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<!DOCTYPE html> <html lang="en"> <head> <title>Depth -- Finds the depth of a module or ideal, and systems of parameters in an ideal</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="Finds the depth of a module or ideal, and systems of parameters in an ideal" href="index.html">Depth</a> :: <a title="Finds the depth of a module or ideal, and systems of parameters in an ideal" href="index.html">Depth</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="_depth_lp__Ideal_cm__Ring_rp.html">next</a> | previous | <a href="_depth_lp__Ideal_cm__Ring_rp.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Depth -- Finds the depth of a module or ideal, and systems of parameters in an ideal</h1> <div> <h2>Description</h2> <div> <p>There are two major groups of routines: one for finding the depth of an ideal or module or ring,or the depth of an ideal on a module; and the other for finding relatively sparse systems of homogeneous parameters in an ideal (there is an inhomogeneous version too.)</p> <p>The depth of an ideal on a module is a fundamental invariant, a kind of arithmetic version of codimension. In many local or graded cases one is interested in the depth of the maximal ideal on the module, and then we speak just of the depth of the module (or, when the module is the ring itself, the depth of the ring.) Over a regular ring, depth is most efficiently computed using the Auslander-Buchsbaum formula depth M = dim R - pdim M, where pdim is the projective dimension. In general, depth(I,M) = min {i | Ext^i(R/I,M) != 0}. Both these methods are incorporated.</p> <p>Depth can be computed from regular sequences, which are systems of parameters. The other routines in this package try probabilistically to find relatively sparse systems of parameters; the maximum length of such a system is equal to the codimension of the ideal, so (inhomogeneousSystemOfParameters,Ideal) and (systemOfParameters, Ideal), without further arguments, look for regular sequences of length codim I.</p> <p>To find such sequences, one can simply take an appropriate number of random linear combinations of the generators of the ideal, and this is what's done by inhomogeneousSystemOfParameters. Since being a system of parameters is a matter of certain elements not being in certain prime ideals, this succeeds with very high probability over any field of reasonable size. But it produces inhomogeneous elements. When the ideal is homogeneous, one generally wants a homogeneous system of parameters; this is provided, again probabilistically, by the routine systemOfParameters.</p> <p>Here is an example computing depths of modules (that is, the depths of the maximal ideal on the module:</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : S = ZZ/101[a,b,c,d] o1 = S o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : K = koszul vars S 1 4 6 4 1 o2 = S <-- S <-- S <-- S <-- S 0 1 2 3 4 o2 : ChainComplex</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : apply(numgens S, i-> depth coker K.dd_(i+1)) o3 = {0, 1, 2, 3} o3 : List</code></pre> </td> </tr> </table> <div> <p>and here is one computing systems of parameters. The "Density" (a number between 0 and 1) is a measure of the sparseness sought, and "Attempts" bounds the number of probabilistic attempts.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : I = ideal"ab,bc,cd2,da" 2 o4 = ideal (a*b, b*c, c*d , a*d) o4 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : codim I o5 = 2</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : setRandomSeed 0 -- setting random seed to 0 o6 = 0</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : systemOfParameters(I, Density => .1, Attempts => 1000, Verbose => true) Attempts: 1000 Density: .1 Seed: null 2 o7 = ideal (a*b, c*d ) o7 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : inhomogeneousSystemOfParameters I o8 = | cd2+bc bc+ad | 1 2 o8 : Matrix S <-- S</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> <div> <p>The systemOfParameters code could be improved by working one degree at a time, using a knowledge of the codim of the ideal generated by elements of degrees <=d for each d.