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<head><title>ambient(Module) -- ambient free module</title>
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<div><h1>ambient(Module) -- ambient free 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>ambient M</tt></div>
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<li><span>Function: <a href="_ambient.html" title="ambient free module of a subquotient, or ambient ring">ambient</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="___Module.html">module</a></span>, a free module</span></li>
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<div class="single"><h2>Description</h2>
<div>If a module is a submodule or quotient of a free module F, or is a subquotient of F (that is, a submodule of a quotient of F), then this routine yields the free module F.<table class="examples"><tr><td><pre>i1 : R = QQ[x_1 .. x_5]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : N = image matrix{{x_1,x_2},{x_2,x_3}}
o2 = image | x_1 x_2 |
| x_2 x_3 |
2
o2 : R-module, submodule of R</pre>
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<tr><td><pre>i3 : ambient N
2
o3 = R
o3 : R-module, free</pre>
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<tr><td><pre>i4 : ambient cokernel vars R
1
o4 = R
o4 : R-module, free</pre>
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<tr><td><pre>i5 : ambient kernel vars R
5
o5 = R
o5 : R-module, free, degrees {1, 1, 1, 1, 1}</pre>
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<tr><td><pre>i6 : M = image vars R ++ cokernel vars R
o6 = subquotient (| x_1 x_2 x_3 x_4 x_5 0 |, | 0 0 0 0 0 |)
| 0 0 0 0 0 1 | | x_1 x_2 x_3 x_4 x_5 |
2
o6 : R-module, subquotient of R</pre>
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<tr><td><pre>i7 : ambient M
2
o7 = R
o7 : R-module, free</pre>
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This module is always the common target free module of the generator and relation matrices of M<table class="examples"><tr><td><pre>i8 : ambient M == target generators M
o8 = true</pre>
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<tr><td><pre>i9 : ambient M == target relations M
o9 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_subquotient_spmodules.html" title="the way Macaulay2 represents modules">subquotient modules</a> -- the way Macaulay2 represents modules</span></li>
<li><span><a href="_cover_lp__Module_rp.html" title="get the covering free module">cover(Module)</a> -- get the covering free module</span></li>
<li><span><a href="_super.html" title="get the ambient module">super(Module)</a> -- get the ambient module</span></li>
<li><span><a href="_generators_lp__Module_rp.html" title="the generator matrix of a module">generators(Module)</a> -- the generator matrix of a module</span></li>
<li><span><a href="_relations.html" title="the defining relations">relations(Module)</a> -- the defining relations</span></li>
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