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<head><title>Bruns -- produces an ideal with three generators whose 2nd syzygy module is isomorphic to a given module</title>
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<div><h1>Bruns -- produces an ideal with three generators whose 2nd syzygy module is isomorphic to a given module</h1>
<div class="single"><h2>Description</h2>
<div><p><em>Bruns</em> is a package of functions for transforming syzygies.</p>
<p>A well-known paper of Winfried Bruns, entitled <b>”Jede” freie Auflösung ist freie Auflösung eines drei-Erzeugenden Ideals </b> (J. Algebra 39 (1976), no. 2, 429-439), shows that every second syzygy module is the second syzygy module of an ideal with three generators.</p>
<p>The general context of this result uses the theory of ”basic elements”, a commutative algebra version of the general position arguments of the algebraic geometers. The ”Syzygy Theorem” of Evans and Griffiths (<b>Syzygies.</b> London Mathematical Society Lecture Note Series, 106. Cambridge University Press, Cambridge, 1985) asserts that if a module M over a regular local ring S containing a field (the field is conjecturally not necessary), or a graded module over a polynomial ring S, is a k-th syzygy module but not a free module, then M has rank at least k. The theory of basic elements shows that if M is a k-th syzygy of rank >k, then for a ”sufficiently general” element m of M the module M/Sm is again a k-th syzygy.</p>
<p>The idea of Bruns’ theorem is that if M is a second syzygy module, then factoring out (rank M) - 2 general elements gives a second syzygy N of rank 2. It turns out that three general homomorphisms from M to S embed N in S<sup>3</sup> in such a way that the quotient S<sup>3</sup>/N is an ideal generated by three elements.</p>
<div>This package implements this method.</div>
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<div class="single"><h2>Author</h2>
<ul><li><div class="single"><a href="http://www.msri.org/~de">David Eisenbud</a><span> <<a href="mailto:de@msri.org">de@msri.org</a>></span></div>
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<div class="single"><h2>Version</h2>
This documentation describes version <b>2.0</b> of Bruns.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/Bruns.m2">Bruns.m2</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Functions<ul><li><span><a href="_bruns.html" title="Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module">bruns</a> -- Returns an ideal generated by three elements whose 2nd syzygy module is isomorphic to a given module</span></li>
<li><span><a href="_bruns__Ideal.html" title="Returns an ideal generated by three elements whose 2nd syzygy module agrees with the given ideal">brunsIdeal</a> -- Returns an ideal generated by three elements whose 2nd syzygy module agrees with the given ideal</span></li>
<li><span><a href="_elementary.html" title="Elementary moves are used to reduce the target of a syzygy matrix">elementary</a> -- Elementary moves are used to reduce the target of a syzygy matrix</span></li>
<li><span><a href="_evans__Griffith.html" title="Reduces the rank of a syzygy">evansGriffith</a> -- Reduces the rank of a syzygy</span></li>
<li><span><a href="_is__Syzygy.html" title="Tests if a module is a d-th syzygy">isSyzygy</a> -- Tests if a module is a d-th syzygy</span></li>
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