<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>inverse systems</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_inverse_lp__Matrix_rp.html">next</a> | <a href="_inverse.html">previous</a> | <a href="_inverse_lp__Matrix_rp.html">forward</a> | <a href="_inverse.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>inverse systems</h1> <div>Suppose that R = k[x1,...,xn], and that E = k[y1,...,yn] is the injective envelop of k. IE, E is given the R-module structure: x^A . y^B = y^(B-A), if B-A >= 0 in every component, and x^A . y^B = 0 otherwise.<p/> If I is an ideal of R, then the submodule I' = Hom_R(R/I,E) of E is called the (Macaulay) inverse system of I. I is zero-dimensional if and only if I' is finitely generated.<p/> This is a dual operation, since I can be recovered as ann_R(I').<p/> In Macaulay2, currently the computation of the inverse system I' (toDual) and of the ideal I from I' (fromDual) are restricted to the situation where I and I' are homogeneous. As an example, let's compute the ideal corresponding to a cubic.<table class="examples"><tr><td><pre>i1 : R = QQ[a..e];</pre> </td></tr> <tr><td><pre>i2 : g = matrix{{a^3+b^3+c^3+d^3+e^3-d^2*e-a*b*c-a*d*e}} o2 = | a3+b3-abc+c3+d3-ade-d2e+e3 | 1 1 o2 : Matrix R <--- R</pre> </td></tr> <tr><td><pre>i3 : f = fromDual g o3 = | ce be d2+ae+e2 cd bd ad+e2 bc+ae-de b2+ac ab+c2 a2-ae+de | 1 10 o3 : Matrix R <--- R</pre> </td></tr> <tr><td><pre>i4 : I = ideal f 2 2 2 2 o4 = ideal (c*e, b*e, d + a*e + e , c*d, b*d, a*d + e , b*c + a*e - d*e, b ------------------------------------------------------------------------ 2 2 + a*c, a*b + c , a - a*e + d*e) o4 : Ideal of R</pre> </td></tr> </table> The resulting ideal is always zero dimensional, and its Cohen-Macaulay type is the number of generators of the submodule E defined by g. Therefore, if g is a 1 by 1 matrix, then the resulting ideal is Gorenstein.<table class="examples"><tr><td><pre>i5 : res I 1 10 21 21 10 1 o5 = R <-- R <-- R <-- R <-- R <-- R <-- 0 0 1 2 3 4 5 6 o5 : ChainComplex</pre> </td></tr> <tr><td><pre>i6 : betti oo 0 1 2 3 4 5 o6 = total: 1 10 21 21 10 1 0: 1 . . . . . 1: . 10 16 5 . . 2: . . 5 16 10 . 3: . . . . . 1 o6 : BettiTally</pre> </td></tr> </table> <p/> The other direction (starting with an ideal I) is more complicated, since the result may not be finitely generated. So, we must give an integer d as well as the generators of I:<table class="examples"><tr><td><pre>i7 : toDual(3,f) o7 = {12} | a3+b3-abc+c3+d3-ade-d2e+e3 | 1 1 o7 : Matrix R <--- R</pre> </td></tr> </table> The integer d has two interpretations. The most general is that the (finitely generated) intersection of I' and the submodule generated by y1^d ... yn^d is returned. If the ideal I is zero dimensional, d should be an integer such that x^(d+1) is in I = image f for every variable x.<table class="examples"><tr><td><pre>i8 : f = matrix{{a*b,c*d,e^2}} o8 = | ab cd e2 | 1 3 o8 : Matrix R <--- R</pre> </td></tr> <tr><td><pre>i9 : toDual(1,f) o9 = {2} | ace | {2} | bce | {2} | ade | {2} | bde | 4 1 o9 : Matrix R <--- R</pre> </td></tr> <tr><td><pre>i10 : toDual(2,f) o10 = {5} | a2c2e | {5} | b2c2e | {5} | a2d2e | {5} | b2d2e | 4 1 o10 : Matrix R <--- R</pre> </td></tr> <tr><td><pre>i11 : toDual(3,f) o11 = {8} | a3c3e | {8} | b3c3e | {8} | a3d3e | {8} | b3d3e | 4 1 o11 : Matrix R <--- R</pre> </td></tr> <tr><td><pre>i12 : g = toDual(4,f) o12 = {11} | a4c4e | {11} | b4c4e | {11} | a4d4e | {11} | b4d4e | 4 1 o12 : Matrix R <--- R</pre> </td></tr> <tr><td><pre>i13 : fromDual g o13 = | e2 cd ab d5 c5 b5 a5 | 1 7 o13 : Matrix R <--- R</pre> </td></tr> </table> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_to__Dual.html" title="inverse system">toDual</a> -- inverse system</span></li> <li><span><a href="_from__Dual.html" title="ideal from inverse system">fromDual</a> -- ideal from inverse system</span></li> </ul> </div> </div> </body> </html>