# cohomologyTable -- dimensions of cohomology groups

## Synopsis

• Usage:
cohomologyTable(F,l,h) or cohomologyTable(m,E,l,h)
• Inputs:
• F, , a coherent sheaf on a projective scheme
• l, an integer, lower cohomological degree
• h, an integer, upper bound on the cohomological degree
• m, , a presentation matrix for a module
• E, , exterior algebra
• Outputs:

## Description

This function takes as input a coherent sheaf F, two integers l and h, and prints the dimension dim HH^j F(i-j) for h>=i>=l. As a simple example, we compute the dimensions of cohomology groups of the projective plane.
 i1 : S = ZZ/32003[x_0..x_2];  i2 : PP2 = Proj S;  i3 : F =sheaf S^1 1 o3 = OO PP2 o3 : coherent sheaf on PP2 i4 : cohomologyTable(F,-10,5) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 o4 = 2: 55 45 36 28 21 15 10 6 3 1 . . . . . . 1: . . . . . . . . . . . . . . . . 0: . . . . . . . . . . 1 3 6 10 15 21 o4 : CohomologyTally
There is also a built-in sheaf cohomology function HH in Macaulay2. However, these algorithms are much slower than cohomologyTable.

Alternatively, this function takes as input a presentation matrix m of a finitely generated graded S-module Mand an exterior algebra Ewith the same number of variables. In this form, the function is equivalent to the function sheafCohomology in Sheaf Algorithms Using Exterior Algebra.

 i5 : S = ZZ/32003[x_0..x_2];  i6 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true]; i7 : m = koszul (3, vars S); 3 1 o7 : Matrix S <--- S i8 : regularity coker m o8 = 2 i9 : betti tateResolution(m,E,-6,2) 0 1 2 3 4 5 6 7 8 9 10 o9 = total: 15 8 3 1 3 8 15 24 35 48 63 -4: 15 8 3 . . . . . . . . -3: . . . 1 . . . . . . . -2: . . . . 3 8 15 24 35 48 63 o9 : BettiTally i10 : cohomologyTable(m,E,-6,2) -6 -5 -4 -3 -2 -1 0 1 2 3 4 o10 = 2: 63 48 35 24 15 8 3 . . . . 1: . . . . . . . 1 . . . 0: . . . . . . . . 3 8 15 o10 : CohomologyTally

As the third example, we compute the dimensions of cohomology groups of the structure sheaf of an irregular elliptic surface.

 i11 : S = ZZ/32003[x_0..x_4];  i12 : X = Proj S;  i13 : ff = res coker map(S^{1:0},S^{3:-1,2:-2},{{x_0..x_2,x_3^2,x_4^2}});  i14 : alpha = map(S^{1:-2},target ff.dd_3,{{1,4:0,x_0,2:0,x_1,0}})*ff.dd_3; 1 10 o14 : Matrix S <--- S i15 : beta = ff.dd_4//syz alpha; 18 5 o15 : Matrix S <--- S i16 : K = syz syz alpha|beta; 18 21 o16 : Matrix S <--- S i17 : fK = res prune coker K; i18 : s = random(target fK.dd_1,S^{1:-4,3:-5}); 13 4 o18 : Matrix S <--- S i19 : ftphi = res prune coker transpose (fK.dd_1|s); i20 : I = ideal ftphi.dd_2; o20 : Ideal of S i21 : F = sheaf S^1/I;  i22 : cohomologyTable(F,-2,6) -2 -1 0 1 2 3 4 5 6 7 8 o22 = 2: 123 75 39 15 3 . . . . . . 1: . . . 1 2 1 . . . . . 0: . . 1 5 16 39 75 123 183 255 339 o22 : CohomologyTally