# map(Complex,Complex,ZZ) -- make the zero map or identity between chain complexes

## Synopsis

• Function: map
• Usage:
f = map(D, C, 0)
f = map(C, C, 1)
• Inputs:
• Optional inputs:
• Degree => an integer, default value null, the degree of the resulting map
• DegreeLift => ..., default value null, unused
• DegreeMap => ..., default value null, unused
• Outputs:
• f, , the zero map from $C$ to $D$ or the identity map from $C$ to $C$

## Description

A map of complexes $f : C \rightarrow D$ of degree $d$ is a sequence of maps $f_i : C_i \rightarrow D_{d+i}$.

We construct the zero map between two chain complexes.

 i1 : R = QQ[a,b,c] o1 = R o1 : PolynomialRing i2 : C = freeResolution coker vars R 1 3 3 1 o2 = R <-- R <-- R <-- R 0 1 2 3 o2 : Complex i3 : D = freeResolution coker matrix{{a^2, b^2, c^2}} 1 3 3 1 o3 = R <-- R <-- R <-- R 0 1 2 3 o3 : Complex i4 : f = map(D, C, 0) o4 = 0 o4 : ComplexMap i5 : assert isWellDefined f i6 : assert isComplexMorphism f i7 : g = map(C, C, 0, Degree => 13) o7 = 0 o7 : ComplexMap i8 : assert isWellDefined g i9 : assert(degree g == 13) i10 : assert not isComplexMorphism g i11 : assert isCommutative g i12 : assert isHomogeneous g i13 : assert(source g == C) i14 : assert(target g == C)

Using this function to create the identity map is the same as using id _ Complex.

 i15 : assert(map(C, C, 1) === id_C)