# homologyClass -- Computes the element of the homology algebra corresponding to a cycle in a DGAlgebra.

## Synopsis

• Usage:
h = homologyClass(A,z)
• Inputs:
• Outputs:
• h, ,

## Description

This function computes the element in the homology algebra of a cycle in a DGAlgebra. In order to do this, the homologyAlgebra is retrieved (or computed, if it hasn't been already).

 i1 : Q = QQ[x,y,z] o1 = Q o1 : PolynomialRing i2 : I = ideal (x^3,y^3,z^3) 3 3 3 o2 = ideal (x , y , z ) o2 : Ideal of Q i3 : R = Q/I o3 = R o3 : QuotientRing i4 : KR = koszulComplexDGA R o4 = {Ring => R } Underlying algebra => R[T ..T ] 1 3 Differential => {x, y, z} o4 : DGAlgebra i5 : z1 = x^2*T_1 2 o5 = x T 1 o5 : R[T ..T ] 1 3 i6 : z2 = y^2*T_2 2 o6 = y T 2 o6 : R[T ..T ] 1 3 i7 : H = HH(KR) Finding easy relations : -- used 0.0159403 seconds o7 = H o7 : PolynomialRing, 3 skew commutative variables i8 : homologyClass(KR,z1*z2) o8 = X X 1 2 o8 : H

## Ways to use homologyClass :

• "homologyClass(DGAlgebra,RingElement)"

## For the programmer

The object homologyClass is .