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beilinson -- apply the beilinson functor

Synopsis

Description

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 U^a. Note that the U^a for multidegrees a=\{a_1,...,a_t\} with 0 \le a_i \le n_i form a full exceptional series for the derived category of coherent sheaves on the product PP = P^{n_1} \times ... \times P^{n_t} 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 chosen 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 $\mathcal F$ is the sheafication of beilinson( T($\mathcal F$)) of its Tate resolution T($\mathcal F$).

i1 : (S,E) = productOfProjectiveSpaces {2,1}

o1 = (S, E)

o1 : Sequence
 
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
 
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
 
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
 
i5 : T = chainComplex(psi)

      1      1
o5 = E  <-- E
             
     0      1

o5 : ChainComplex
 
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
 
i7 : betti T

            0 1
o7 = total: 1 1
         1: 1 .
         2: . 1

o7 : BettiTally

See also

Ways to use beilinson:

For the programmer

The object beilinson is a method function with options.