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# chainComplex(List) -- make a chain complex

- Function: chainComplex
- Usage:
    `C = chainComplex{f1,f2,f3,...}`
    `C = chainComplex(f1,f2,f3,...)`
- Inputs:
  - f1,f2,f3,..., homomorphisms over the same ring, forming a complex
- Outputs:
  - C, a chain complex, the given complex, where f1 == C.dd_1, f2 == CC.dd_2, etc.

## Description

The maps f1, f2, ... must be defined over the same base ring, and they must form a complex: the target of f(i+1) is the source of fi.

The following example illustrates how chainComplex adjusts the degrees of the modules involved to ensure that sources and targets of the differentials correspond exactly.

    i1 : R = ZZ/101[x,y]

    o1 = R

    o1 : PolynomialRing

    i2 : C = chainComplex{matrix{{x,y}},matrix{{x*y},{-x^2}}}

          1      2      1
    o2 = R  <-- R  <-- R

         0      1      2

    o2 : ChainComplex

We check that this is a complex:

    i3 : C.dd^2 == 0

    o3 = true

The homology of this complex:

    i4 : HH C

    o4 = 0 : cokernel | x y |

         1 : subquotient ({1} | -y |, {1} | xy  |)
                          {1} | x  |  {1} | -x2 |

         2 : image 0

    o4 : GradedModule

## See also

- chain complexes -- legacy implementation of chain complexes

## Ways to use this method:

- chainComplex(List) -- make a chain complex
- `chainComplex(Sequence)`

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldChainComplexes/docs/chainComplex-doc.m2:59:0.