The Beilinson functor is a functor from the category of free E-modules to the category of coherent sheaves which associates to a cyclic free E-module of generated in multidegree a the vector bundle Ua. Note that the Ua for multidegrees a={a1,...,at}with 0 ≤ai ≤ni form a full exceptional series for the derived category of coherent sheaves on the product PP = Pn1 ×... ×Pnt of t projective spaces, see e.g. Tate Resolutions on Products of Projective Spaces.
In the function we compute from a complex of free E-modules the corresponding complex of graded S-modules, whose sheafifications are the corresponding sheaves. The corresponding graded S-module are choosen as quotients of free S-modules in case of the default option BundleType=>PrunedQuotient, or as submodules of free S-modules. The true Beilinson functor is obtained by the sheafication of resulting the complex.
The Beilinson monad of a coherent sheaf F is the the sheafication of beilinson( T(F)) of its Tate resolution T(F).
i1 : (S,E) = productOfProjectiveSpaces {2,1}
o1 = (S, E)
o1 : Sequence
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i2 : psi=random(E^{{-1,0}}, E^{{-2,-1}})
o2 = {1, 0} | 107e_(0,0)e_(1,0)-5570e_(0,1)e_(1,0)+3783e_(0,2)e_(1,0)+4376e_(
------------------------------------------------------------------------
0,0)e_(1,1)+3187e_(0,1)e_(1,1)-5307e_(0,2)e_(1,1) |
1 1
o2 : Matrix E <--- E
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i3 : phi=beilinson psi
o3 = {1, 0} | -5307x_(1,0)-3783x_(1,1) |
{1, 0} | -3187x_(1,0)-5570x_(1,1) |
{1, 0} | 4376x_(1,0)-107x_(1,1) |
o3 : Matrix
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i4 : beilinson(E^{{-1,0}})
o4 = cokernel {1, 0} | x_(0,2) |
{1, 0} | -x_(0,1) |
{1, 0} | x_(0,0) |
3
o4 : S-module, quotient of S
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i5 : T = chainComplex(psi)
1 1
o5 = E <-- E
0 1
o5 : ChainComplex
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i6 : C = beilinson T
1
o6 = cokernel {1, 0} | x_(0,2) | <-- S
{1, 0} | -x_(0,1) |
{1, 0} | x_(0,0) | 1
0
o6 : ChainComplex
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i7 : betti T
0 1
o7 = total: 1 1
1: 1 .
2: . 1
o7 : BettiTally
|