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<head><title>Singular Book 2.1.24 -- submodules, presentation of a module</title>
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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_basic_spcommutative_spalgebra.html" title="">basic commutative algebra</a> > <a href="___M2__Singular__Book.html" title="Macaulay2 examples for the Singular book">M2SingularBook</a> > <a href="___Singular_sp__Book_sp2.1.24.html" title="submodules, presentation of a module">Singular Book 2.1.24</a></div>
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<div><h1>Singular Book 2.1.24 -- submodules, presentation of a module</h1>
<div><table class="examples"><tr><td><pre>i1 : A = QQ[x,y,z]
o1 = A
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : N = image matrix{{x*y,0},{0,x*z},{y*z,z^2}}
o2 = image | xy 0 |
| 0 xz |
| yz z2 |
3
o2 : A-module, submodule of A</pre>
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The submodule is generated by the two columns of this matrix.<table class="examples"><tr><td><pre>i3 : N + x*N
o3 = image | xy 0 x2y 0 |
| 0 xz 0 x2z |
| yz z2 xyz xz2 |
3
o3 : A-module, submodule of A</pre>
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It is easy to go between matrices and submodules. Use <a href="_generators_lp__Module_rp.html" title="the generator matrix of a module">generators(Module)</a> and <a href="_image.html" title="image of a map">image(Matrix)</a>(<tt>gens</tt> and <tt>generators</tt> are synonyms). There is no automatic conversion between modules and matrices in Macaulay2.<table class="examples"><tr><td><pre>i4 : f = matrix{{x*y,x*z},{y*z,z^2}}
o4 = | xy xz |
| yz z2 |
2 2
o4 : Matrix A <--- A</pre>
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<tr><td><pre>i5 : M = image f
o5 = image | xy xz |
| yz z2 |
2
o5 : A-module, submodule of A</pre>
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<tr><td><pre>i6 : g = gens M
o6 = | xy xz |
| yz z2 |
2 2
o6 : Matrix A <--- A</pre>
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<tr><td><pre>i7 : f == g
o7 = true</pre>
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In Macaulay2, matrices are not automatically either presentation matrices or generating matrices for a module. You use whichever you have in mind.<table class="examples"><tr><td><pre>i8 : N = cokernel f
o8 = cokernel | xy xz |
| yz z2 |
2
o8 : A-module, quotient of A</pre>
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<tr><td><pre>i9 : presentation N
o9 = | xy xz |
| yz z2 |
2 2
o9 : Matrix A <--- A</pre>
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<tr><td><pre>i10 : presentation M
o10 = {2} | -z |
{2} | y |
2 1
o10 : Matrix A <--- A</pre>
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Notice that the presentation of N requires no computation, whereas the presentation of M requires a syzygy computation.<p/>
<a href="_kernel_lp__Matrix_rp.html" title="kernel of a matrix">kernel(Matrix)</a> gives a submodule, while <a href="_syz_lp__Matrix_rp.html" title="compute the syzygy matrix">syz(Matrix)</a> returns the matrix.<table class="examples"><tr><td><pre>i11 : syz f
o11 = {2} | -z |
{2} | y |
2 1
o11 : Matrix A <--- A</pre>
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<tr><td><pre>i12 : kernel f
o12 = image {2} | -z |
{2} | y |
2
o12 : A-module, submodule of A</pre>
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