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# homologyAlgebra -- Compute the homology algebra of a DGAlgebra.

## Synopsis

• Usage:
HA = homologyAlgebra(A)
• Inputs:
• Optional inputs:
• GenDegreeLimit => ..., default value infinity, Option to specify the maximum degree to look for generators
• RelDegreeLimit => ..., default value infinity, Option to specify the maximum degree to look for relations
• Verbosity => ..., default value 0, Option to specify the maximum degree to look for generators when computing the deviations
• Outputs:

## Description

 i1 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4} o1 = R o1 : QuotientRing i2 : A = koszulComplexDGA(R) o2 = {Ring => R } Underlying algebra => R[T ..T ] 1 4 Differential => {a, b, c, d} o2 : DGAlgebra i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A)) o3 = {1, 4, 6, 4, 1} o3 : List i4 : HA = homologyAlgebra(A) -- used 0.0199964s (cpu); 0.0196215s (thread); 0s (gc) Finding easy relations : o4 = HA o4 : PolynomialRing, 4 skew commutative variable(s)

Note that HA is a graded commutative polynomial ring (i.e. an exterior algebra) since R is a complete intersection.

 i5 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3} o5 = R o5 : QuotientRing i6 : A = koszulComplexDGA(R) o6 = {Ring => R } Underlying algebra => R[T ..T ] 1 4 Differential => {a, b, c, d} o6 : DGAlgebra i7 : apply(maxDegree A + 1, i -> numgens prune homology(i,A)) o7 = {1, 5, 10, 10, 4} o7 : List i8 : HA = homologyAlgebra(A) -- used 0.201499s (cpu); 0.101594s (thread); 0s (gc) Finding easy relations : o8 = HA o8 : QuotientRing i9 : numgens HA o9 = 19 i10 : HA.cache.cycles 3 3 3 3 2 3 3 3 2 3 3 3 3 2 3 3 o10 = {a T , b T , c T , d T , a b c d T , a b c d T T , a b c d T T , 1 2 3 4 1 1 2 1 2 ----------------------------------------------------------------------- 2 3 3 3 2 3 3 3 2 3 3 3 3 2 3 3 a b c d T T , a b c d T T , a b c d T T T , a b c d T T T , 1 3 1 4 1 2 3 1 2 3 ----------------------------------------------------------------------- 3 3 2 3 2 3 3 3 3 2 3 3 2 3 3 3 a b c d T T T , a b c d T T T , a b c d T T T , a b c d T T T , 1 2 3 1 2 4 1 2 4 1 3 4 ----------------------------------------------------------------------- 2 3 3 3 3 2 3 3 3 3 2 3 3 3 3 2 a b c d T T T T , a b c d T T T T , a b c d T T T T , a b c d T T T T } 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 o10 : List
 i11 : Q = ZZ/101[x,y,z] o11 = Q o11 : PolynomialRing i12 : I = ideal{y^3,z*x^2,y*(z^2+y*x),z^3+2*x*y*z,x*(z^2+y*x),z*y^2,x^3,z*(z^2+2*x*y)} 3 2 2 2 3 2 2 2 3 o12 = ideal (y , x z, x*y + y*z , 2x*y*z + z , x y + x*z , y z, x , 2x*y*z + ----------------------------------------------------------------------- 3 z ) o12 : Ideal of Q i13 : R = Q/I o13 = R o13 : QuotientRing i14 : A = koszulComplexDGA(R) o14 = {Ring => R } Underlying algebra => R[T ..T ] 1 3 Differential => {x, y, z} o14 : DGAlgebra i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A)) o15 = {1, 7, 7, 1} o15 : List i16 : HA = homologyAlgebra(A) -- used 0.0519989s (cpu); 0.0523448s (thread); 0s (gc) Finding easy relations : o16 = HA o16 : QuotientRing

One can check that HA has Poincare duality since R is Gorenstein.

If your DGAlgebra has generators in even degrees, then one must specify the options GenDegreeLimit and RelDegreeLimit.

 i17 : R = ZZ/101[a,b,c,d] o17 = R o17 : PolynomialRing i18 : S = R/ideal{a^4,b^4,c^4,d^4} o18 = S o18 : QuotientRing i19 : A = acyclicClosure(R,EndDegree=>3) o19 = {Ring => R } Underlying algebra => R[T ..T ] 1 4 Differential => {a, b, c, d} o19 : DGAlgebra i20 : B = A ** S o20 = {Ring => S } Underlying algebra => S[T ..T ] 1 4 Differential => {a, b, c, d} o20 : DGAlgebra i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14) -- used 0.0199994s (cpu); 0.0178351s (thread); 0s (gc) Finding easy relations : o21 = HB o21 : PolynomialRing, 4 skew commutative variable(s)

## Ways to use homologyAlgebra :

• homologyAlgebra(DGAlgebra)

## For the programmer

The object homologyAlgebra is .