</p> </div> </div> <div> <h2>See also</h2> <ul> <li><span><a title="depth of an object" href="../../Macaulay2Doc/html/_depth.html">depth</a> -- depth of an object</span></li> <li><span><a title="generates an inhomogeneous system of parameters" href="_inhomogeneous__System__Of__Parameters.html">inhomogeneousSystemOfParameters</a> -- generates an inhomogeneous system of parameters</span></li> <li><span><a title="finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal" href="_system__Of__Parameters.html">systemOfParameters</a> -- finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal</span></li> </ul> </div> <div> <div> <div> <h2>Authors</h2> <ul> <li><a href="http://www.math.ohio-state.edu/~snapp/">Bart Snapp</a><span> <<a href="mailto:snapp%40math.ohio-state.edu">snapp@math.ohio-state.edu</a>></span></li> <li><a href="http://www.msri.org/~de">David Eisenbud</a><span> <<a href="mailto:de%40msri.org">de@msri.org</a>></span></li> <li><a href="http://math.adelphi.edu/~bstone/">Branden Stone</a><span> <<a href="mailto:bstone%40adelphi.edu">bstone@adelphi.edu</a>></span></li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>1.0</b> of Depth, released <b>September 2010, rev May 2016</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{DepthSource, title = {{Depth: aids in computations related to depth. Version~1.0}}, author = {Bart Snapp and David Eisenbud and Branden Stone}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Functions and commands <ul> <li><span><a title="generates an inhomogeneous system of parameters" href="_inhomogeneous__System__Of__Parameters.html">inhomogeneousSystemOfParameters</a> -- generates an inhomogeneous system of parameters</span></li> <li><span><a title="whether a ring or module is Cohen-Macaulay" href="_is__C__M.html">isCM</a> -- whether a ring or module is Cohen-Macaulay</span></li> <li><span><a title="whether a list is regular over a ring or module" href="_is__Regular__Sequence.html">isRegularSequence</a> -- whether a list is regular over a ring or module</span></li> <li><span><a title="how much of a list is regular" href="_regular__Sequence__Check.html">regularSequenceCheck</a> -- how much of a list is regular</span></li> <li><span><a title="finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal" href="_system__Of__Parameters.html">systemOfParameters</a> -- finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>depth(Ideal,Ideal)</kbd> -- see <span><a title="computes the depth of a ring" href="_depth_lp__Ideal_cm__Ring_rp.html">depth(Ideal,Ring)</a> -- computes the depth of a ring</span></span></li> <li><span><kbd>depth(Ideal,Module)</kbd> -- see <span><a title="computes the depth of a ring" href="_depth_lp__Ideal_cm__Ring_rp.html">depth(Ideal,Ring)</a> -- computes the depth of a ring</span></span></li> <li><span><a title="computes the depth of a ring" href="_depth_lp__Ideal_cm__Ring_rp.html">depth(Ideal,Ring)</a> -- computes the depth of a ring</span></li> <li><span><kbd>depth(Module)</kbd> -- see <span><a title="computes the depth of a ring" href="_depth_lp__Ideal_cm__Ring_rp.html">depth(Ideal,Ring)</a> -- computes the depth of a ring</span></span></li> <li><span><kbd>depth(Ring)</kbd> -- see <span><a title="computes the depth of a ring" href="_depth_lp__Ideal_cm__Ring_rp.html">depth(Ideal,Ring)</a> -- computes the depth of a ring</span></span></li> <li><span><kbd>inhomogeneousSystemOfParameters(Ideal)</kbd> -- see <span><a title="generates an inhomogeneous system of parameters" href="_inhomogeneous__System__Of__Parameters.html">inhomogeneousSystemOfParameters</a> -- generates an inhomogeneous system of parameters</span></span></li> <li><span><kbd>inhomogeneousSystemOfParameters(Ideal,Ring)</kbd> -- see <span><a title="generates an inhomogeneous system of parameters" href="_inhomogeneous__System__Of__Parameters.html">inhomogeneousSystemOfParameters</a> -- generates an inhomogeneous system of parameters</span></span></li> <li><span><kbd>inhomogeneousSystemOfParameters(Ring)</kbd> -- see <span><a title="generates an inhomogeneous system of parameters" href="_inhomogeneous__System__Of__Parameters.html">inhomogeneousSystemOfParameters</a> -- generates an inhomogeneous system of parameters</span></span></li> <li><span><kbd>isCM(Module)</kbd> -- see <span><a title="whether a ring or module is Cohen-Macaulay" href="_is__C__M.html">isCM</a> -- whether a ring or module is Cohen-Macaulay</span></span></li> <li><span><kbd>isCM(Ring)</kbd> -- see <span><a title="whether a ring or module is Cohen-Macaulay" href="_is__C__M.html">isCM</a> -- whether a ring or module is Cohen-Macaulay</span></span></li> <li><span><kbd>isRegularSequence(List)</kbd> -- see <span><a title="whether a list is regular over a ring or module" href="_is__Regular__Sequence.html">isRegularSequence</a> -- whether a list is regular over a ring or module</span></span></li> <li><span><kbd>isRegularSequence(List,Module)</kbd> -- see <span><a title="whether a list