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<div><h1>lift -- lift to another ring</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>lift(f,R)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, a <a href="___Ring__Element.html">ring element</a>, <a href="___Ideal.html">ideal</a>, or <a href="___Matrix.html">matrix</a></span></li>
<li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span></span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span>a <a href="___Ring__Element.html">ring element</a>, <a href="___Ideal.html">ideal</a>, or <a href="___Matrix.html">matrix</a>, over the ring <tt>R</tt></span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>Verify => </tt><span><span>a <a href="___Boolean.html">Boolean value</a></span>, <span>default value true</span>, whether to give an error message if lifting is not possible, or, alternatively, to return <a href="_null.html" title="the unique member of the empty class">null</a></span></span></li>
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<div class="single"><h2>Description</h2>
<div><p>(Disambiguation: for division of matrices, which is thought of as lifting one homomorphism over another, see instead <a href="___Matrix_sp_sl_sl_sp__Matrix.html" title="factor a map through another">Matrix // Matrix</a>. For lifting a map between modules to a map between their free resolutions, see <a href="_extend.html" title="extend a module map to a chain map, if possible">extend</a>.)</p>
<p>The ring <tt>R</tt> should be one of the base rings associated with the ring of <tt>f</tt>. An error is raised if <tt>f</tt> cannot be lifted to <tt>R</tt>.</p>
<p>The first example is lifting from the fraction field of R to R.</p>
<table class="examples"><tr><td><pre>i1 : lift(4/2,ZZ)
o1 = 2</pre>
</td></tr>
<tr><td><pre>i2 : R = ZZ[x];</pre>
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<tr><td><pre>i3 : f = ((x+1)^3*(x+4))/((x+4)*(x+1))
2
o3 = x + 2x + 1
o3 : frac(R)</pre>
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<tr><td><pre>i4 : lift(f,R)
2
o4 = x + 2x + 1
o4 : R</pre>
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<p>Another use of lift is to take polynomials in a quotient ring and lift them to the polynomial ring.</p>
<table class="examples"><tr><td><pre>i5 : A = QQ[a..d];</pre>
</td></tr>
<tr><td><pre>i6 : B = A/(a^2-b,c^2-d-a-3);</pre>
</td></tr>
<tr><td><pre>i7 : f = c^5
2
o7 = 2a*c*d + c*d + 6a*c + b*c + 6c*d + 9c
o7 : B</pre>
</td></tr>
<tr><td><pre>i8 : lift(f,A)
2
o8 = 2a*c*d + c*d + 6a*c + b*c + 6c*d + 9c
o8 : A</pre>
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<tr><td><pre>i9 : jf = jacobian ideal f
o9 = {1} | 2cd+6c |
{1} | c |
{1} | 2ad+d2+6a+b+6d+9 |
{1} | 2ac+2cd+6c |
4 1
o9 : Matrix B <--- B</pre>
</td></tr>
<tr><td><pre>i10 : lift(jf,A)
o10 = {1} | 2cd+6c |
{1} | c |
{1} | 2ad+d2+6a+b+6d+9 |
{1} | 2ac+2cd+6c |
4 1
o10 : Matrix A <--- A</pre>
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<p>Elements may be lifted to any base ring, if such a lift exists.</p>
<table class="examples"><tr><td><pre>i11 : use B;</pre>
</td></tr>
<tr><td><pre>i12 : g = (a^2+2*a-3)-(a+1)^2
o12 = -4
o12 : B</pre>
</td></tr>
<tr><td><pre>i13 : lift(g,A)
o13 = -4
o13 : A</pre>
</td></tr>
<tr><td><pre>i14 : lift(g,QQ)
o14 = -4
o14 : QQ</pre>
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<tr><td><pre>i15 : lift(lift(g,QQ),ZZ)
o15 = -4</pre>
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<p>The functions <tt>lift</tt> and <a href="_substitute.html" title="substituting values for variables">substitute</a> are useful to move numbers from one kind of coefficient ring to another.</p>
<table class="examples"><tr><td><pre>i16 : lift(3.0,ZZ)
o16 = 3</pre>
</td></tr>
<tr><td><pre>i17 : lift(3.0,QQ)
o17 = 3
o17 : QQ</pre>
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<p>A continued fraction method is used to lift a real number to a rational number, whereas <a href="_promote.html" title="promote to another ring">promote</a> uses the internal binary representation.</p>
<table class="examples"><tr><td><pre>i18 : lift(123/2341.,QQ)
123
o18 = ----
2341
o18 : QQ</pre>
</td></tr>
<tr><td><pre>i19 : promote(123/2341.,QQ)
7572049608428139
o19 = ------------------
144115188075855872
o19 : QQ</pre>
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<tr><td><pre>i20 : factor oo
3*811*39877*78045679
o20 = --------------------
57
2
o20 : Expression of class Divide</pre>
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<p>For numbers and ring elements, an alternate syntax with <a href="_^.html" title="a binary operator, usually used for powers">^</a> is available, analogous to the use of <a href="__us.html" title="a binary operator, used for subscripting and access to elements">_</a> for <a href="_promote.html" title="promote to another ring">promote</a>.</p>
<table class="examples"><tr><td><pre>i21 : .0001^QQ
1
o21 = -----
10000
o21 : QQ</pre>
</td></tr>
<tr><td><pre>i22 : .0001_QQ
7378697629483821
o22 = --------------------
73786976294838206464
o22 : QQ</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_base__Rings.html" title="store the list of base rings of a ring">baseRings</a> -- store the list of base rings of a ring</span></li>
<li><span><a href="_liftable.html" title="whether lifting to another ring is possible">liftable</a> -- whether lifting to another ring is possible</span></li>
<li><span><a href="_promote.html" title="promote to another ring">promote</a> -- promote to another ring</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>lift</tt> :</h2>
<ul><li><span>lift(BettiTally,type of ZZ), see <span><a href="___Betti__Tally.html" title="the class of all Betti tallies">BettiTally</a> -- the class of all Betti tallies</span></span></li>
<li>lift(CC,type of QQ)</li>
<li>lift(CC,type of ZZ)</li>
<li>lift(Ideal,type of QQ)</li>
<li>lift(Ideal,type of RingElement)</li>
<li>lift(Ideal,type of ZZ)</li>
<li>lift(Matrix,type of Number)</li>
<li>lift(Matrix,type of QQ,type of QQ)</li>
<li>lift(Matrix,type of QQ,type of ZZ)</li>
<li>lift(Matrix,type of RingElement)</li>
<li>lift(Matrix,type of ZZ,type of ZZ)</li>
<li>lift(QQ,type of QQ)</li>
<li>lift(QQ,type of ZZ)</li>
<li>lift(RR,type of QQ)</li>
<li>lift(RR,type of ZZ)</li>
<li>lift(ZZ,type of ZZ)</li>
</ul>
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