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# isHomologyAlgebraTrivial -- Determines if the homology algebra of a DGAlgebra is trivial

## Synopsis

• Usage:
isTriv = isHomologyAlgebraTrivial(A)
• Inputs:
• Optional inputs:
• GenDegreeLimit => ..., default value infinity, Option to specify the maximum degree to look for generators
• Outputs:
• isTriv, ,

## Description

This function computes the homology algebra of the DGAlgebra A and determines if the multiplication on H(A) is trivial.

 i1 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4} o1 = R o1 : QuotientRing i2 : S = R/ideal{a^3*b^3*c^3*d^3} o2 = S o2 : QuotientRing i3 : A = acyclicClosure(R,EndDegree=>3) o3 = {Ring => R } Underlying algebra => R[T ..T ] 1 8 3 3 3 3 Differential => {a, b, c, d, a T , b T , c T , d T } 1 2 3 4 o3 : DGAlgebra i4 : B = A ** S o4 = {Ring => S } Underlying algebra => S[T ..T ] 1 8 3 3 3 3 Differential => {a, b, c, d, a T , b T , c T , d T } 1 2 3 4 o4 : DGAlgebra i5 : isHomologyAlgebraTrivial(B,GenDegreeLimit=>6) o5 = true

The command returns true since R --> S is Golod. Notice we also used the option GenDegreeLimit here.

 i6 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4} o6 = R o6 : QuotientRing i7 : A = koszulComplexDGA(R) o7 = {Ring => R } Underlying algebra => R[T ..T ] 1 4 Differential => {a, b, c, d} o7 : DGAlgebra i8 : isHomologyAlgebraTrivial(A) o8 = false

The command returns false, since R is Gorenstein, and so HA has Poincare Duality, hence the multiplication is far from trivial.

## Ways to use isHomologyAlgebraTrivial :

• isHomologyAlgebraTrivial(DGAlgebra)

## For the programmer

The object isHomologyAlgebraTrivial is .