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<head><title>resolution(Module) -- compute a projective resolution of a module</title>
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<div><h1>resolution(Module) -- compute a projective resolution of a module</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>resolution M</tt><br/><tt>res M</tt></div>
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<li><span>Function: <a href="_resolution.html" title="projective resolution">resolution</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Chain__Complex.html">chain complex</a></span>, a free resolution of <tt>M</tt></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_resolution_lp..._cm_sp__Degree__Limit_sp_eq_gt_sp..._rp.html">DegreeLimit => ...</a>, -- compute only up to this degree</span></li>
<li><span><a href="_resolution_lp..._cm_sp__Hard__Degree__Limit_sp_eq_gt_sp..._rp.html">HardDegreeLimit => ...</a>, </span></li>
<li><span><a href="_resolution_lp..._cm_sp__Length__Limit_sp_eq_gt_sp..._rp.html">LengthLimit => ...</a>, -- stop when the resolution reaches this length</span></li>
<li><span><a href="_resolution_lp..._cm_sp__Pair__Limit_sp_eq_gt_sp..._rp.html">PairLimit => ...</a>, -- stop when this number of pairs has been handled</span></li>
<li><span><a href="_resolution_lp..._cm_sp__Sort__Strategy_sp_eq_gt_sp..._rp.html">SortStrategy => ...</a>, </span></li>
<li><span><a href="_resolution_lp..._cm_sp__Stop__Before__Computation_sp_eq_gt_sp..._rp.html">StopBeforeComputation => ...</a>, -- whether to stop the computation immediately</span></li>
<li><span><a href="_resolution_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html">Strategy => ...</a>, </span></li>
<li><span><a href="_resolution_lp..._cm_sp__Syzygy__Limit_sp_eq_gt_sp..._rp.html">SyzygyLimit => ...</a>, -- stop when this number of syzygies are obtained</span></li>
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<div class="single"><h2>Description</h2>
<div>Warning: the resolution can have free modules with unexpected ranks when the module <tt>M</tt> is not homogeneous. Here is an example where even the lengths of the resolutions differ. We compute a resolution of the kernel of a ring map in two ways. The ring <tt>R</tt> is constructed naively, but the ring <tt>S</tt> is constructed with variables of the right degrees so the ring map <tt>g</tt> will turn out to be homogeneous.<table class="examples"><tr><td><pre>i1 : k = ZZ/101; T = k[v..z];</pre>
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<tr><td><pre>i3 : m = matrix {{x,y,z,x^2*v,x*y*v,y^2*v,z*v,x*w,y^3*w,z*w}}
o3 = | x y z vx2 vxy vy2 vz wx wy3 wz |
1 10
o3 : Matrix T <--- T</pre>
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<tr><td><pre>i4 : n = rank source m
o4 = 10</pre>
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<tr><td><pre>i5 : R = k[u_1 .. u_n]
o5 = R
o5 : PolynomialRing</pre>
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<tr><td><pre>i6 : S = k[u_1 .. u_n,Degrees => degrees source m]
o6 = S
o6 : PolynomialRing</pre>
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<tr><td><pre>i7 : f = map(T,R,m)
2 2 3
o7 = map(T,R,{x, y, z, v*x , v*x*y, v*y , v*z, w*x, w*y , w*z})
o7 : RingMap T <--- R</pre>
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<tr><td><pre>i8 : g = map(T,S,m)
2 2 3
o8 = map(T,S,{x, y, z, v*x , v*x*y, v*y , v*z, w*x, w*y , w*z})
o8 : RingMap T <--- S</pre>
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<tr><td><pre>i9 : res ker f
1 17 57 76 46 12 1
o9 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5 6 7
o9 : ChainComplex</pre>
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<tr><td><pre>i10 : res ker g
1 14 35 35 15 2
o10 = S <-- S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5 6
o10 : ChainComplex</pre>
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<tr><td><pre>i11 : isHomogeneous f
o11 = false</pre>
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<tr><td><pre>i12 : isHomogeneous g
o12 = true</pre>
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<table class="examples"><tr><td><pre>i13 : R = ZZ/32003[a..d]/(a^2+b^2+c^2+d^2);</pre>
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<tr><td><pre>i14 : M = coker vars R
o14 = cokernel | a b c d |
1
o14 : R-module, quotient of R</pre>
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<tr><td><pre>i15 : C = resolution(M, LengthLimit=>6)
1 4 7 8 8 8 8
o15 = R <-- R <-- R <-- R <-- R <-- R <-- R
0 1 2 3 4 5 6
o15 : ChainComplex</pre>
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For an overview of resolutions, in order of increasing detail, see<ul><li><span><a href="___Hilbert_spfunctions_spand_spfree_spresolutions.html" title="including degree and betti numbers">Hilbert functions and free resolutions</a></span></li>
<li><span><a href="_free_spresolutions_spof_spmodules.html" title="">free resolutions of modules</a></span></li>
<li><span><a href="_computing_spresolutions.html" title="">computing resolutions</a> -- most detailed</span></li>
</ul>
Some useful related functions<ul><li><span><a href="_betti_lp__Graded__Module_rp.html" title="display of degrees in a graded module">betti(GradedModule)</a> -- display of degrees in a graded module</span></li>
<li><span><a href="_status.html" title="status of a resolution computation">status(Resolution)</a> -- status of a resolution computation</span></li>
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