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<head><title>logCohomology(RingElement) -- logarithmic cohomology groups in two variables</title>
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<div><h1>logCohomology(RingElement) -- logarithmic cohomology groups in two variables</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>logCohomology f</tt></div>
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<li><span>Function: <a href="_log__Cohomology_lp__Ring__Element_rp.html" title="logarithmic cohomology groups in two variables">logCohomology</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, polynomial in two variables</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, with entries {VResolution, Input, TransferCycles, CohomologyGroups, PreCycles, OmegaRes, LocalizeMap, BFunction} </span></li>
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<div class="single"><h2>Description</h2>
<div>For a polynomial f in two variables executes the algorithm described in Castro-Jimenez and Takayama "The Computation of the Logarithmic Cohomology for Plane Curves" (arXiv:0712.0001).<table class="examples"><tr><td><pre>i1 : S=QQ[x,y];</pre>
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<tr><td><pre>i2 : f=x*y*(x-y);</pre>
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<tr><td><pre>i3 : logCohomology(f)
o3 = HashTable{BFunction => (s - 1) }
1
CohomologyGroups => HashTable{0 => QQ }
3
1 => QQ
2
2 => QQ
Input => cokernel | xdx+ydy+3 xydy-y2dy+x-2y |
LocalizeMap => LocMap
1 2 1
OmegaRes => (QQ[x, y, dx, dy]) <-- (QQ[x, y, dx, dy]) <-- (QQ[x, y, dx, dy]) <-- 0
0 1 2 3
PreCycles => HashTable{0 => | x | }
| 1 |
1 => | -x -y 0 |
| 0 0 -1 |
| 0 0 0 |
2 => | -y -x |
TransferCycles => HashTable{0 => | -x2y+xy2 | }
1 => | xy y2 -xy+y2 |
| x2 xy 0 |
2 => | -y -x |
1 3 2
VResolution => (QQ[x, y, dx, dy]) <-- (QQ[x, y, dx, dy]) <-- (QQ[x, y, dx, dy]) <-- 0
0 1 2 3
o3 : HashTable</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_de__Rham.html" title="deRham cohomology groups for the complement of a hypersurface">deRham</a> -- deRham cohomology groups for the complement of a hypersurface</span></li>
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