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<div><h1>Module _ Array -- inclusion from summand</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>M_[i,j,...,k]</tt></div>
</dd></dl>
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</li>
<li><span>Operator: <a href="__us.html" title="a binary operator, used for subscripting and access to elements">_</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span>, or <span>a <a href="___Chain__Complex.html">chain complex</a></span></span></li>
<li><span><tt>[i,j,...,k]</tt>, an array of indices</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, or <span>a <a href="___Chain__Complex__Map.html">chain complex map</a></span></span></li>
</ul>
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</li>
</ul>
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<div class="single"><h2>Description</h2>
<div><p/>
The module or chain complex <tt>M</tt> should be a direct sum, and the result is the map corresponding to inclusion from the sum of the components numbered or named <tt>i, j, ..., k</tt>. Free modules are regarded as direct sums of modules.<p/>
<table class="examples"><tr><td><pre>i1 : M = ZZ^2 ++ ZZ^3
5
o1 = ZZ
o1 : ZZ-module, free</pre>
</td></tr>
<tr><td><pre>i2 : M_[0]
o2 = | 1 0 |
| 0 1 |
| 0 0 |
| 0 0 |
| 0 0 |
5 2
o2 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i3 : M_[1]
o3 = | 0 0 0 |
| 0 0 0 |
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
5 3
o3 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i4 : M_[1,0]
o4 = | 0 0 0 1 0 |
| 0 0 0 0 1 |
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 1 0 0 |
5 5
o4 : Matrix ZZ <--- ZZ</pre>
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<p/>
If the components have been given names (see <a href="_direct__Sum.html" title="direct sum of modules or maps">directSum</a>), use those instead.<table class="examples"><tr><td><pre>i5 : R = QQ[a..d];</pre>
</td></tr>
<tr><td><pre>i6 : M = (a => image vars R) ++ (b => coker vars R)
o6 = subquotient (| a b c d 0 |, | 0 0 0 0 |)
| 0 0 0 0 1 | | a b c d |
2
o6 : R-module, subquotient of R</pre>
</td></tr>
<tr><td><pre>i7 : M_[a]
o7 = {1} | 1 0 0 0 |
{1} | 0 1 0 0 |
{1} | 0 0 1 0 |
{1} | 0 0 0 1 |
{0} | 0 0 0 0 |
o7 : Matrix</pre>
</td></tr>
<tr><td><pre>i8 : isWellDefined oo
o8 = true</pre>
</td></tr>
<tr><td><pre>i9 : M_[b]
o9 = {1} | 0 |
{1} | 0 |
{1} | 0 |
{1} | 0 |
{0} | 1 |
o9 : Matrix</pre>
</td></tr>
<tr><td><pre>i10 : isWellDefined oo
o10 = true</pre>
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<p/>
This works the same way for chain complexes.<table class="examples"><tr><td><pre>i11 : C = res coker vars R
1 4 6 4 1
o11 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o11 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i12 : D = (a=>C) ++ (b=>C)
2 8 12 8 2
o12 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o12 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i13 : D_[a]
2 1
o13 = 0 : R <--------- R : 0
| 1 |
| 0 |
8 4
1 : R <------------------- R : 1
{1} | 1 0 0 0 |
{1} | 0 1 0 0 |
{1} | 0 0 1 0 |
{1} | 0 0 0 1 |
{1} | 0 0 0 0 |
{1} | 0 0 0 0 |
{1} | 0 0 0 0 |
{1} | 0 0 0 0 |
12 6
2 : R <----------------------- R : 2
{2} | 1 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 |
{2} | 0 0 1 0 0 0 |
{2} | 0 0 0 1 0 0 |
{2} | 0 0 0 0 1 0 |
{2} | 0 0 0 0 0 1 |
{2} | 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 |
8 4
3 : R <------------------- R : 3
{3} | 1 0 0 0 |
{3} | 0 1 0 0 |
{3} | 0 0 1 0 |
{3} | 0 0 0 1 |
{3} | 0 0 0 0 |
{3} | 0 0 0 0 |
{3} | 0 0 0 0 |
{3} | 0 0 0 0 |
2 1
4 : R <------------- R : 4
{4} | 1 |
{4} | 0 |
5 : 0 <----- 0 : 5
0
o13 : ChainComplexMap</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_direct__Sum.html" title="direct sum of modules or maps">directSum</a> -- direct sum of modules or maps</span></li>
<li><span><a href="___Matrix_sp^_sp__Array.html" title="component of map corresponding to summand of target">Matrix ^ Array</a> -- component of map corresponding to summand of target</span></li>
<li><span><a href="___Module_sp^_sp__Array.html" title="projection onto summand">Module ^ Array</a> -- projection onto summand</span></li>
<li><span><a href="___Module_sp_us_sp__List.html" title="map from free module to some generators">Module _ List</a> -- map from free module to some generators</span></li>
</ul>
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