I want to define a monad bundle on P1xP1 and compute its sheaf cohomology groups. As a toy example, say I want to compute the cohomology groups of the kernel of {{x,y,u,v}}, where (x:y) and (u:v) are coordinates on the two P1 factors respectively.
I found two ways to define P1xP1. However, in both cases the last line attempting to compute the cohomology gives the error "expected degree length 1".
Way 1:
needsPackage "SegreClasses"
R = makeProductRing(QQ,{1,1})
F = sheaf ker matrix {{a,b,c,d}}
HH^0(F)
Way 2:
needsPackage "TateOnProducts"
(S,E) = productOfProjectiveSpaces{1,1}
F = sheaf ker matrix {{x_(0,0),x_(1,0),x_(0,1),x_(1,1)}}
HH^0(F)
Alternatively, I could view P1xP1 as a smooth quadric in P3 under the Segre embedding. But I don't know how to define the sheaf in terms of the coordinates on P3. How can I compute the cohomology groups I'm after?