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NormalToricVarieties :: HH^ZZ(NormalToricVariety,CoherentSheaf)

HH^ZZ(NormalToricVariety,CoherentSheaf) -- compute the cohomology of a coherent sheaf

Synopsis

Description

The cohomology functor HHi(X,-) from the category of sheaves of abelian groups to the category of abelian groups is the right derived functor of the global sections functor.

As a simple example, we compute the dimensions of the cohomology groups for some line bundles on the projective plane.

i1 : PP2 = projectiveSpace 2;
i2 : HH^0(PP2,OO_PP2(1))

       3
o2 = QQ

o2 : QQ-module, free
i3 : apply(10, i -> HH^2(PP2,OO_PP2(-i)))

                 1    3    6    10    15    21    28
o3 = {0, 0, 0, QQ , QQ , QQ , QQ  , QQ  , QQ  , QQ  }

o3 : List
i4 : loadPackage "BoijSoederberg";
i5 : loadPackage "BGG";
i6 : cohomologyTable(CoherentSheaf,NormalToricVariety,ZZ,ZZ):=CohomologyTally=>(
           (F,X,lo,hi) -> new CohomologyTally from select(flatten apply(1+dim X, 
               j -> apply(toList(lo-j..hi), i -> {(j,i),rank HH^j(X,F(i))})), 
             p -> p#1 != 0));
i7 : cohomologyTable(OO_PP2^1,PP2,-10,10)

        -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2  3  4  5  6  7  8  9 10
o7 = 2:  55 45 36 28 21 15 10  6  3  1 . . .  .  .  .  .  .  .  .  .
     1:   .  .  .  .  .  .  .  .  .  . . . .  .  .  .  .  .  .  .  .
     0:   .  .  .  .  .  .  .  .  .  . 1 3 6 10 15 21 28 36 45 55 66

o7 : CohomologyTally
Compare this table with the first example in cohomologyTable.

For a second example, we compute the dimensions of the cohomology groups for some line bundles on a Hirzebruch surface

i8 : cohomologyTable(ZZ,CoherentSheaf,List,List):=(k,F,lo,hi)->(
           new CohomologyTally from select(flatten apply(toList(lo#0..hi#0),
               j -> apply(toList(lo#1..hi#1), 
                 i -> {(j,i-j), rank HH^k(variety F, F(i,j))})), 
             p -> p#1 != 0));
i9 : FF2 = hirzebruchSurface 2;
i10 : cohomologyTable(0,OO_FF2^1,{-7,-7},{7,7})

         -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4   5   6   7
o10 = 7: 20 25 30 36 42 49 56 64 72 80 88 96 104 112 120
      6: 12 16 20 25 30 36 42 49 56 63 70 77  84  91  98
      5:  6  9 12 16 20 25 30 36 42 48 54 60  66  72  78
      4:  2  4  6  9 12 16 20 25 30 35 40 45  50  55  60
      3:  .  1  2  4  6  9 12 16 20 24 28 32  36  40  44
      2:  .  .  .  1  2  4  6  9 12 15 18 21  24  27  30
      1:  .  .  .  .  .  1  2  4  6  8 10 12  14  16  18
      0:  .  .  .  .  .  .  .  1  2  3  4  5   6   7   8

o10 : CohomologyTally
i11 : cohomologyTable(1,OO_FF2^1,{-7,-7},{7,7})

          -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6  7
o11 =  7: 12  9  6  4  2  1  . . . . . . . .  .
       6: 12  9  6  4  2  1  . . . . . . . .  .
       5: 12  9  6  4  2  1  . . . . . . . .  .
       4: 12  9  6  4  2  1  . . . . . . . .  .
       3: 12  9  6  4  2  1  . . . . . . . .  .
       2: 12  9  6  4  2  1  . . . . . . . .  .
       1: 10  8  6  4  2  1  . . . . . . . .  .
       0:  6  5  4  3  2  1  . . . . . . . .  .
      -1:  .  .  .  .  .  .  . . . . . . . .  .
      -2:  .  .  .  .  .  .  . . . 1 2 3 4 5  6
      -3:  .  .  .  .  .  .  . . . 1 2 4 6 8 10
      -4:  .  .  .  .  .  .  . . . 1 2 4 6 9 12
      -5:  .  .  .  .  .  .  . . . 1 2 4 6 9 12
      -6:  .  .  .  .  .  .  . . . 1 2 4 6 9 12
      -7:  .  .  .  .  .  .  . . . 1 2 4 6 9 12

o11 : CohomologyTally
i12 : cohomologyTable(2,OO_FF2^1,{-7,-7},{7,7})

          -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5 6 7
o12 = -2:  8  7  6  5  4  3  2  1  .  .  .  .  . . .
      -3: 18 16 14 12 10  8  6  4  2  1  .  .  . . .
      -4: 30 27 24 21 18 15 12  9  6  4  2  1  . . .
      -5: 44 40 36 32 28 24 20 16 12  9  6  4  2 1 .
      -6: 60 55 50 45 40 35 30 25 20 16 12  9  6 4 2
      -7: 78 72 66 60 54 48 42 36 30 25 20 16 12 9 6

o12 : CohomologyTally

When F is free, the algorithm based on Diane Maclagan, Gregory G. Smith, Multigraded Castelnuovo-Mumford regularity, J. Reine Angew. Math. 571 (2004), 179-212. The general case uses the methods described in David Eisenbud, Mircea Mustata, Mike Stillman, Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583-600.

See also