-- -*- M2-comint -*- {* hash: 1897382560 *}
i1 : S = ZZ/101[x,y,z];
i2 : FF=res ((ideal vars S)^3);
i3 : M=coker (FF.dd_2)
o3 = cokernel {3} | -y 0 0 -z 0 0 0 0 0 0 0 0 0 0 0 |
{3} | x -y 0 0 -z 0 0 0 0 0 0 0 0 0 0 |
{3} | 0 x -y 0 0 0 -z 0 0 0 0 0 0 0 0 |
{3} | 0 0 x 0 0 0 0 0 -z 0 0 0 0 0 0 |
{3} | 0 0 0 x y -y 0 0 0 -z 0 0 0 0 0 |
{3} | 0 0 0 0 0 x y -y 0 0 -z 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 x y 0 0 0 -z 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 x y -y 0 -z 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 x y 0 -z |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 x y |
10
o3 : S-module, quotient of S
i4 : universalEmbedding M
o4 = | x3 x2y xy2 y3 x2z xyz y2z xz2 yz2 z3 |
o4 : Matrix
i5 : x = symbol x;
i6 : R=QQ[x_1..x_8];
i7 : m1=genericMatrix(R,x_1,2,2); m2=genericMatrix(R,x_5,2,2);
2 2
o7 : Matrix R <--- R
2 2
o8 : Matrix R <--- R
i9 : m=m1*m2
o9 = | x_1x_5+x_3x_6 x_1x_7+x_3x_8 |
| x_2x_5+x_4x_6 x_2x_7+x_4x_8 |
2 2
o9 : Matrix R <--- R
i10 : d1=minors(2,m1); d2=minors(2,m2);
o10 : Ideal of R
o11 : Ideal of R
i12 : M=matrix{{0,d1_0,m_(0,0),m_(0,1)}, {0,0,m_(1,0),m_(1,1)}, {0,0,0,d2_0}, {0,0,0,0}}
o12 = | 0 -x_2x_3+x_1x_4 x_1x_5+x_3x_6 x_1x_7+x_3x_8 |
| 0 0 x_2x_5+x_4x_6 x_2x_7+x_4x_8 |
| 0 0 0 -x_6x_7+x_5x_8 |
| 0 0 0 0 |
4 4
o12 : Matrix R <--- R
i13 : M=M-(transpose M);
4 4
o13 : Matrix R <--- R
i14 : N= coker(res coker transpose M).dd_2
o14 = cokernel {0} | -x_2x_5-x_4x_6 -x_2x_7-x_4x_8 x_6x_7-x_5x_8 0 |
{2} | x_2x_3-x_1x_4 0 x_1x_7+x_3x_8 x_2x_7+x_4x_8 |
{2} | -x_1x_5-x_3x_6 -x_1x_7-x_3x_8 0 -x_6x_7+x_5x_8 |
{2} | 0 x_2x_3-x_1x_4 -x_1x_5-x_3x_6 -x_2x_5-x_4x_6 |
4
o14 : R-module, quotient of R
i15 : universalEmbedding(N)
o15 = {-1} | x_1 -x_6 -x_2 -x_8 |
{-1} | x_3 x_5 -x_4 x_7 |
o15 : Matrix
i16 : p = 3;
i17 : S = ZZ/p[x,y,z];
i18 : R = S/((ideal(x^p,y^p))+(ideal(x,y,z))^(p+1))
o18 = R
o18 : QuotientRing
i19 : I = module ideal(z)
o19 = image | z |
1
o19 : R-module, submodule of R
i20 : betti Hom(I,R^1)
0 1
o20 = total: 3 14
0: 3 3
1: . 2
2: . 9
o20 : BettiTally
i21 : ui = universalEmbedding I
o21 = | z |
| y |
| x |
o21 : Matrix
i22 : kernel ui
o22 = image 0
1
o22 : R-module, submodule of R
i23 : inci = map(R^1,I,matrix{{z}})
o23 = | z |
o23 : Matrix
i24 : kernel inci
o24 = image 0
1
o24 : R-module, submodule of R
i25 : gi = map(R^2, I, matrix{{x},{y}})
o25 = | x |
| y |
o25 : Matrix
i26 : kernel gi
o26 = image 0
1
o26 : R-module, submodule of R
i27 : u= map(R^3,R^{-1},ui)
o27 = | z |
| y |
| x |
3 1
o27 : Matrix R <--- R
i28 : inc=map(R^1, R^{-1}, matrix{{z}})
o28 = | z |
1 1
o28 : Matrix R <--- R
i29 : g=map(R^2, R^{-1}, matrix{{x},{y}})
o29 = | x |
| y |
2 1
o29 : Matrix R <--- R
i30 : A=symmetricKernel u
3 2 2 2 2 2 2
o30 = ideal (z w , y*z w , x*z w , y z*w , x*y*z*w , x z*w , x*y w , x y*w ,
0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 3 3 3 4
z w , y*z*w , x*z*w , y w , x*y*w , x w , z*w , y*w , x*w , w )
0 0 0 0 0 0 0 0 0 0
o30 : Ideal of R[w ]
0
i31 : B1=symmetricKernel inc
3 2 2 2 2 2 2
o31 = ideal (z w , y*z w , x*z w , y z*w , x*y*z*w , x z*w , x*y w , x y*w ,
0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 3 3 3 4
z w , y*z*w , x*z*w , y w , x*y*w , x w , z*w , y*w , x*w , w )
0 0 0 0 0 0 0 0 0 0
o31 : Ideal of R[w ]
0
i32 : B=(map(ring A, ring B1)) B1
3 2 2 2 2 2 2
o32 = ideal (z w , y*z w , x*z w , y z*w , x*y*z*w , x z*w , x*y w , x y*w ,
0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 3 3 3 4
z w , y*z*w , x*z*w , y w , x*y*w , x w , z*w , y*w , x*w , w )
0 0 0 0 0 0 0 0 0 0
o32 : Ideal of R[w ]
0
i33 : C1 = symmetricKernel g
3 3 2 2 2 2 2
o33 = ideal (w , z w , y*z w , x*z w , y z*w , x*y*z*w , x z*w , x*y w ,
0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 2
x y*w , z w , y*z*w , x*z*w , y w , x*y*w , x w )
0 0 0 0 0 0 0
o33 : Ideal of R[w ]
0
i34 : C=(map(ring A, ring C1)) C1
3 3 2 2 2 2 2
o34 = ideal (w , z w , y*z w , x*z w , y z*w , x*y*z*w , x z*w , x*y w ,
0 0 0 0 0 0 0 0
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 2
x y*w , z w , y*z*w , x*z*w , y w , x*y*w , x w )
0 0 0 0 0 0 0
o34 : Ideal of R[w ]
0
i35 : A==B
o35 = true
i36 : A==C
o36 = false
i37 :
