Is there an algorithm to find relations in polynomial algebra?
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In the context of equivariant cohomology for a GKM-manifold, I'm trying to compute the cohomology of the variety given by $x_1y_1+x_2y_2+x_3y_3=0$ with the action of $(mathbb{C}^*)^3$:
$$
((t_1,t_2,t_3),[x_1,x_2,x_3,y_1,y_2,y_3]) mapsto [t_1t_2t_3x_1,t_1t_2x_2,t_1t_3x_3,y_1,t_3y_2,t_2y_3].
$$
Thanks to a theorem of Goresky, Kottwitz, MacPherson, I can see the equivariant cohomology ring of the variety embedded as an algebra in $bigoplus^6 mathbb{C}[t_1,t_2,t_3]$.
I've found the generators for this algebra, that are
begin{align*}
a&=[1,1,1,1,1,1];\
b&=[0,t_3,t_2,t_3-t_1,t_2-t_1,t_3+t_2-t_1];\
c&=[0,0,t_2(t_2-t_3),t_1(t_1-t_3),(t_2-t_1)(t_2-t_3),t_2(t_2-t_1)];\
d&=[0,0,0,t_1(t_1-t_3),t_1(t_1-t_2),(t_1-t_3)(t_1-t_2)];\
e&=[0,0,0,0,t_1(t_1-t_2)(t_2-t_3),t_2(t_1-t_2)(t_1-t_3)];\
f&=[0,0,0,0,0,t_2t_3(t_1-t_2)(t_1-t_3)].
end{align*}
but I'm in trouble to fins the relations between them, so there exists an algorithm that can compute them for me?
abstract-algebra algebraic-geometry group-actions
asked Nov 22 at 10:10
Bobech
699
add a comment |
up vote
0
down vote
favorite
In the context of equivariant cohomology for a GKM-manifold, I'm trying to compute the cohomology of the variety given by $x_1y_1+x_2y_2+x_3y_3=0$ with the action of $(mathbb{C}^*)^3$:
$$
((t_1,t_2,t_3),[x_1,x_2,x_3,y_1,y_2,y_3]) mapsto [t_1t_2t_3x_1,t_1t_2x_2,t_1t_3x_3,y_1,t_3y_2,t_2y_3].
$$
Thanks to a theorem of Goresky, Kottwitz, MacPherson, I can see the equivariant cohomology ring of the variety embedded as an algebra in $bigoplus^6 mathbb{C}[t_1,t_2,t_3]$.
I've found the generators for this algebra, that are
begin{align*}
a&=[1,1,1,1,1,1];\
b&=[0,t_3,t_2,t_3-t_1,t_2-t_1,t_3+t_2-t_1];\
c&=[0,0,t_2(t_2-t_3),t_1(t_1-t_3),(t_2-t_1)(t_2-t_3),t_2(t_2-t_1)];\
d&=[0,0,0,t_1(t_1-t_3),t_1(t_1-t_2),(t_1-t_3)(t_1-t_2)];\
e&=[0,0,0,0,t_1(t_1-t_2)(t_2-t_3),t_2(t_1-t_2)(t_1-t_3)];\
f&=[0,0,0,0,0,t_2t_3(t_1-t_2)(t_1-t_3)].
end{align*}
but I'm in trouble to fins the relations between them, so there exists an algorithm that can compute them for me?
abstract-algebra algebraic-geometry group-actions
asked Nov 22 at 10:10
Bobech
699
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In the context of equivariant cohomology for a GKM-manifold, I'm trying to compute the cohomology of the variety given by $x_1y_1+x_2y_2+x_3y_3=0$ with the action of $(mathbb{C}^*)^3$:
$$
((t_1,t_2,t_3),[x_1,x_2,x_3,y_1,y_2,y_3]) mapsto [t_1t_2t_3x_1,t_1t_2x_2,t_1t_3x_3,y_1,t_3y_2,t_2y_3].
$$
Thanks to a theorem of Goresky, Kottwitz, MacPherson, I can see the equivariant cohomology ring of the variety embedded as an algebra in $bigoplus^6 mathbb{C}[t_1,t_2,t_3]$.
I've found the generators for this algebra, that are
begin{align*}
a&=[1,1,1,1,1,1];\
b&=[0,t_3,t_2,t_3-t_1,t_2-t_1,t_3+t_2-t_1];\
c&=[0,0,t_2(t_2-t_3),t_1(t_1-t_3),(t_2-t_1)(t_2-t_3),t_2(t_2-t_1)];\
d&=[0,0,0,t_1(t_1-t_3),t_1(t_1-t_2),(t_1-t_3)(t_1-t_2)];\
e&=[0,0,0,0,t_1(t_1-t_2)(t_2-t_3),t_2(t_1-t_2)(t_1-t_3)];\
f&=[0,0,0,0,0,t_2t_3(t_1-t_2)(t_1-t_3)].
end{align*}
but I'm in trouble to fins the relations between them, so there exists an algorithm that can compute them for me?
abstract-algebra algebraic-geometry group-actions
asked Nov 22 at 10:10
Bobech
699
In the context of equivariant cohomology for a GKM-manifold, I'm trying to compute the cohomology of the variety given by $x_1y_1+x_2y_2+x_3y_3=0$ with the action of $(mathbb{C}^*)^3$:
$$
((t_1,t_2,t_3),[x_1,x_2,x_3,y_1,y_2,y_3]) mapsto [t_1t_2t_3x_1,t_1t_2x_2,t_1t_3x_3,y_1,t_3y_2,t_2y_3].
$$
Thanks to a theorem of Goresky, Kottwitz, MacPherson, I can see the equivariant cohomology ring of the variety embedded as an algebra in $bigoplus^6 mathbb{C}[t_1,t_2,t_3]$.
I've found the generators for this algebra, that are
begin{align*}
a&=[1,1,1,1,1,1];\
b&=[0,t_3,t_2,t_3-t_1,t_2-t_1,t_3+t_2-t_1];\
c&=[0,0,t_2(t_2-t_3),t_1(t_1-t_3),(t_2-t_1)(t_2-t_3),t_2(t_2-t_1)];\
d&=[0,0,0,t_1(t_1-t_3),t_1(t_1-t_2),(t_1-t_3)(t_1-t_2)];\
e&=[0,0,0,0,t_1(t_1-t_2)(t_2-t_3),t_2(t_1-t_2)(t_1-t_3)];\
f&=[0,0,0,0,0,t_2t_3(t_1-t_2)(t_1-t_3)].
end{align*}
but I'm in trouble to fins the relations between them, so there exists an algorithm that can compute them for me?
abstract-algebra algebraic-geometry group-actions
abstract-algebra algebraic-geometry group-actions
asked Nov 22 at 10:10
Bobech
699
asked Nov 22 at 10:10
Bobech
699
asked Nov 22 at 10:10
Bobech
699
asked Nov 22 at 10:10
Bobech
699
asked Nov 22 at 10:10
Bobech
699
699
add a comment |
add a comment |
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