# 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 . ${\ displaystyle R}$

## definition

### Chain complex

A chain complex consists of a sequence

${\ displaystyle \, C_ {n}, \, n \ in \ mathbb {Z}}$

of modules (Abelian groups, objects of an Abelian category A ) and a sequence ${\ displaystyle R}$

${\ displaystyle d_ {n}: C_ {n} \ rightarrow C_ {n-1}}$

of module homomorphisms (group homomorphisms, morphisms in A ), so that ${\ displaystyle R}$

${\ displaystyle \, d_ {n} \ circ d_ {n + 1} = 0}$

holds for all n . The operator is called the boundary operator . Elements of hot n-chains . Elements of ${\ displaystyle \ mathrm {d} _ {n}}$${\ displaystyle C_ {n}}$

${\ displaystyle Z_ {n} (C, d): = \ ker d_ {n} \ subseteq C_ {n}}$ or. ${\ displaystyle B_ {n} (C, d): = \ mathop {\ mathrm {im}} d_ {n + 1} \ subseteq C_ {n}}$

are called n-cycles or n-boundaries . Because of the condition , each edge is a cycle. The quotient${\ displaystyle \, d_ {n} d_ {n + 1} = 0}$

${\ displaystyle \, H_ {n} (C, d): = Z_ {n} (C, d) / B_ {n} (C, d)}$

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 . ${\ displaystyle \, (C, d)}$

### Coquette complex

A coquette complex consists of a sequence

${\ displaystyle C ^ {n}, \, n \ in \ mathbb {Z}}$

of modules (Abelian groups, objects of an Abelian category A ) and a sequence ${\ displaystyle R}$

${\ displaystyle \, d ^ {n}: C ^ {n} \ rightarrow C ^ {n + 1}}$

of module homomorphisms (group homomorphisms, morphisms in A ) so that ${\ displaystyle R}$

${\ displaystyle \, d ^ {n} \ circ d ^ {n-1} = 0}$

holds for all n . Elements of hot n-coquettes . Elements of ${\ displaystyle \, C ^ {n}}$

${\ displaystyle \, Z ^ {n}: = \ ker d ^ {n} \ subseteq C ^ {n}}$ or. ${\ displaystyle B ^ {n}: = \ operatorname {im} d ^ {n-1} \ subseteq C ^ {n}}$

are called n-coccycles or n-corands . Because of the condition , every Korand is a cocycle. The quotient ${\ displaystyle \, d ^ {n} d ^ {n-1} = 0}$

${\ displaystyle \, H ^ {n} (C, d): = Z ^ {n} (C, d) / B ^ {n} (C, d)}$

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 . ${\ displaystyle \, (C, d)}$

### Double complex

A 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 ${\ displaystyle D _ {**}}$${\ displaystyle D _ {**}}$

${\ displaystyle D_ {p, q} \ in \ operatorname {ob} A \ ,, \ quad p, q \ in \ mathbb {Z}}$

along with morphisms

${\ displaystyle D_ {p, q} {\ xrightarrow {d}} D_ {p-1, q}}$   and   ${\ displaystyle D_ {p, q} {\ xrightarrow {d '}} D_ {p, q-1} \ quad \ forall \, p, q \ in \ mathbb {Z}}$

which meet the following three conditions:

${\ displaystyle d \ circ d = 0 \ quad d '\ circ d' = 0 \ quad d \ circ d '+ d' \ circ d = 0 \ ,.}$

The total complex of     the double     complex is given by the chain complex ${\ displaystyle \ operatorname {Tot} (D) _ {*}}$${\ displaystyle D _ {**}}$

${\ displaystyle \ operatorname {Tot} (D) _ {n} = \ bigoplus _ {p + q = n} D_ {p, q}}$

with the following border illustration: for     with     is ${\ displaystyle x \ in D_ {p, q}}$${\ displaystyle p + q = n}$

${\ displaystyle d_ {n} (x) = d (x) + d '(x) \ in D_ {p-1, q} \ oplus D_ {p, q-1} \ subseteq \ operatorname {Tot} (D ) _ {n-1} \ ,.}$

Double complexes are needed, among other things, in order to prove that the value of is independent of whether one M dissolves or N . ${\ displaystyle \ operatorname {Tor} _ {*} ^ {R} (M, N)}$

## 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.${\ displaystyle (C _ {\ bullet}, d _ {\ bullet})}$${\ displaystyle i}$${\ displaystyle H_ {i} (C _ {\ bullet}, d _ {\ bullet}) = 0}$
• A chain complex is called acyclic if all of its homology groups disappear, i.e. if it is exact.

## Chain homomorphism

One function

${\ displaystyle f: (A _ {\ bullet}, d_ {A, \ bullet}) \ to (B _ {\ bullet}, d_ {B, \ bullet})}$

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: ${\ displaystyle f_ {n}: A_ {n} \ rightarrow B_ {n}}$${\ displaystyle d}$

${\ displaystyle d_ {B, n} \ circ f_ {n} = f_ {n-1} \ circ d_ {A, n}}$.

The same applies to the coquette homomorphism

${\ displaystyle d_ {B} ^ {n} \ circ f_ {n} = f_ {n + 1} \ circ d_ {A} ^ {n}}$.

This condition ensures that cycle maps to cycle and edges to edges. ${\ displaystyle f}$

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 ${\ displaystyle (C, d)}$${\ displaystyle R}$${\ displaystyle R}$

${\ displaystyle \ chi (C, d) = \ sum _ {i} (- 1) ^ {i} \ dim _ {K} \ mathrm {H} ^ {i} (C, d) \ in \ mathbb { Z}.}$

If the individual components are also finite-dimensional and only finitely many of them are nontrivial, so is ${\ displaystyle C ^ {i}}$

${\ displaystyle \ chi (C, d) = \ sum _ {i} (- 1) ^ {i} \ dim _ {K} C ^ {i} \ in \ mathbb {Z}.}$

In the special case of a complex with only two nontrivial entries, this statement is the ranking . ${\ displaystyle C ^ {0} \ to C ^ {1}}$

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 ${\ displaystyle C ^ {i}}$ ${\ displaystyle R}$

${\ displaystyle \ chi (C, d) = \ sum _ {i} (- 1) ^ {i} [C ^ {i}] \ in K_ {0} (R).}$

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. ${\ displaystyle R}$${\ displaystyle K_ {0} (R) \ cong \ mathbb {Z}}$${\ displaystyle [R ^ {n}] {\ mathrel {\ hat {=}}} n}$

## Examples

${\ displaystyle (C, d) = (\ ldots \ to 0 \ to 0 \ to A \ to B \ to 0 \ to 0 \ to \ ldots).}$
If one sets the indices so that it is in degree 0 and in degree 1, then is ${\ displaystyle A}$${\ displaystyle B}$
${\ displaystyle H ^ {0} (C, d) = \ ker f}$ and ${\ displaystyle H ^ {1} (C, d) = \ mathrm {coker} \, f.}$
The Euler characteristic
${\ displaystyle \ dim \ ker f- \ dim \ mathrm {coker} \, f}$
of is in the theory of Fredholm operators of Fredholm index of called. This denotes the coke core of .${\ displaystyle (C, d)}$${\ displaystyle f}$${\ displaystyle \ mathrm {coker} \, f}$${\ displaystyle f}$

## Individual evidence

1. 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 .
2. 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 .
3. J. Cuntz, R. Meyer, J. Rosenberg: Topological and Bivariant K-Theory , Birkhäuser Verlag (2007), ISBN 3-764-38398-4 , definition 1.31