The first version computes a basic Beilinson bundle, i.e. the pullback of a Beilinson bundle from a single factor of a the product PP = Pn0 ×... ×Pn(r-1) of r projective spaces.
The second version computes the tensor product of the basic bundles beilinsonBundle(ai,i,E) for i from 0 to r-1. See also Tate Resolutions on Products of Projective Spaces.
The vector bundle B is represented by its S-module of global sections, which is either the quotient or a submodule of a free S-modules depending on the value of the option BundleType.
The results are stashed in E.TateData.BeilinsonBundles, so they are not recomputed.
i1 : (S,E) = productOfProjectiveSpaces {2,3}
o1 = (S, E)
o1 : Sequence
|
i2 : B1=beilinsonBundle(1,0,E)
o2 = cokernel {1, 0} | x_(0,2) |
{1, 0} | -x_(0,1) |
{1, 0} | x_(0,0) |
3
o2 : S-module, quotient of S
|
i3 : B2=beilinsonBundle(1,1,E)
o3 = cokernel {0, 1} | x_(1,2) x_(1,3) 0 0 |
{0, 1} | -x_(1,1) 0 x_(1,3) 0 |
{0, 1} | x_(1,0) 0 0 x_(1,3) |
{0, 1} | 0 -x_(1,1) -x_(1,2) 0 |
{0, 1} | 0 x_(1,0) 0 -x_(1,2) |
{0, 1} | 0 0 x_(1,0) x_(1,1) |
6
o3 : S-module, quotient of S
|
i4 : B=beilinsonBundle({1,1},E); betti B
0 1
o5 = total: 18 18
2: 18 18
o5 : BettiTally
|
i6 : B1**B2 == B o6 = true |