MEGA 2007Effective Methods in Algebraic Geometry Strobl, Austria, June 25th - 29th |
 |
Electronic Proceedings
Abstract
| Title | Suslin's lemma for elimination |
| Keywords | Suslin's lemma, Quillen-Suslin theorem,resultant, radical of an ideal, commutative algebra, algebraic geometry, constructive mathematics, computer algebra |
| Abstract | The purpose of this paper is to study algorithmically an important
lemma of Suslin which turned out to be useful for eliminating variables and decisive in Suslin's second solution of Serre's conjecture, that is, in his elementary proof that finitely generated projective modules over $\K[X_1,\ldots,X_n]$, $\K$ a principal domain, are free. This lemma says that for a commutative ring $\A$, if $\gen{v_1(X),\ldots,v_n(X)} = \A[X]$ where $v_1$ is monic and $n \geq 3$, then there exist $\gamma_1,\ldots,\gamma_{\ell} \in {\rm E}_{n-1}(\A[X])$ such that $\gen{\Res(v_1, e_1.\gamma_1 \tra{ (v_2,\ldots,v_n))},\ldots, \Res (v_1, e_1.\gamma_{\ell} \tra{ (v_2,\ldots,v_n))}} = \A$. \n In fact, this lemma is the only non constructive step in Suslin's elementary proof of Serre's conjecture. The problem with Suslin's proof is that it does not give an explicit way to find the elementary operations $\gamma_i$ since it reasons modulo each maximal ideal of $\A$. Of course this important for concrete applications in circuits, systems controls, signal processing, and other areas. We will give an algorithm realizing Suslin's lemma for any ring $\A$. A detailed example of a unimodular completion of a vector in $ {\rm Um}_{3}(\mathbb{Z}[x]) $ will be given. In the particular case where the basic ring $\A$ contains an infinite field, we give a more precise and simpler version of this lemma. As a matter of fact, using a new definition of the resultant, we will give a simplified formulation of Suslin's lemma. As application to our study of Suslin's lemma, we give two simple algorithms for unimodular completion (the Quillen-Suslin theorem). The first one is over rings $\A$ containing an infinite field. This algorithm has been implemented with the computer algebra system Maple. The second one is over any ring $\A$ and is partially implemented. |
The Institute is named after the famous Austrian mathematician Johann Radon (1887-1956)
Medieninhaber:
Österreichische Akademie der Wissenschaften
Juristische Person öffentlichen Rechts (BGBl 569/1921 idF BGBl I 130/2003)
Dr. Ignaz Seipel-Platz 2, 1010 Wien
Diese Website dient zur Information über die wissenschaftlichen Aktivitäten der Österreichischen Akademie der Wissenschaften und setzt somit den gesetzlichen Auftrag um, die Wissenschaft in jeder Hinsicht zu fördern.
This RICAM page was made with 100% valid HTML & CSS - Send comments to Webmaster
Today's date and time is 02/09/12 - 10:52 CET and this file ( /mega2007/openconf/electronic/31-abs.html ) was last modified on 06/19/07 - 14:54 CEST