isGolodAInf -- Determines if the ring is Golod or not

Synopsis

Usage:

h = isGolod R
Inputs:
- R, a ring,
Outputs:
- h, a Boolean value, Whether or not the ring is Golod

Description

This function computes the A-infinity multiplications to all required orders, and reduces them modulo the maximal ideal. If all reductions are zero, then the ring R is Golod.

Below is an example of an artinian ring R (based on an example of Roos and Katthan) which has minimal multiplications of order two, but is not Golod.

i1 : kk = ZZ/101

o1 = kk

o1 : QuotientRing

i2 : S = kk[x,y,z,u]

o2 = S

o2 : PolynomialRing

i3 : I = ideal(u^3, x*y^2, (x+y)*z^2, x^2*u+z*u^2, y^2*u+x*z*u, y^2*z+y*z^2)

             3     2     2      2   2       2   2            2       2
o3 = ideal (u , x*y , x*z  + y*z , x u + z*u , y u + x*z*u, y z + y*z )

o3 : Ideal of S

i4 : J = trim (I + (ideal vars S)^6)

             3   2            2       2     2      2   2       2     2   5  
o4 = ideal (u , y u + x*z*u, x u + z*u , x*z  + y*z , y z + y*z , x*y , z u,
     ------------------------------------------------------------------------
      6   4      5    6   5    6
     z , x y*z, x z, y , x y, x )

o4 : Ideal of S

i5 : hasMinimalMult(quotient J, 2)

o5 = true

i6 : isGolodAInf quotient J

o6 = false

Ways to use isGolodAInf :

isGolodAInf(Ring)

For the programmer

The object isGolodAInf is a method function.