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.