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<!DOCTYPE html> <html lang="en"> <head> <title>Example 4 -- invariants of principal jets of monomial ideals</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="compute jets of various algebraic, geometric and combinatorial objects" href="index.html">Jets</a> » <a title="invariants of principal jets of monomial ideals" href="___Example_sp4.html">Example 4</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="___Storing_sp__Computations.html">next</a> | <a href="___Example_sp3.html">previous</a> | <a href="___Storing_sp__Computations.html">forward</a> | <a href="___Example_sp3.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>Example 4 -- invariants of principal jets of monomial ideals</h1> <div> <div> <p>This follows Examples 7.5 and 7.7 in <a href="https://arxiv.org/abs/2407.01836">F. Galetto, N. Iammarino, and T. Yu, Jets and principal components of monomial ideals, and very well-covered graphs</a>.</p> <p>Consider the following squarefree monomial ideal in a standard graded polynomial ring.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[v..z] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = ideal(v*w*x,x*y,y*z) o2 = ideal (v*w*x, x*y, y*z) o2 : Ideal of R</code></pre> </td> </tr> </table> <div> <p>This is the Stanley-Reisner ideal of a simplicial complex $\Delta$ whose $f$-vector we compute below.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i3 : needsPackage "SimplicialComplexes" o3 = SimplicialComplexes o3 : Package</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : Δ = simplicialComplex I o4 = simplicialComplex | wxz vxz vwz vwy | o4 : SimplicialComplex</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : f = matrix{fVector(Δ)} o5 = | 1 5 8 4 | 1 4 o5 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> </table> <div> <p>Next, we construct the ideal $\mathcal{P}_1 (I)$ of principal 1-jets of $I$ (see <a title="compute principal component of jets" href="_principal__Component.html">principalComponent</a> for details). This is also the Stanley-Reisner ideal of a simplicial complex $\Gamma_1$ and we can compute its $f$-vector.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i6 : P1 = principalComponent(1,I) o6 = ideal (y0*z0, x0*y0, y0*z1, z0*y1, x0*y1, y0*x1, y1*z1, x1*y1, v0*w0*x0, ------------------------------------------------------------------------ v0*w0*x1, v0*x0*w1, w0*x0*v1, v0*w1*x1, w0*v1*x1, x0*v1*w1, v1*w1*x1) o6 : Ideal of QQ[v0, w0, x0, y0, z0][v1, w1, x1, y1, z1]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : phi = last flattenRing ring P1; o7 : RingMap QQ[v1, w1, x1, y1, z1, v0, w0, x0, y0, z0] <-- QQ[v0, w0, x0, y0, z0][v1, w1, x1, y1, z1]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : Γ1 = simplicialComplex phi P1 o8 = simplicialComplex | w1x1z1w0x0z0 v1x1z1v0x0z0 v1w1z1v0w0z0 v1w1y1v0w0y0 | o8 : SimplicialComplex</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : F = matrix{fVector Γ1} o9 = | 1 10 37 64 56 24 4 | 1 7 o9 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> </table> <div> <p>The $f$-vector of $\Gamma_1$ can be obtained by multiplying the $f$-vector of $\Delta$ with a <a title="arrange values of lifting function in a matrix" href="_lifting__Matrix.html">liftingMatrix</a> of the appropriate size.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i10 : L = liftingMatrix(1,4,7) o10 = | 1 0 0 0 0 0 0 | | 0 2 1 0 0 0 0 | | 0 0 4 4 1 0 0 | | 0 0 0 8 12 6 1 | 4 7 o10 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : F == f*L o11 = true</code></pre> </td> </tr> </table> <div> <p>There is a similar relation between the Betti numbers of the Stanley-Reisner rings $\Bbbk [\Delta]$ and $\Bbbk [\Gamma_1]$. First, we compute the Betti diagram of $\Bbbk [\Delta]$ and turn it into a matrix by sliding the $i$-th row $i$ units to the right.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i12 : betti res I 0 1 2 o12 = total: 1 3 2 0: 1 . . 1: . 2 1 2: . 1 1 o12 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : b = mutableMatrix(ZZ,3,5);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i14 : scanPairs(betti res I, (k,v) -> b_(k_2-k_0,k_2) = v);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i15 : b = matrix b o15 = | 1 0 0 0 0 | | 0 0 2 1 0 | | 0 0 0 1 1 | 3 5 o15 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> </table> <div> <p>Next, we do the same with the Betti diagram of $\Bbbk [\Gamma_1]$.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i16 : betti res P1 0 1 2 3 4 5 6 o16 = total: 1 16 44 52 31 9 1 0: 1 . . . . . . 1: . 8 16 14 6 1 . 2: . 8 28 38 25 8 1 o16 : BettiTally</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i17 : B = mutableMatrix(ZZ,3,9);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i18 : scanPairs(betti res P1, (k,v) -> B_(k_2-k_0,k_2) = v);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i19 : B = matrix B o19 = | 1 0 0 0 0 0 0 0 0 | | 0 0 8 16 14 6 1 0 0 | | 0 0 0 8 28 38 25 8 1 | 3 9 o19 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> </table> <div> <p>The matrix containing the Betti numbers of $\Bbbk [\Gamma_1]$ can be obtained by multiplying the matrix containing the Betti numbers of $\Bbbk [\Delta]$ with a <a title="arrange values of lifting function in a matrix" href="_lifting__Matrix.html">liftingMatrix</a> of the appropriate size.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i20 : L = liftingMatrix(1,5,9) o20 = | 1 0 0 0 0 0 0 0 0 | | 0 2 1 0 0 0 0 0 0 | | 0 0 4 4 1 0 0 0 0 | | 0 0 0 8 12 6 1 0 0 | | 0 0 0 0 16 32 24 8 1 | 5 9 o20 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i21 : B == b*L o21 = true</code></pre> </td> </tr> </table> </div> <div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Jets.m2:1666:0</span>.</p> </div> </div> </div> </body> </html>