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

Synopsis

Function: cohomology
Usage:

HH^i (X, F)
Inputs:
- i, an integer,
- X, a normal toric variety,
- F, a coherent sheaf, on X
Optional inputs:
- Degree (missing documentation) => ..., default value 0,
Outputs:
- a module, the i-th cohomology group of F

Description

The cohomology functor $HH^i (X,-)$ 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 = toricProjectiveSpace 2;

i2 : HH^0 (PP2, OO_PP2(1))

       3
o2 = QQ

o2 : QQ-module, free

i3 : matrix table (reverse toList (0..2), toList (-10..5),  (i,j) -> rank HH^i (PP2, OO_PP2(j-i)))

o3 = | 55 45 36 28 21 15 10 6 3 1 0 0 0 0  0  0  |
     | 0  0  0  0  0  0  0  0 0 0 0 0 0 0  0  0  |
     | 0  0  0  0  0  0  0  0 0 0 1 3 6 10 15 21 |

              3       16
o3 : Matrix ZZ  <-- ZZ

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

i4 : FF2 = hirzebruchSurface 2;

i5 : HH^0 (FF2, OO_FF2(1,1))

       6
o5 = QQ

o5 : QQ-module, free

i6 : matrix table (reverse toList (-7..7), toList (-7..7),  (i,j) -> rank HH^0 (FF2, OO_FF2(j,i)))

o6 = | 20 25 30 36 42 49 56 64 72 80 88 96 104 112 120 |
     | 12 16 20 25 30 36 42 49 56 63 70 77 84  91  98  |
     | 6  9  12 16 20 25 30 36 42 48 54 60 66  72  78  |
     | 2  4  6  9  12 16 20 25 30 35 40 45 50  55  60  |
     | 0  1  2  4  6  9  12 16 20 24 28 32 36  40  44  |
     | 0  0  0  1  2  4  6  9  12 15 18 21 24  27  30  |
     | 0  0  0  0  0  1  2  4  6  8  10 12 14  16  18  |
     | 0  0  0  0  0  0  0  1  2  3  4  5  6   7   8   |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0   0   0   |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0   0   0   |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0   0   0   |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0   0   0   |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0   0   0   |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0   0   0   |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0   0   0   |

              15       15
o6 : Matrix ZZ   <-- ZZ

i7 : matrix table (reverse toList (-7..7), toList (-7..7),  (i,j) -> rank HH^1 (FF2, OO_FF2(j,i)))

o7 = | 12 9 6 4 2 1 0 0 0 0 0 0 0 0 0  |
     | 12 9 6 4 2 1 0 0 0 0 0 0 0 0 0  |
     | 12 9 6 4 2 1 0 0 0 0 0 0 0 0 0  |
     | 12 9 6 4 2 1 0 0 0 0 0 0 0 0 0  |
     | 12 9 6 4 2 1 0 0 0 0 0 0 0 0 0  |
     | 12 9 6 4 2 1 0 0 0 0 0 0 0 0 0  |
     | 10 8 6 4 2 1 0 0 0 0 0 0 0 0 0  |
     | 6  5 4 3 2 1 0 0 0 0 0 0 0 0 0  |
     | 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0  |
     | 0  0 0 0 0 0 0 0 0 1 2 3 4 5 6  |
     | 0  0 0 0 0 0 0 0 0 1 2 4 6 8 10 |
     | 0  0 0 0 0 0 0 0 0 1 2 4 6 9 12 |
     | 0  0 0 0 0 0 0 0 0 1 2 4 6 9 12 |
     | 0  0 0 0 0 0 0 0 0 1 2 4 6 9 12 |
     | 0  0 0 0 0 0 0 0 0 1 2 4 6 9 12 |

              15       15
o7 : Matrix ZZ   <-- ZZ

i8 : matrix table (reverse toList (-7..7), toList (-7..7),  (i,j) -> rank HH^2 (FF2, OO_FF2(j,i)))

o8 = | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
     | 0  0  0  0  0  0  0  0  0  0  0  0  0  0 0 |
     | 8  7  6  5  4  3  2  1  0  0  0  0  0  0 0 |
     | 18 16 14 12 10 8  6  4  2  1  0  0  0  0 0 |
     | 30 27 24 21 18 15 12 9  6  4  2  1  0  0 0 |
     | 44 40 36 32 28 24 20 16 12 9  6  4  2  1 0 |
     | 60 55 50 45 40 35 30 25 20 16 12 9  6  4 2 |
     | 78 72 66 60 54 48 42 36 30 25 20 16 12 9 6 |

              15       15
o8 : Matrix ZZ   <-- ZZ

When F is free, the algorithm based on [Diane Maclagan and 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, and Mike Stillman, Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583-600].

Ways to use this method: