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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_modules.html" title="">modules</a> > <a href="_module_sphomomorphisms.html" title="">module homomorphisms</a></div>
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<div><h1>module homomorphisms</h1>
<div>A homomorphism <tt>f : M --> N</tt> is represented as a matrix from the generators of M to the generators of N.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y]/(y^2-x^3);</pre>
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<tr><td><pre>i2 : M = module ideal(x,y)
o2 = image | x y |
1
o2 : R-module, submodule of R</pre>
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One homomorphism <tt>F : M --> R</tt> is <tt>x |--> y, y |--> x^2</tt> (this is multiplication by the fraction <tt>y/x</tt>). We write this in the following way.<table class="examples"><tr><td><pre>i3 : F = map(R^1,M,matrix{{y,x^2}})
o3 = | y x2 |
o3 : Matrix</pre>
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Notice that as is usual in Macaulay2, the target comes before the source.<p/>
Macaulay2 doesn't display the source and target, unless they are both free modules. Use <a href="_target.html" title="target of a map">target</a> and <a href="_source.html" title="source of a map">source</a> to get them. The <a href="_matrix.html" title="make a matrix">matrix</a> routine recovers the matrix of free modules between the generators of the source and target.<table class="examples"><tr><td><pre>i4 : source F
o4 = image | x y |
1
o4 : R-module, submodule of R</pre>
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<tr><td><pre>i5 : target F == R^1
o5 = true</pre>
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<tr><td><pre>i6 : matrix F
o6 = | y x2 |
1 2
o6 : Matrix R <--- R</pre>
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Macaulay2 also does not check that the homomorphism is well-defined (i.e. the relations of the source map into the relations of the target). Use <a href="_is__Well__Defined.html" title="whether a map is well defined">isWellDefined</a> to check. This generally requires a Gröbner basis computation (which is performed automatically, if it is required and has not already been done).<table class="examples"><tr><td><pre>i7 : isWellDefined F
o7 = true</pre>
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<tr><td><pre>i8 : isIsomorphism F
o8 = false</pre>
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The image of <tt>F</tt> lies in the submodule <tt>M</tt> of <tt>R^1</tt>. Suppose we wish to define this new map <tt>G : M --> M</tt>. How does one do this?<p/>
To obtain the map <tt>M --> M</tt>, we use <a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a>. In order to do this, we need the inclusion map of <tt>M</tt> into <tt>R^1</tt>.<table class="examples"><tr><td><pre>i9 : inc = inducedMap(R^1, M)
o9 = | x y |
o9 : Matrix</pre>
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Now we use // to lift <tt>F : M --> R^1</tt> along <tt>inc : M --> R^1</tt>, to obtain a map <tt>G : M --> M</tt>, such that <tt>inc * G == F</tt>.<table class="examples"><tr><td><pre>i10 : G = F // inc
o10 = {1} | 0 x |
{1} | 1 0 |
o10 : Matrix</pre>
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<tr><td><pre>i11 : target G == M and source G == M
o11 = true</pre>
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<tr><td><pre>i12 : inc * G == F
o12 = true</pre>
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Let's make sure that this map <tt>G</tt> is welldefined.<table class="examples"><tr><td><pre>i13 : isWellDefined G
o13 = true</pre>
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<tr><td><pre>i14 : isIsomorphism G
o14 = false</pre>
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<tr><td><pre>i15 : prune coker G
o15 = cokernel | y x |
1
o15 : R-module, quotient of R</pre>
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<tr><td><pre>i16 : kernel G == 0
o16 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_is__Well__Defined.html" title="whether a map is well defined">isWellDefined(Matrix)</a> -- whether a map is well defined</span></li>
<li><span><a href="_is__Isomorphism.html" title="whether a map is an isomorphism">isIsomorphism(Matrix)</a> -- whether a map is an isomorphism</span></li>
<li><span><a href="_is__Injective.html" title="whether a map is injective">isInjective(Matrix)</a> -- whether a map is injective</span></li>
<li><span><a href="_is__Surjective.html" title="whether a map is surjective">isSurjective(Matrix)</a> -- whether a map is surjective</span></li>
<li><span><a href="_kernel_lp__Matrix_rp.html" title="kernel of a matrix">kernel(Matrix)</a> -- kernel of a matrix</span></li>
<li><span><a href="_cokernel.html" title="cokernel of a map of modules, graded modules, or chaincomplexes">cokernel(Matrix)</a> -- cokernel of a map of modules, graded modules, or chaincomplexes</span></li>
<li><span><a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a> -- factor a map through another</span></li>
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