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<!DOCTYPE html> <html lang="en"> <head> <title>LocalRing -- The class of all local rings</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="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Localizations of polynomial rings at prime ideals" href="index.html">LocalRings</a> » <a title="The class of all local rings" href="___Local__Ring.html">LocalRing</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 | previous | <a href="_local__Ring.html">forward</a> | <a href="index.html">backward</a> | <a href="index.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>LocalRing -- The class of all local rings</h1> <div> <h2>Description</h2> <div> <p>Currently only localizations at prime ideals of a polynomial ring are supported.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : S = QQ[x,y,z,w];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = ideal"xz-y2,yw-z2,xw-yz"; -- The twisted cubic curve o2 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : R = S_I o3 = R 2 2 o3 : LocalRing, maximal ideal (- y + x*z, - z + y*w, - y*z + x*w)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : K = frac(S/I) o4 = K o4 : FractionField</code></pre> </td> </tr> </table> <div> <p>The maximal ideal and a residue map to the residue field are stored in the ring.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : max R 2 2 o5 = ideal (- y + x*z, - z + y*w, - y*z + x*w) o5 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : R.maxIdeal 2 2 o6 = ideal (- y + x*z, - z + y*w, - y*z + x*w) o6 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : R.residueMap o7 = map (K, R, {x, y, z, w}) o7 : RingMap K <-- R</code></pre> </td> </tr> </table> <div> <p>Objects over the base ring can be localized easily.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i8 : I ** R 2 2 o8 = ideal (- y + x*z, - z + y*w, - y*z + x*w) o8 : Ideal of R</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="the class of all ordered monoid rings" href="../../Macaulay2Doc/html/___Polynomial__Ring.html">PolynomialRing</a> -- the class of all ordered monoid rings</span></li> <li><span><a title="Computes the Hilbert-Samuel Function of Modules over Local Rings" href="_hilbert__Samuel__Function.html">hilbertSamuelFunction</a> -- Computes the Hilbert-Samuel Function of Modules over Local Rings</span></li> <li><span><a title="compute the length of a module" href="../../Macaulay2Doc/html/_length_lp__Module_rp.html">length(Module)</a> -- compute the length of a module</span></li> </ul> </div> <div> <h3>Menu</h3> <ul> <li><span><a title="Constructor for local rings" href="_local__Ring.html">localRing</a> -- Constructor for local rings</span></li> <li><span><a title="Lifts various objects over R_P to R." href="_lift__Up.html">liftUp</a> -- Lifts various objects over R_P to R.</span></li> <li><span><a title="produce the ring from which a ring was formed" href="_base__Ring_lp__Local__Ring_rp.html">baseRing(LocalRing)</a> -- produce the ring from which a ring was formed</span></li> <li><span><a title="get the characteristic of the ring or field" href="_char_lp__Local__Ring_rp.html">char(LocalRing)</a> -- get the characteristic of the ring or field</span></li> <li><span><a title="get the coefficient ring" href="_coefficient__Ring_lp__Local__Ring_rp.html">coefficientRing(LocalRing)</a> -- get the coefficient ring</span></li> <li><span><a title="the length of the degree vector" href="_degree__Length_lp__Local__Ring_rp.html">degreeLength(LocalRing)</a> -- the length of the degree vector</span></li> <li><span><a title="degrees of generators" href="_degrees_lp__Local__Ring_rp.html">degrees(LocalRing)</a> -- degrees of generators</span></li> <li><span><a title="compute the Krull dimension" href="_dim_lp__Local__Ring_rp.html">dim(LocalRing)</a> -- compute the Krull dimension</span></li> <li><span><a title="construct a fraction field" href="_frac_lp__Local__Ring_rp.html">frac(LocalRing)</a> -- construct a fraction field</span></li> <li><span><a title="the list of generators of a ring" href="_generators_lp__Local__Ring_rp.html">generators(LocalRing)</a> -- the