Chain complex
A (co) chain complex in mathematics is a sequence of Abelian groups or modules or - even more generally - objects in an Abelian category , which are linked in a chain-like manner by illustrations .
definition
Chain complex
A chain complex 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
holds for all n . The operator is called the boundary operator . Elements of hot n-chains . Elements of
- or.
are called n-cycles or n-boundaries . Because of the condition , each edge is a cycle. The quotient
is called the n th homology group (homology object) of , its elements are called homology classes . Cycles that are in the same homology class are called homologous .
Coquette complex
A coquette complex 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
holds for all n . Elements of hot n-coquettes . Elements of
- or.
are called n-coccycles or n-corands . Because of the condition , every Korand is a cocycle. The quotient
is called the n th cohomology group (cohomology object) of , its elements cohomology classes . Coccycles that are in the same cohomology class are called cohomologous .
Double complex
A double-complex in the abelian category A is essentially a chain complex in the abelian category of chain complexes in A . Somewhat more precisely consists of objects
along with morphisms
- and
which meet the following three conditions:
The total complex of the double complex is given by the chain complex
with the following border illustration: for with is
Double complexes are needed, among other things, in order to prove that the value of is independent of whether one M dissolves or N .
properties
- A chain complex is exactly at the point if is, correspondingly for coquette complexes. The (co) homology thus measures how much a (co) chain complex deviates from the exactness.
- A chain complex is called acyclic if all of its homology groups disappear, i.e. if it is exact.
Chain homomorphism
One function
is called (Ko) -chain homomorphism , or simply chain mapping , if it consists of a sequence of group homomorphisms , which is exchanged with the boundary operator . That means for the chain homomorphism:
- .
The same applies to the coquette homomorphism
- .
This condition ensures that cycle maps to cycle and edges to edges.
Chain complexes together with the chain homomorphisms form the category Ch (MOD R) of the chain complexes.
Euler characteristic
It is a coquette complex of modules over a ring . If only finitely many cohomology groups are nontrivial, and if these are finite-dimensional, then the Euler characteristic of the complex is defined as the integer
If the individual components are also finite-dimensional and only finitely many of them are nontrivial, so is
In the special case of a complex with only two nontrivial entries, this statement is the ranking .
In a somewhat more general way, a chain complex is called perfect if only finitely many components are nontrivial and each component is a finitely generated projective module . The dimension is then to be replaced by the associated equivalence class in the K _{0} group of and it is defined as the Euler characteristic _{}
If every projective module is free , for example if there is a body or a main ideal ring , then one can speak of dimensions and receive with . Then this more general definition coincides with the one given first.
Examples
- Simplicial complex
- The singular chain complex for the definition of the singular homology and the singular cohomology of topological spaces .
- Group (co) homology .
- Each homomorphism defines a coquette complex
- If one sets the indices so that it is in degree 0 and in degree 1, then is
- and
- The Euler characteristic
- of is in the theory of Fredholm operators of Fredholm index of called. This denotes the coke core of .
- An elliptical complex or a Dirac complex is a coquette complex that is important in global analysis . These occur, for example, in connection with the Atiyah-Bott fixed point theorem .
literature
- Peter John Hilton, Urs Stammbach : A Course in Homological Algebra ( Graduate Texts in Mathematics 4). Springer, New York a. a. 1971, ISBN 0-387-90033-0 .
Individual evidence
- ↑ P. 7–8 in Charles A. Weibel : An introduction to homological algebra (= Cambridge Studies in Advanced Mathematics . Volume 38 ). Cambridge University Press, 1994, ISBN 0-521-43500-5 .
- ↑ Section 2.7 in Charles A. Weibel : An introduction to homological algebra (= Cambridge Studies in Advanced Mathematics . Volume 38 ). Cambridge University Press, 1994, ISBN 0-521-43500-5 .
- ↑ J. Cuntz, R. Meyer, J. Rosenberg: Topological and Bivariant K-Theory , Birkhäuser Verlag (2007), ISBN 3-764-38398-4 , definition 1.31