# making chain complexes by hand

A new chain complex can be made with C = new ChainComplex. This will automatically initialize C.dd, in which the differentials are stored. The modules can be installed with statements like C#i=M and the differentials can be installed with statements like C.dd#i=d. The ring is installed with C.ring = R. It's up to the user to ensure that the composite of consecutive differential maps is zero.
 i1 : R = QQ[x,y,z]; i2 : d1 = matrix {{x,y}}; 1 2 o2 : Matrix R <--- R
We take care to use map to ensure that the target of d2 is exactly the same as the source of d1.
 i3 : d2 = map(source d1, ,{{y*z},{-x*z}}); 2 1 o3 : Matrix R <--- R i4 : d1 * d2 == 0 o4 = true
Now we make the chain complex, as explained above.
 i5 : C = new ChainComplex; C.ring = R; i7 : C#0 = target d1; C#1 = source d1; C#2 = source d2; i10 : C.dd#1 = d1; C.dd#2 = d2; 1 2 o10 : Matrix R <--- R 2 1 o11 : Matrix R <--- R
Our complex is ready to use.
 i12 : C 1 2 1 o12 = R <-- R <-- R 0 1 2 o12 : ChainComplex i13 : HH_0 C o13 = cokernel | x y | 1 o13 : R-module, quotient of R i14 : prune HH_1 C o14 = cokernel {2} | z | 1 o14 : R-module, quotient of R
The chain complex we've just made is simple, in the sense that it's a homological chain complex with nonzero modules in degrees 0, 1, ..., n. Such a chain complex can be made also with chainComplex. It goes to a bit of extra trouble to adjust the differentials to match the degrees of the basis elements.
 i15 : D = chainComplex(matrix{{x,y}}, matrix {{y*z},{-x*z}}) 1 2 1 o15 = R <-- R <-- R 0 1 2 o15 : ChainComplex i16 : degrees source D.dd_2 o16 = {{3}} o16 : List