list of generators of a ring</span></li> <li><span><a title="whether a ring is commutative" href="_is__Commutative_lp__Local__Ring_rp.html">isCommutative(LocalRing)</a> -- whether a ring is commutative</span></li> <li><span><a title="whether a local ring is well defined" href="_is__Well__Defined_lp__Local__Ring_rp.html">isWellDefined(LocalRing)</a> -- whether a local ring is well defined</span></li> <li><span><a title="number of generators of a polynomial ring" href="_numgens_lp__Local__Ring_rp.html">numgens(LocalRing)</a> -- number of generators of a polynomial ring</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Functions and methods returning a local ring:</h2> <ul> <li><span><a title="Constructor for local rings" href="_local__Ring.html">localRing</a> -- Constructor for local rings</span></li> <li><span><kbd>PolynomialRing _ Ideal</kbd> -- see <span><a title="Constructor for local rings" href="_local__Ring.html">localRing</a> -- Constructor for local rings</span></span></li> <li><span><kbd>PolynomialRing _ RingElement</kbd> -- see <span><a title="Constructor for local rings" href="_local__Ring.html">localRing</a> -- Constructor for local rings</span></span></li> </ul> <h2>Methods that use a local ring:</h2> <ul> <li><span><a title="produce the ring from which a ring was formed" href="_base__Ring_lp__Local__Ring_rp.html">baseRing(LocalRing)</a> -- produce the ring from which a ring was formed</span></li> <li><span><a title="get the characteristic of the ring or field" href="_char_lp__Local__Ring_rp.html">char(LocalRing)</a> -- get the characteristic of the ring or field</span></li> <li><span><a title="get the coefficient ring" href="_coefficient__Ring_lp__Local__Ring_rp.html">coefficientRing(LocalRing)</a> -- get the coefficient ring</span></li> <li><span><a title="the length of the degree vector" href="_degree__Length_lp__Local__Ring_rp.html">degreeLength(LocalRing)</a> -- the length of the degree vector</span></li> <li><span><a title="degrees of generators" href="_degrees_lp__Local__Ring_rp.html">degrees(LocalRing)</a> -- degrees of generators</span></li> <li><span><a title="compute the Krull dimension" href="_dim_lp__Local__Ring_rp.html">dim(LocalRing)</a> -- compute the Krull dimension</span></li> <li><span><a title="construct a fraction field" href="_frac_lp__Local__Ring_rp.html">frac(LocalRing)</a> -- construct a fraction field</span></li> <li><span><a title="the list of generators of a ring" href="_generators_lp__Local__Ring_rp.html">generators(LocalRing)</a> -- the list of generators of a ring</span></li> <li><span><a title="whether a ring is commutative" href="_is__Commutative_lp__Local__Ring_rp.html">isCommutative(LocalRing)</a> -- whether a ring is commutative</span></li> <li><span><a title="whether a local ring is well defined" href="_is__Well__Defined_lp__Local__Ring_rp.html">isWellDefined(LocalRing)</a> -- whether a local ring is well defined</span></li> <li><kbd>max(LocalRing)</kbd></li> <li><span><a title="number of generators of a polynomial ring" href="_numgens_lp__Local__Ring_rp.html">numgens(LocalRing)</a> -- number of generators of a polynomial ring</span></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="The class of all local rings" href="___Local__Ring.html">LocalRing</a> is <span>a <a title="the class of all mutable types" href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a title="the class of rings handled by the engine" href="../../Macaulay2Doc/html/___Engine__Ring.html">EngineRing</a> < <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">Ring</a> < <a title="the class of all mutable types" href="../../Macaulay2Doc/html/___Type.html">Type</a> < <a title="the class of all mutable hash tables" href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html">MutableHashTable</a> < <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">HashTable</a> < <a title="the class of all things" href="../../Macaulay2Doc/html/___Thing.html">Thing</a>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">LocalRings/doc.m2:234:0</span>.</p> </div> </div> </div> </body> </html>