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<!DOCTYPE html> <html lang="en"> <head> <title>rightKernelBergman -- Methods for computing kernels of matrices over noncommutative rings using Bergman</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 href="index.html">NCAlgebra</a> :: <a title="Methods for computing kernels of matrices over noncommutative rings using Bergman" href="_right__Kernel__Bergman.html">rightKernelBergman</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> <a href="_ring_lp__N__C__Ideal_rp.html">next</a> | <a href="_right__Kernel.html">previous</a> | <a href="_ring_lp__N__C__Ideal_rp.html">forward</a> | <a href="_right__Kernel.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>rightKernelBergman -- Methods for computing kernels of matrices over noncommutative rings using Bergman</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">rightKernelBergman(M,DegreeLimit=>n)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">M</span>, <span>an instance of the type <a title="Type of a matrix over a noncommutative ring" href="___N__C__Matrix.html">NCMatrix</a></span>, a homogeneous matrix interpreted as a map of free right modules</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">DegreeLimit</span><span class="tt"> => </span><span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, <span>default value 10</span>, the maximum degree in which to compute the kernel</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>an instance of the type <a title="Type of a matrix over a noncommutative ring" href="___N__C__Matrix.html">NCMatrix</a></span>, the kernel of the matrix (considered as a right module map) to degree n</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Let M be a matrix with homogeneous entries in an NCRing. If the degrees of the entries of M satisfy certain consistency conditions, one can define a graded homomorphism of free right modules via left multiplication by M. If isHomogeneous(M) returns true, these conditions have been verified for M and M is a valid input for rightKernelBergman. Otherwise, an error is returned stating that M is not homogeneous. To set the isHomogeneous flag to true, use <a title="Weights entries of a matrix to make associated map of free modules graded" href="_assign__Degrees.html">assignDegrees</a>.</p> <p></p> </div> <div> <p>For valid inputs, this method computes the first n homogeneous components of the (right) kernel of the homomorphism determined by M. If n is not specified by the user, the default maximum degree is 10. The method returns a minimal set of generators for the kernel in these degrees.</p> <p>The results of this command are cached in the input matrix M in M.cache#rightKernel, and the maximum degree used in this computation is in M.cache#rightKernelDegreeLimit.</p> <p></p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z}) --Calling Bergman for NCGB calculation. Complete! o1 = B o1 : NCQuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : A = ambient B o2 = A o2 : NCPolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : g = -y^3-x*y*z+y*x*z+x^3 3 3 o3 = -y +yxz-xyz+x o3 : A</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : C = A/(ideal B + ncIdeal g) --Calling Bergman for NCGB calculation. Complete! o4 = C o4 : NCQuotientRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : M3 = ncMatrix {{x,y,z,0}, {-y*z-2*x^2,-y*x,z*x-x*z,x},{x*y-2*y*x,x*z,-x^2,y}, {-y^2-z*x,x^2,-x*y,z}} o5 = | x y z 0 | | -y*z-2*x^2 -y*x y^2-2*x*z x | | -2*y*x+x*y x*z -x^2 y | | -2*y^2+x*z x^2 -x*y z | o5 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : assignDegrees(M3,{1,0,0,0},{2,2,2,1}) o6 = | x y z 0 | | -y*z-2*x^2 -y*x y^2-2*x*z x | | -2*y*x+x*y x*z -x^2 y | | -2*y^2+x*z x^2 -x*y z | o6 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : ker1M3 = rightKernelBergman(M3) --Calling Bergman for NCGB calculation. Complete! --Calling Bergman for NCGB calculation. Complete! o7 = | -z -x y -y*z-x^2 | | y z x y^2 | | -x y -z 2*y*x-x*y | | -2*y^2 -2*x^2 -2*y*x+2*x*y -2*x*y*z | o7 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : M3*ker1M3 == 0 o8 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : ker2M3 = rightKernelBergman(ker1M3) --Calling Bergman for NCGB calculation. Complete! --Calling Bergman for NCGB calculation. Complete! --Calling Bergman for NCGB calculation. Complete! --Calling Bergman for NCGB calculation. Complete! o9 = | -y x^2 -x*z -x*y | | x y^2 -y*x+2*x*y -y*z+2*x^2 | | z -x*y x^2 -x*z | | 0 -z -y -x | o9 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : ker1M3*ker2M3 == 0 o10 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : ker3M3 = rightKernelBergman(ker2M3) --Calling Bergman for NCGB calculation. Complete! --Calling Bergman for NCGB calculation. Complete! o11 = | 0 -2*y*x -2*y^2+2*x*z -y*x*z+x^3 | | -y -z -x -x*y | | -z x y x*z | | x y -z 0 | o11 : NCMatrix</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : ker2M3*ker3M3 == 0 o12 = true</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="whether something is homogeneous (graded)" href="../../Macaulay2Doc/html/_is__Homogeneous.html">isHomogeneous</a> -- whether something is homogeneous (graded)</span></li> <li><span><a title="Weights entries of a matrix to make associated map of free modules graded" href="_assign__Degrees.html">assignDegrees</a> -- Weights entries of a matrix to make associated map of free modules graded</span></li> <li><span><a title="Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman" href="_right__Kernel.html">rightKernel</a> -- Method for computing kernels of matrices over noncommutative rings in a given degree without using Bergman</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">rightKernelBergman</span>:</h2> <ul> <li><kbd>rightKernelBergman(NCMatrix)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Methods for computing kernels of matrices over noncommutative rings using Bergman" href="_right__Kernel__Bergman.html">rightKernelBergman</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">NCAlgebra/NCAlgebraDoc.m2:2358:0</span>.</p> </div> </div> </div> </body> </html>