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<div><h1>manipulating modules</h1>
<div>Suppose we have a module that is represented as an image of a matrix, and we want to represent it as a cokernel of a matrix. This task may be accomplished with <a href="_prune.html" title="prune, e.g., compute a minimal presentation">prune</a>.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y];</pre>
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<tr><td><pre>i2 : I = ideal vars R
o2 = ideal (x, y)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : M = image vars R
o3 = image | x y |
1
o3 : R-module, submodule of R</pre>
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<tr><td><pre>i4 : N = prune M
o4 = cokernel {1} | -y |
{1} | x |
2
o4 : R-module, quotient of R</pre>
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The isomorphism between them may be found under the key <tt>pruningMap</tt>.<table class="examples"><tr><td><pre>i5 : f = N.cache.pruningMap
o5 = {1} | 1 0 |
{1} | 0 1 |
o5 : Matrix</pre>
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<tr><td><pre>i6 : isIsomorphism f
o6 = true</pre>
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<tr><td><pre>i7 : f^-1
o7 = {1} | 1 0 |
{1} | 0 1 |
o7 : Matrix</pre>
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The matrix form of <tt>f</tt> looks nondescript, but the map knows its source and target<table class="examples"><tr><td><pre>i8 : source f
o8 = cokernel {1} | -y |
{1} | x |
2
o8 : R-module, quotient of R</pre>
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<tr><td><pre>i9 : target f
o9 = image | x y |
1
o9 : R-module, submodule of R</pre>
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It's a 2 by 2 matrix because <tt>M</tt> and <tt>N</tt> are both represented as modules with two generators.<p/>
Functions for finding related modules:<ul><li><span><a href="_ambient.html" title="ambient free module of a subquotient, or ambient ring">ambient</a> -- ambient free module of a subquotient, or ambient ring</span></li>
<li><span><a href="_cover.html" title="get the covering free module">cover</a> -- get the covering free module</span></li>
<li><span><a href="_super.html" title="get the ambient module">super</a> -- get the ambient module</span></li>
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<table class="examples"><tr><td><pre>i10 : super M
1
o10 = R
o10 : R-module, free</pre>
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<tr><td><pre>i11 : cover N
2
o11 = R
o11 : R-module, free, degrees {1, 1}</pre>
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Some simple operations on modules:<ul><li><span><a href="___Module_sp^_sp__Z__Z.html" title="direct sum">Module ^ ZZ</a> -- direct sum</span></li>
<li><span><a href="___Module_sp_pl_pl_sp__Module.html" title="direct sum of modules">Module ++ Module</a> -- direct sum of modules</span></li>
<li><span><a href="___Module_sp_st_st_sp__Module.html" title="tensor product">Module ** Module</a> -- tensor product</span></li>
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<table class="examples"><tr><td><pre>i12 : M ++ N
o12 = subquotient ({0} | x y 0 0 |, {0} | 0 |)
{1} | 0 0 1 0 | {1} | -y |
{1} | 0 0 0 1 | {1} | x |
3
o12 : R-module, subquotient of R</pre>
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<tr><td><pre>i13 : M ** N
o13 = cokernel {2} | -y 0 -y 0 |
{2} | x 0 0 -y |
{2} | 0 -y x 0 |
{2} | 0 x 0 x |
4
o13 : R-module, quotient of R</pre>
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Ideals and modules behave differently when making powers:<table class="examples"><tr><td><pre>i14 : M^3
o14 = image | x y 0 0 0 0 |
| 0 0 x y 0 0 |
| 0 0 0 0 x y |
3
o14 : R-module, submodule of R</pre>
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<tr><td><pre>i15 : I^3
3 2 2 3
o15 = ideal (x , x y, x*y , y )
o15 : Ideal of R</pre>
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