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DGAlgebras :: isGolod

isGolod -- Determines if a ring is Golod

Synopsis

Description

This function determines if the Koszul complex of a ring R admits a trivial Massey operation. If one exists, then R is Golod.

i1 : R = ZZ/101[a,b,c,d]/ideal{a^4+b^4+c^4+d^4}

o1 = R

o1 : QuotientRing
i2 : isGolod(R)
Computing generators in degree 1 :      -- used 0.0257067 seconds
Computing generators in degree 2 :      -- used 0.0229142 seconds
Computing generators in degree 3 :      -- used 0.0233473 seconds
Computing generators in degree 4 :      -- used 0.0217527 seconds

o2 = true

Hypersurfaces are Golod, but

i3 : R = ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4}

o3 = R

o3 : QuotientRing
i4 : isGolod(R)
Computing generators in degree 1 :      -- used 0.0280362 seconds
Computing generators in degree 2 :      -- used 0.0604247 seconds
Computing generators in degree 3 :      -- used 0.0575311 seconds
Computing generators in degree 4 :      -- used 0.177327 seconds

o4 = false

complete intersections of higher codimension are not. Here is another example:

i5 : Q = ZZ/101[a,b,c,d]

o5 = Q

o5 : PolynomialRing
i6 : R = Q/(ideal vars Q)^2

o6 = R

o6 : QuotientRing
i7 : isGolod(R)
Computing generators in degree 1 :      -- used 0.029894 seconds
Computing generators in degree 2 :      -- used 0.0831173 seconds
Computing generators in degree 3 :      -- used 0.117811 seconds
Computing generators in degree 4 :      -- used 0.239788 seconds

o7 = true

The above is a (CM) ring minimal of minimal multiplicity, hence Golod.

Caveat

Currently, it does not try to find a full trivial Massey operation for the ring R, it just computes them to second order. Since there is not currently an example of a ring that is not Golod yet has trivial product on H(KR), this is ok for now.

Ways to use isGolod :