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<head><title>eulers(CoherentSheaf) -- list the sectional Euler characteristics</title>
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<div><h1>eulers(CoherentSheaf) -- list the sectional Euler characteristics</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>eulers E</tt></div>
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<li><span>Function: <a href="_eulers.html" title="list the sectional Euler characteristics">eulers</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>E</tt>, <span>a <a href="___Coherent__Sheaf.html">coherent sheaf</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___List.html">list</a></span>, the successive sectional Euler characteristics of a coherent sheaf, or a module.</span></li>
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<div class="single"><h2>Description</h2>
<div>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 <tt>E</tt><p/>
The Horrocks-Mumford bundle on the projective fourspace:<table class="examples"><tr><td><pre>i1 : R = QQ[x_0..x_4];</pre>
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<tr><td><pre>i2 : a = {1,0,0,0,0}
o2 = {1, 0, 0, 0, 0}
o2 : List</pre>
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<tr><td><pre>i3 : b = {0,1,0,0,1}
o3 = {0, 1, 0, 0, 1}
o3 : List</pre>
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<tr><td><pre>i4 : c = {0,0,1,1,0}
o4 = {0, 0, 1, 1, 0}
o4 : List</pre>
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<tr><td><pre>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</pre>
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<tr><td><pre>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</pre>
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<tr><td><pre>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</pre>
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<tr><td><pre>i8 : M = M1 | M2 | M3;
5 15
o8 : Matrix R <--- R</pre>
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<tr><td><pre>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</pre>
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<tr><td><pre>i10 : N = transpose submatrix(C.dd_3,{10..28},{2..36});
35 19
o10 : Matrix R <--- R</pre>
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<tr><td><pre>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</pre>
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<tr><td><pre>i12 : Pfour = Proj(R)
o12 = Pfour
o12 : ProjectiveVariety</pre>
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<tr><td><pre>i13 : HorrocksMumford = sheaf(coker D.dd_3);</pre>
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<tr><td><pre>i14 : HH^0(HorrocksMumford(1))
o14 = 0
o14 : QQ-module</pre>
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<tr><td><pre>i15 : HH^0(HorrocksMumford(2))
4
o15 = QQ
o15 : QQ-module, free</pre>
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<tr><td><pre>i16 : eulers(HorrocksMumford(2))
o16 = {2, 12, 12, 7, 2}
o16 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_genera.html" title="list of the successive linear sectional arithmetic genera">genera</a> -- list of the successive linear sectional arithmetic genera</span></li>
<li><span><a href="_genus.html" title="arithmetic genus">genus</a> -- arithmetic genus</span></li>
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