equivariantCohomology -- compute the equivariant cohomology of a toric space (normal toric variety and moment-angle complex)

Usage:

equivariantCohomology(X)

equivariantCohomology(Z)
Inputs:
- X, a normal toric variety,
- Z, a moment-angle complex,
Outputs:
- a module, the torus equivariant cohomology of Z, as a module over polynomial ring

Description

Compute the equivariant cohomology of certain classes of spaces with torus actions: normal toric varieties and moment-angle complex. The equivariant cohomology is computed as a module over polynomial ring in which the coefficients are dependent on the context. For normal toric varieties, the underlying ring is the polynomial ring QQ[t_1, .. t_r] where r is the dimension of the toric variety. For moment-angle complexes, the underlying ring is the polynomial ring k[x_1, ..., x_m] where k is the coefficient ring of the polynomial ring over which the underlying simplicial complex was created.

The equivariant cohomology of a moment-angle complex is free over the polynomial ring when the simplicial complex is a full simplex.

i1 : needsPackage "SimplicialComplexes"

o1 = SimplicialComplexes

o1 : Package

i2 : R = QQ[x,y,z]

o2 = R

o2 : PolynomialRing

i3 : K = simplicialComplex {x*y*z}

o3 = simplicialComplex | xyz |

o3 : SimplicialComplex

i4 : Z = momentAngleComplex K

o4 = MomentAngleComplex{MACSimplicialComplex => simplicialComplex | xyz |}

o4 : MomentAngleComplex

i5 : M = equivariantCohomology Z

               1
o5 = (QQ[x..z])

o5 : QQ[x..z]-module, free

i6 : isFreeModule M

o6 = true

If there is any missing simplex, then the equivariant cohomology is not free.

i7 : needsPackage "SimplicialComplexes"

o7 = SimplicialComplexes

o7 : Package

i8 : R = QQ[x,y,z]

o8 = R

o8 : PolynomialRing

i9 : K = simplicialComplex {x*y, y*z, x*z}

o9 = simplicialComplex | yz xz xy |

o9 : SimplicialComplex

i10 : Z = momentAngleComplex K

o10 = MomentAngleComplex{MACSimplicialComplex => simplicialComplex | yz xz xy |}

o10 : MomentAngleComplex

i11 : M = equivariantCohomology Z

o11 = cokernel | xyz |

                                             1
o11 : QQ[x..z]-module, quotient of (QQ[x..z])

i12 : isFreeModule M

o12 = false

We can also compute the equivariant cohomology of a normal toric variety. In the example below, we compute the equivariant cohomology of $\mathbb{CP}^2$ with respect to the standard torus ($T^2$) action.

i13 : needsPackage "NormalToricVarieties"

o13 = NormalToricVarieties

o13 : Package

i14 : rayList = {{1, 0}, {0, 1}, {}}

o14 = {{1, 0}, {0, 1}, {}}

o14 : List

Ways to use equivariantCohomology:

equivariantCohomology(MomentAngleComplex)
equivariantCohomology(NormalToricVariety)

For the programmer

The object equivariantCohomology is a method function.

The source of this document is in ToricTopology/Documentation.m2:240:0.