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Serre's Problem on Projective Modules
02 Dec 2010-
TL;DR: The "Classical" Results on Serre's Conjecture: 1955-1976 as mentioned in this paper The classical results on the Serre-Conjecture are the following: Horrocks' theorem and Quillen's Methods.
Abstract: to Serre's Conjecture: 1955-1976.- Foundations.- The "Classical" Results on Serre's Conjecture.- The Basic Calculus of Unimodular Rows.- Horrocks' Theorem.- Quillen's Methods.- K1-Analogue of Serre's Conjecture.- The Quadratic Analogue of Serre's Conjecture.- References for Chapters I-VII.- Appendix: Complete Intersections and Serre's Conjecture.- New Developments (since 1977).- References for Chapter VIII.
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29 Jul 2015TL;DR: A detailed introduction to various basic concepts, methods, principles, and results of commutative algebra can be found in this paper, where the authors take a constructive viewpoint and study algorithmic approaches alongside several abstract classical theories, such as Galois theory, Dedekind rings, Prfer rings, finitely generated projective modules, dimension theory of Commutative rings and others.
Abstract: Translated from the popular French edition, this book offers a detailed introduction to various basic concepts, methods, principles, and results of commutative algebra. It takes a constructive viewpoint in commutative algebra and studies algorithmic approaches alongside several abstract classical theories. Indeed, it revisits these traditional topics with a new and simplifying manner, making the subject both accessible and innovative. The algorithmic aspects of such naturally abstract topics as Galois theory, Dedekind rings, Prfer rings, finitely generated projective modules, dimension theory of commutative rings, and others in the current treatise, are all analysed in the spirit of the great developers of constructive algebra in the nineteenth century. This updated and revised edition contains over 350 well-arranged exercises, together with their helpful hints for solution. A basic knowledge of linear algebra, group theory, elementary number theory as well as the fundamentals of ring and module theory is required. Commutative Algebra: Constructive Methods will be useful for graduate students, and also researchers, instructors and theoretical computer scientists.
60 citations
TL;DR: A reasonably detailed review is given of several fundamental theoretical issues that occur in the use of Grobner bases in multidimensional signals and systems applications, including the primeness of multivariate polynomial matrices, multivariate unimodularPolynomial matrix completion, and prime factorization of multidity matrices.
Abstract: This paper is a tutorial on Grobner bases and a survey on the applications of Grobner bases in the broad field of signals and systems. A reasonably detailed review is given of several fundamental theoretical issues that occur in the use of Grobner bases in multidimensional signals and systems applications. These topics include the primeness of multivariate polynomial matrices, multivariate unimodular polynomial matrix completion, and prime factorization of multivariate polynomial matrices. A brief review is also presented on the wide-ranging applications of Grobner bases in multidimensional as well as one-dimensional circuits, networks, control, coding, signals, and systems and other related areas like robotics and applied mechanics. The impact and scope of Grobner bases in signals and systems are highlighted with respect to what has already been accomplished as a stepping stone to expanding future research.
60 citations
TL;DR: In this paper, it was shown that BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra.
Abstract: In Bennett et al. [BGG reciprocity for current algebras, Adv. Math. 231 (2012), 276–305] it was conjectured that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We associate a current algebra to any indecomposable affine Lie algebra and show that, in this generality, the BGG reciprocity is true for the corresponding category of representations.
58 citations
Cites background from "Serre's Problem on Projective Modul..."
...Since Aλ is a polynomial ring it follows from the the famous result of Quillen (see [23] for an exposition) that W (λ, r) + μ is in fact a free Aλ–submodule of W (λ, r)....
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01 Jan 2007
TL;DR: In this article, a constructive version of the Quillen-Suslin theorem, also known as Serre's conjecture, is used to effectively compute flat outputs and injective parametrizations of flat multidimensional linear systems.
Abstract: The purpose of this paper is to give four new applications of the Quillen-Suslin theorem to mathematical systems theory. Using a constructive version of the Quillen-Suslin theorem, also known as Serre's conjecture, we show how to effectively compute flat outputs and injective parametrizations of flat multidimensional linear systems. We prove that a flat multidimensional linear system is algebraically equivalent to the controllable 1-D dimensional linear systems obtained by setting all but one functional operator to zero in the polynomial matrix defining the system. In particular, we show that a flat ordinary differential time-delay linear system is algebraically equivalent to the corresponding ordinary differential system without delay, i.e., the controllable ordinary differential linear system obtained by setting all the delay amplitudes to zero. We also give a constructive proof of a generalization of Serre's conjecture known as Lin-Bose's conjecture. Moreover, we show how to constructively compute (weakly) left-/right-/doubly coprime factorizations of rational transfer matrices over a commutative polynomial ring. The Quillen-Suslin theorem also plays a central part in the so-called decomposition problem of linear functional systems studied in the literature of symbolic computation. In particular, we show how the basis computation of certain free modules, coming from projectors of the endomorphism ring of the module associated with the system, allows us to obtain unimodular matrices which transform the system matrix into an equivalent block-triangular or a block-diagonal form. Finally, we demonstrate the package QuillenSuslin which, to our knowledge, contains the first implementation of the Quillen-Suslin theorem in a computer algebra system as well as the different algorithms developed in the paper.
53 citations
Cites background or methods from "Serre's Problem on Projective Modul..."
...We refer the interested reader to Lam’s nice books [23, 24] concerning S erre’s conjecture....
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...We refer to [23, 24] for the best introductions nowadays available on this su bject....
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...It is well known that the Quillen-Suslin theorem is a particular case of this problem when M is a projective D-module ([23, 24, 52, 53])....
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...Using the famous Quillen-Suslin theorem ([52, 55]), also known as Serr e’s conjecture ([23, 24]), we then know that free D-modules are projective ones....
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...4 (Serre) [10, 23, 24] Let M be a finitely generated projective D = k[x1, ....
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27 Jul 2010
TL;DR: The main purpose of these lectures was to introduce the French community of symbolic computation to the constructive approach to algebraic analysis and particularly algebraic D-modules, its applications to mathematical systems theory and its implementations in computer algebra systems such as Maple or GAP4 as mentioned in this paper.
Abstract: This text is an extension of lectures notes I prepared for les Journees Nationales de Calcul Formel held at the CIRM, Luminy (France) on May 3-7, 2010. The main purpose of these lectures was to introduce the French community of symbolic computation to the constructive approach to algebraic analysis and particularly to algebraic D-modules, its applications to mathematical systems theory and its implementations in computer algebra systems such as Maple or GAP4. Since algebraic analysis is a mathematical theory which uses different techniques coming from module theory, homological algebra, sheaf theory, algebraic geometry, and microlocal analysis, it can be difficult to enter this fascinating new field of mathematics. Indeed, there are very few introducing texts. We are quickly led to Bjork's books which, at first glance, may look difficult for the members of the symbolic computation community and for applied mathematicians. I believe that the main issue is less the technical difficulty of the existing references than the lack of friendly introduction to the topic, which could have offered a general idea of it, shown which kind of results and applications we can expect and how to handle the different computations on explicit examples. To a very small extent, these lectures notes were planned to fill this gap, at least for the basic ideas of algebraic analysis. Since, we can only teach well what we have clearly understood, I have chosen to focus on my work on the constructive aspects of algebraic analysis and its applications.
52 citations
Cites background from "Serre's Problem on Projective Modul..."
...In 1976, this remark, called “Serre’s conjecture” ([55]), was independently solved by Quillen ([107]) and Suslin ([115])....
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