex

A chain complex is a sequence

Of -modules ( abelian groups, objects of an abelian category A) and a sequence

Of module homomorphisms ( group homomorphisms, morphisms in A), so that

Is valid for all n. The operator is called a boundary operator. Elements of hot n- chains. elements of

Hot n -cycle and n- edges. Due to the condition of each edge is a cycle. The quotient

Is called n-th homology group ( homology object) of its elements are called homology classes. Cycles that are in the same homology class are called homologous.

Kokettenkomplex

A Kokettenkomplex consists of a sequence

Of -modules ( abelian groups, objects of an abelian category A) and a sequence

Of module homomorphisms ( group homomorphisms, morphisms in A), so that

Is valid for all n. Elements of hot n- cochains. elements of

Hot n- cocycle and n- Koränder. Due to the condition of every coboundary is a cocycle. The quotient

Is called n-th cohomology group ( Kohomologieobjekt ) of their elements Kohomologieklassen. Cocycle, which lie in the same cohomology class, called kohomolog.

Properties

  • A chain complex is exact then exactly at the point when, according to Kokettenkomplexe. Thus, the ( co) homology measures how strongly a ( co) chain complex differs from the accuracy.

Kettenhomomorphismus

A function

Means ( Ko) - Kettenhomomorphismus, if it exists a sequence of group homomorphisms, which commutes with the boundary operator. That is for the Kettenhomomorphismus:

For the Kokettenhomomorphismus shall apply mutatis mutandis

This condition ensures that maps to Cycles Cycles and edges to edges.

Chain complexes, together with the category Kettenhomomorphismen Ch ( mod r ) of the chain complexes.

Euler characteristic

It was a Kokettenkomplex of - modules over a ring. If there are only finitely many non-trivial cohomology groups, and these are finite, then the Euler characteristic of the complex is defined as the integer

Are the individual components is finite and only finitely many of them are not trivial, as is also

In the special case of a complex with only two non-trivial entries, this statement is the rank theorem.

Examples

  • Simplicial
  • The singular chain complex to define the singular homology and singular cohomology of topological spaces.
  • Groups ( co ) homology.
  • Every homomorphism defines a Kokettenkomplex
  • An elliptical complex or complex is a Dirac Kokettenkomplex, which is in the global analysis of significance. These occur, for example in connection with the Atiyah - Bott fixed-point theorem.
Function (mathematics) Quotient space (linear algebra) Group cohomology Hodge theory Atiyah–Bott fixed-point theorem
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