Let S be the homogeneous coordinate ring of PN, and x0,...,xN be the coordinates. Let π:X→Pn be a Noether normalization. Note that giving a coherent sheaf F on X is equivalent to giving a sheaf G (=π*F) on Pn together with multiplication maps Xi (=π* (⋅xi)) : G→G(1) such that Xi Xj = Xj Xi for every i, j, and f(X0, ..., Xn)=0 for every f ∈I. In other words, {X0,...,XN} gives an action which makes G into an OX-module.
This method checks first that actionList is composed of commuting matrices, and then checks whether f(X0,...,Xn)=0 for each generator f of I.
The following is an example when C is a conic, F=OC, and π is a linear projection at the coordinate point [0:0:1]. In the case, the pushforward π*F = OP1 ⊕OP1(-1).
i1 : S=QQ[x_0..x_2]; R=QQ[y_0,y_1]; |
i3 : I=ideal(x_0*x_1-x_2^2); o3 : Ideal of S |
i4 : M=R^{{1:0},{1:-1}};
|
i5 : X0=map(M**R^{1},M,{{y_0,0},{0,y_0}})
o5 = {-1} | y_0 0 |
{0} | 0 y_0 |
2 2
o5 : Matrix R <--- R
|
i6 : X1=map(M**R^{1},M,{{y_1,0},{0,y_1}})
o6 = {-1} | y_1 0 |
{0} | 0 y_1 |
2 2
o6 : Matrix R <--- R
|
i7 : X2=map(M**R^{1},M,{{0,y_0*y_1},{1,0}})
o7 = {-1} | 0 y_0y_1 |
{0} | 1 0 |
2 2
o7 : Matrix R <--- R
|
i8 : isAction(I,{X0,X1,X2})
o8 = true
|