# HH_ZZ Complex -- homology or cohomology module of a complex

## Synopsis

• Function: homology
• Usage:
HH_i C
HH^i C
• Inputs:
• Outputs:
• , the $i$-th homology or cohomology of the complex

## Description

The $i$-th homology of a complex $C$ is the quotient (ker dd^C_i/image dd^C_(i+1)).

The first example is the complex associated to a triangulation of the real projective plane, having 6 vertices, 15 edges, and 10 triangles.

 i1 : d1 = matrix { {1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {-1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0}, {0, -1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 1, 0, 0, 0}, {0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 1, 0}, {0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0, 1}, {0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, -1}}; 6 15 o1 : Matrix ZZ <--- ZZ i2 : d2 = matrix { {-1, -1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, -1, -1, 0, 0, 0, 0, 0, 0}, {1, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, -1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, -1, -1, 0, 0, 0}, {-1, 0, 0, 0, 0, 0, 0, -1, 0, 0}, {0, -1, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 1, 0, 0}, {0, 0, -1, 0, 0, 0, 0, 0, -1, 0}, {0, 0, 0, 0, 0, -1, 0, 0, 1, 0}, {0, 0, 0, -1, 0, 0, -1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, -1, -1}, {0, 0, 0, 0, 0, 0, 0, -1, 0, 1}, {0, 0, 0, 0, -1, 0, 0, 0, 0, -1}}; 15 10 o2 : Matrix ZZ <--- ZZ i3 : C = complex {d1,d2} 6 15 10 o3 = ZZ <-- ZZ <-- ZZ 0 1 2 o3 : Complex i4 : dd^C 6 15 o4 = 0 : ZZ <---------------------------------------------------- ZZ : 1 | 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 | | -1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 | | 0 -1 0 0 0 -1 0 0 0 1 1 1 0 0 0 | | 0 0 -1 0 0 0 -1 0 0 -1 0 0 1 1 0 | | 0 0 0 -1 0 0 0 -1 0 0 -1 0 -1 0 1 | | 0 0 0 0 -1 0 0 0 -1 0 0 -1 0 -1 -1 | 15 10 1 : ZZ <------------------------------------- ZZ : 2 | -1 -1 0 0 0 0 0 0 0 0 | | 0 0 -1 -1 0 0 0 0 0 0 | | 1 0 1 0 0 0 0 0 0 0 | | 0 1 0 0 -1 0 0 0 0 0 | | 0 0 0 1 1 0 0 0 0 0 | | 0 0 0 0 0 -1 -1 0 0 0 | | -1 0 0 0 0 0 0 -1 0 0 | | 0 -1 0 0 0 1 0 0 0 0 | | 0 0 0 0 0 0 1 1 0 0 | | 0 0 -1 0 0 0 0 0 -1 0 | | 0 0 0 0 0 -1 0 0 1 0 | | 0 0 0 -1 0 0 -1 0 0 0 | | 0 0 0 0 0 0 0 0 -1 -1 | | 0 0 0 0 0 0 0 -1 0 1 | | 0 0 0 0 -1 0 0 0 0 -1 | o4 : ComplexMap i5 : HH C o5 = cokernel | 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 | <-- subquotient (| 0 1 0 0 0 0 0 0 0 0 |, | -1 -1 0 0 0 0 0 0 0 0 |) <-- image 0 | -1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 | | 1 0 0 0 0 0 0 0 0 0 | | 0 0 -1 -1 0 0 0 0 0 0 | | 0 -1 0 0 0 -1 0 0 0 1 1 1 0 0 0 | | 0 -1 1 0 -1 0 1 0 1 0 | | 1 0 1 0 0 0 0 0 0 0 | 2 | 0 0 -1 0 0 0 -1 0 0 -1 0 0 1 1 0 | | 0 0 0 0 0 1 0 0 0 0 | | 0 1 0 0 -1 0 0 0 0 0 | | 0 0 0 -1 0 0 0 -1 0 0 -1 0 -1 0 1 | | -1 0 -1 0 1 -1 -1 0 -1 0 | | 0 0 0 1 1 0 0 0 0 0 | | 0 0 0 0 -1 0 0 0 -1 0 0 -1 0 -1 -1 | | 0 0 0 0 1 0 0 0 0 0 | | 0 0 0 0 0 -1 -1 0 0 0 | | 0 0 0 1 0 0 0 0 1 1 | | -1 0 0 0 0 0 0 -1 0 0 | 0 | 0 1 -1 0 0 0 0 0 -1 0 | | 0 -1 0 0 0 1 0 0 0 0 | | 0 0 1 -1 -1 0 0 0 0 -1 | | 0 0 0 0 0 0 1 1 0 0 | | 0 0 0 0 0 0 0 1 0 0 | | 0 0 -1 0 0 0 0 0 -1 0 | | 0 0 0 0 0 0 0 0 0 1 | | 0 0 0 0 0 -1 0 0 1 0 | | 1 0 0 0 1 0 0 -1 0 -1 | | 0 0 0 -1 0 0 -1 0 0 0 | | 0 -1 1 0 -1 0 1 1 2 1 | | 0 0 0 0 0 0 0 0 -1 -1 | | 0 0 0 1 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 -1 0 1 | | 0 0 0 0 -1 1 1 1 1 2 | | 0 0 0 0 -1 0 0 0 0 -1 | 1 o5 : Complex i6 : prune HH_0 C 1 o6 = ZZ o6 : ZZ-module, free i7 : prune HH_1 C o7 = cokernel | 2 | 1 o7 : ZZ-module, quotient of ZZ i8 : prune HH_2 C o8 = 0 o8 : ZZ-module

The $i$-th cohomology of a complex $C$ is the $(-i)$-th homology of $C$.

 i9 : S = ZZ/101[a..d, DegreeRank=>4]; i10 : I = intersect(ideal(a,b),ideal(c,d)) o10 = ideal (b*d, a*d, b*c, a*c) o10 : Ideal of S i11 : C = dual freeResolution (S^1/I) 1 4 4 1 o11 = S <-- S <-- S <-- S -3 -2 -1 0 o11 : Complex i12 : prune HH^1 C o12 = 0 o12 : S-module i13 : prune HH^2 C o13 = cokernel {-1, -1, 0, 0} | b a 0 0 | {0, 0, -1, -1} | 0 0 d c | 2 o13 : S-module, quotient of S i14 : prune HH^3 C o14 = cokernel {-1, -1, -1, -1} | d c b a | 1 o14 : S-module, quotient of S