is regular over a ring or module" href="_is__Regular__Sequence.html">isRegularSequence</a> -- whether a list is regular over a ring or module</span></span></li> <li><span><kbd>isRegularSequence(List,Ring)</kbd> -- see <span><a title="whether a list is regular over a ring or module" href="_is__Regular__Sequence.html">isRegularSequence</a> -- whether a list is regular over a ring or module</span></span></li> <li><span><kbd>isRegularSequence(Matrix)</kbd> -- see <span><a title="whether a list is regular over a ring or module" href="_is__Regular__Sequence.html">isRegularSequence</a> -- whether a list is regular over a ring or module</span></span></li> <li><span><kbd>isRegularSequence(Matrix,Module)</kbd> -- see <span><a title="whether a list is regular over a ring or module" href="_is__Regular__Sequence.html">isRegularSequence</a> -- whether a list is regular over a ring or module</span></span></li> <li><span><kbd>isRegularSequence(Matrix,Ring)</kbd> -- see <span><a title="whether a list is regular over a ring or module" href="_is__Regular__Sequence.html">isRegularSequence</a> -- whether a list is regular over a ring or module</span></span></li> <li><span><kbd>regularSequenceCheck(List,Module)</kbd> -- see <span><a title="how much of a list is regular" href="_regular__Sequence__Check.html">regularSequenceCheck</a> -- how much of a list is regular</span></span></li> <li><span><kbd>regularSequenceCheck(List,Ring)</kbd> -- see <span><a title="how much of a list is regular" href="_regular__Sequence__Check.html">regularSequenceCheck</a> -- how much of a list is regular</span></span></li> <li><span><kbd>regularSequenceCheck(Matrix,Module)</kbd> -- see <span><a title="how much of a list is regular" href="_regular__Sequence__Check.html">regularSequenceCheck</a> -- how much of a list is regular</span></span></li> <li><span><kbd>regularSequenceCheck(Matrix,Ring)</kbd> -- see <span><a title="how much of a list is regular" href="_regular__Sequence__Check.html">regularSequenceCheck</a> -- how much of a list is regular</span></span></li> <li><span><kbd>systemOfParameters(Ideal)</kbd> -- see <span><a title="finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal" href="_system__Of__Parameters.html">systemOfParameters</a> -- finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal</span></span></li> <li><span><kbd>systemOfParameters(Ring)</kbd> -- see <span><a title="finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal" href="_system__Of__Parameters.html">systemOfParameters</a> -- finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal</span></span></li> <li><span><kbd>systemOfParameters(ZZ,Ideal)</kbd> -- see <span><a title="finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal" href="_system__Of__Parameters.html">systemOfParameters</a> -- finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal</span></span></li> </ul> </li> <li>Symbols <ul> <li><span><kbd>Attempts</kbd> -- see <span><a title="generates an inhomogeneous system of parameters" href="_inhomogeneous__System__Of__Parameters.html">inhomogeneousSystemOfParameters</a> -- generates an inhomogeneous system of parameters</span></span></li> <li><span><kbd>Bound</kbd> -- see <span><a title="generates an inhomogeneous system of parameters" href="_inhomogeneous__System__Of__Parameters.html">inhomogeneousSystemOfParameters</a> -- generates an inhomogeneous system of parameters</span></span></li> <li><span><kbd>Maximal</kbd> -- see <span><a title="generates an inhomogeneous system of parameters" href="_inhomogeneous__System__Of__Parameters.html">inhomogeneousSystemOfParameters</a> -- generates an inhomogeneous system of parameters</span></span></li> <li><span><kbd>Sparseness</kbd> -- see <span><a title="generates an inhomogeneous system of parameters" href="_inhomogeneous__System__Of__Parameters.html">inhomogeneousSystemOfParameters</a> -- generates an inhomogeneous system of parameters</span></span></li> <li><span><kbd>Seed</kbd> -- see <span><a title="finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal" href="_system__Of__Parameters.html">systemOfParameters</a> -- finds a relatively sparse homogeneous system of parameters of minimal degree in an ideal</span></span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="Finds the depth of a module or ideal, and systems of parameters in an ideal" href="index.html">Depth</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Depth.m2</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Depth.m2:572:0</span>.</p> </div> </div> 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