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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_varieties.html" title="">varieties</a> > <a href="_coherent_spsheaves.html" title="">coherent sheaves</a></div>
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<div><h1>coherent sheaves</h1>
<div>The main reason to implement algebraic varieties is support the computation of sheaf cohomology of coherent sheaves, which doesn't have an immediate description in terms of graded modules.<p/>
In this example, we use <a href="_cotangent__Sheaf.html" title="cotangent sheaf of a projective variety">cotangentSheaf</a> to produce the cotangent sheaf on a K3 surface and compute its sheaf cohomology.<table class="examples"><tr><td><pre>i1 : R = QQ[a,b,c,d]/(a^4+b^4+c^4+d^4);</pre>
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<tr><td><pre>i2 : X = Proj R
o2 = X
o2 : ProjectiveVariety</pre>
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<tr><td><pre>i3 : Omega = cotangentSheaf X
o3 = cokernel {2} | c 0 0 d 0 a3 b3 0 |
{2} | a d 0 0 b3 -c3 0 0 |
{2} | -b 0 d 0 a3 0 c3 0 |
{2} | 0 b a 0 -d3 0 0 c3 |
{2} | 0 -c 0 a 0 -d3 0 b3 |
{2} | 0 0 -c -b 0 0 d3 a3 |
6
o3 : coherent sheaf on X, quotient of OO (-2)
X</pre>
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<tr><td><pre>i4 : HH^1(Omega)
20
o4 = QQ
o4 : QQ-module, free</pre>
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Use the function <a href="_sheaf.html" title="make a coherent sheaf">sheaf</a> to convert a graded module to a coherent sheaf, and <a href="_module.html" title="make or get a module">module</a> to get the graded module back again.<table class="examples"><tr><td><pre>i5 : F = sheaf coker matrix {{a,b}}
o5 = cokernel | a b |
1
o5 : coherent sheaf on X, quotient of OO
X</pre>
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<tr><td><pre>i6 : module F
o6 = cokernel | a b |
1
o6 : R-module, quotient of R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___H__H^__Z__Z_sp__Coherent__Sheaf.html" title="cohomology of a coherent sheaf on a projective variety">HH^ZZ CoherentSheaf</a> -- cohomology of a coherent sheaf on a projective variety</span></li>
<li><span><a href="___H__H^__Z__Z_sp__Sum__Of__Twists.html" title="coherent sheaf cohomology module">HH^ZZ SumOfTwists</a> -- coherent sheaf cohomology module</span></li>
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