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torAlgebra(Ring,Ring) -- Computes Tor_R(S,k) up to a specified generating and relating degree.

Synopsis

• Function: torAlgebra
• Usage:
TorRS = torAlgebra(R,S,GenDegreeLimit=>m,RelDegreeLimit=>n)
• 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
• Outputs:

Description

 i1 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4} o1 = R o1 : QuotientRing i2 : M = coker matrix {{a^3*b^3*c^3*d^3}}; i3 : S = R/ideal{a^3*b^3*c^3*d^3} o3 = S o3 : QuotientRing i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8) -- used 0.528106s (cpu); 0.42708s (thread); 0s (gc) Finding easy relations : o4 = HB o4 : QuotientRing i5 : numgens HB o5 = 35 i6 : apply(5,i -> #(flatten entries getBasis(i,HB))) o6 = {1, 1, 4, 10, 20} o6 : List i7 : Mres = res(M, LengthLimit=>8) 1 1 4 10 20 35 56 84 120 o7 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R 0 1 2 3 4 5 6 7 8 o7 : ChainComplex

Note that in this example, $Tor_*^R(S,k)$ has trivial multiplication, since the map from R to S is a Golod homomorphism by a theorem of Levin and Avramov.