# HH^ZZ ChainComplex -- cohomology of a chain complex

## Synopsis

• Function: cohomology
• Usage:
HH^i C
• Inputs:
• Optional inputs:
• Degree (missing documentation) => ..., default value 0,
• Outputs:
• , HH^i C -- homology at the i-th spot of the chain complex C.

## Description

By definition, this is the same as computing HH_(-i) C.

 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 i3 : M = HH^1 C o3 = 0 o3 : R-module i4 : prune M o4 = 0 o4 : R-module

Here is another example computing simplicial cohomology (for a hollow tetrahedron):

 i5 : needsPackage "SimplicialComplexes" o5 = SimplicialComplexes o5 : Package i6 : R = QQ[a..d] o6 = R o6 : PolynomialRing i7 : D = simplicialComplex {a*b*c,a*b*d,a*c*d,b*c*d} o7 = | bcd acd abd abc | o7 : SimplicialComplex i8 : C = chainComplex D 1 4 6 4 o8 = QQ <-- QQ <-- QQ <-- QQ -1 0 1 2 o8 : ChainComplex i9 : HH_2 C o9 = image | -1 | | 1 | | -1 | | 1 | 4 o9 : QQ-module, submodule of QQ i10 : prune oo 1 o10 = QQ o10 : QQ-module, free