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TorAlgebra :: torAlgClass

torAlgClass -- the class (w.r.t. multiplication in homology) of a local ring

Synopsis

Description

Classifies the local ring obtained by localizing R at the irrelevant maximal ideal as belonging to one of the (parametrized) classes B, C(c), G(r), H(p,q), S, or T, provided that it is codepth at most 3.

i1 : Q = QQ[x,y,z];
i2 : torAlgClass Q

o2 = C(0)
i3 : torAlgClass (Q/ideal (x*y))

o3 = C(1)
i4 : torAlgClass (Q/ideal (x^2,y^2))

o4 = C(2)
i5 : torAlgClass (Q/ideal (x^2,y^2,x*y))

o5 = S
i6 : torAlgClass (Q/ideal (x^2,x*y,y*z,z^2))

o6 = B
i7 : torAlgClass (Q/ideal (x^2,y^2,z^2))

o7 = C(3)
i8 : torAlgClass (Q/ideal (x*y,y*z,x^3,x^2*z,x*z^2-y^3,z^3))

o8 = G(3)
i9 : torAlgClass (Q/ideal (x*z+y*z,x*y+y*z,x^2-y*z,y*z^2+z^3,y^3-z^3))

o9 = G(5), Gorenstein
i10 : torAlgClass (Q/ideal (x^2,y^2,z^2,x*y))

o10 = H(3,2)
i11 : torAlgClass (Q/ideal (x^2,y^2,z^2,x*y*z))

o11 = T

If the local ring is Gorenstein or Golod of codepth 4, then it is classified as belonging to one of the (parametrized) classes C(4), GH(p), GS, GT, or codepth 4 Golod.

i12 : Q = QQ[w,x,y,z];
i13 : torAlgClass (Q/ideal (w^2,x^2,y^2,z^2))

o13 = C(4)
i14 : torAlgClass (Q/ideal (y*z,x*z,x*y+z^2,x^2,w*x+y^2+z^2,w^2+w*y+y^2+z^2))

o14 = GH(5)
i15 : torAlgClass (Q/ideal (z^2,x*z,w*z+y*z,y^2,x*y,w*y,x^2,w*x+y*z,w^2+y*z))

o15 = GS
i16 : torAlgClass (Q/ideal (x^2,y^2,z^2,x*w,y*w,z*w,w^3-x*y*z))

o16 = GT
i17 : torAlgClass (Q/(ideal (w,x,y,z))^2)

o17 = codepth 4 Golod

If the local ring has codepth at least 5, then it is classified as belonging to one of the classes C(c), if it is complete intersection, codepth c Gorenstein, if it is Gorenstein and not complete intersection, codepth c Golod, if it is Golod, and no class otherwise.

i18 : Q = QQ[u,v,w,x,y,z];
i19 : torAlgClass (Q/ideal (u^2,v^2,w^2,x^2+y^2, x^2+z^2))

o19 = C(5)
i20 : torAlgClass (Q/ideal (w^2,v*w,z*w,y*w,v^2,z*v+x*w,y*v,x*v,z^2+x*w,y*z,x*z,y^2+x*w,x*y,x^2))

o20 = codepth 5 Gorenstein
i21 : torAlgClass (Q/ideal (x^2*y^2,x^2*z,y^2*z,u^2*z,v^2*z,w^2*z))

o21 = codepth 5 Golod
i22 : torAlgClass (Q/ideal (u^2,v^2,w^2,x^2,z^2,x*y^15))

o22 = codepth 6 no class

If the defining ideal of R is not contained in the irrelevant maximal ideal, then the resulting local ring is zero, and the function returns zero ring.

i23 : Q = QQ[x,y,z];
i24 : torAlgClass (Q/ideal (x^2-1))

o24 = zero ring