# eulers(CoherentSheaf) -- list the sectional Euler characteristics

## Synopsis

• Function: eulers
• Usage:
eulers E
• Inputs:
• E,
• Outputs:
• a list, the successive sectional Euler characteristics of a coherent sheaf, or a module.

## Description

Computes a list of the successive sectional Euler characteristics of a coherent sheaf, the i-th entry on the list being the Euler characteristic of the i-th generic hyperplane restriction of E

The Horrocks-Mumford bundle on the projective fourspace:

 i1 : R = QQ[x_0..x_4]; i2 : a = {1,0,0,0,0} o2 = {1, 0, 0, 0, 0} o2 : List i3 : b = {0,1,0,0,1} o3 = {0, 1, 0, 0, 1} o3 : List i4 : c = {0,0,1,1,0} o4 = {0, 0, 1, 1, 0} o4 : List i5 : M1 = matrix table(5,5, (i,j)-> x_((i+j)%5)*a_((i-j)%5)) o5 = | x_0 0 0 0 0 | | 0 x_2 0 0 0 | | 0 0 x_4 0 0 | | 0 0 0 x_1 0 | | 0 0 0 0 x_3 | 5 5 o5 : Matrix R <--- R i6 : M2 = matrix table(5,5, (i,j)-> x_((i+j)%5)*b_((i-j)%5)) o6 = | 0 x_1 0 0 x_4 | | x_1 0 x_3 0 0 | | 0 x_3 0 x_0 0 | | 0 0 x_0 0 x_2 | | x_4 0 0 x_2 0 | 5 5 o6 : Matrix R <--- R i7 : M3 = matrix table(5,5, (i,j)-> x_((i+j)%5)*c_((i-j)%5)) o7 = | 0 0 x_2 x_3 0 | | 0 0 0 x_4 x_0 | | x_2 0 0 0 x_1 | | x_3 x_4 0 0 0 | | 0 x_0 x_1 0 0 | 5 5 o7 : Matrix R <--- R i8 : M = M1 | M2 | M3; 5 15 o8 : Matrix R <--- R i9 : betti (C=res coker M) 0 1 2 3 4 5 o9 = total: 5 15 29 37 20 2 0: 5 15 10 2 . . 1: . . 4 . . . 2: . . 15 35 20 . 3: . . . . . 2 o9 : BettiTally i10 : N = transpose submatrix(C.dd_3,{10..28},{2..36}); 35 19 o10 : Matrix R <--- R i11 : betti (D=res coker N) 0 1 2 3 4 5 o11 = total: 35 19 19 35 20 2 -5: 35 15 . . . . -4: . 4 . . . . -3: . . . . . . -2: . . . . . . -1: . . . . . . 0: . . 4 . . . 1: . . 15 35 20 . 2: . . . . . 2 o11 : BettiTally i12 : Pfour = Proj(R) o12 = Pfour o12 : ProjectiveVariety i13 : HorrocksMumford = sheaf(coker D.dd_3); i14 : HH^0(HorrocksMumford(1)) o14 = 0 o14 : QQ-module i15 : HH^0(HorrocksMumford(2)) 4 o15 = QQ o15 : QQ-module, free i16 : eulers(HorrocksMumford(2)) o16 = {2, 12, 12, 7, 2} o16 : List