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 |