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Andrew Snowden - One of the best experts on this subject based on the ideXlab platform.
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The spectrum of a twisted Commutative Algebra
arXiv: Commutative Algebra, 2020Co-Authors: Andrew SnowdenAbstract:A twisted Commutative Algebra is (for us) a Commutative $\mathbf{Q}$-Algebra equipped with an action of the infinite general linear group. In such Algebras the "$\mathbf{GL}$-prime" ideals assume the duties fulfilled by prime ideals in ordinary Commutative Algebra, and so it is crucial to understand them. Unfortunately, distinct $\mathbf{GL}$-primes can have the same radical, which obstructs one from studying them geometrically. We show that this problem can be eliminated by working with super vector spaces: doing so provides enough geometry to distinguish $\mathbf{GL}$-primes. This yields an effective method for analyzing $\mathbf{GL}$-primes.
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Generalizations of Stillman’s Conjecture via Twisted Commutative Algebra
International Mathematics Research Notices, 2019Co-Authors: Daniel Erman, Andrew SnowdenAbstract:Abstract Combining recent results on Noetherianity of twisted Commutative Algebras by Draisma and the resolution of Stillman’s conjecture by Ananyan–Hochster, we prove a broad generalization of Stillman’s conjecture. Our theorem yields an array of boundedness results in Commutative Algebra that only depend on the degrees of the generators of an ideal and not the number of variables in the ambient polynomial ring.
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Generalizations of Stillman's conjecture via twisted Commutative Algebras
arXiv: Commutative Algebra, 2018Co-Authors: Daniel Erman, Andrew SnowdenAbstract:Combining recent results on noetherianity of twisted Commutative Algebras by Draisma and the resolution of Stillman's conjecture by Ananyan-Hochster, we prove a broad generalization of Stillman's conjecture. Our theorem yields an array of boundedness results in Commutative Algebra that only depend on the degrees of the generators of an ideal, and not the number of variables in the ambient polynomial ring.
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Regularity bounds for twisted Commutative Algebras
arXiv: Commutative Algebra, 2017Co-Authors: Andrew SnowdenAbstract:Let A be the twisted Commutative Algebra freely generated by d indeterminates of degree 1. We show that the regularity of an A-module can be bounded from the first floor(d^2/4) + 2 terms of its minimal free resolution. This extends results of Church and Ellenberg from the d=1 case.
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Introduction to twisted Commutative Algebras
arXiv: Commutative Algebra, 2012Co-Authors: Andrew SnowdenAbstract:This article is an expository account of the theory of twisted Commutative Algebras, which simply put, can be thought of as a theory for handling Commutative Algebras with large groups of linear symmetries. Examples include the coordinate rings of determinantal varieties, Segre-Veronese embeddings, and Grassmannians. The article is meant to serve as a gentle introduction to the papers of the two authors on the subject, and also to point out some literature in which these Algebras appear. The first part reviews the representation theory of the symmetric groups and general linear groups. The second part introduces a related category and develops its basic properties. The third part develops some basic properties of twisted Commutative Algebras from the perspective of classical Commutative Algebra and summarizes some of the results of the authors. We have tried to keep the prerequisites to this article at a minimum. The article is aimed at graduate students interested in Commutative Algebra, Algebraic combinatorics, or representation theory, and the interactions between these subjects.
David Eisenbud - One of the best experts on this subject based on the ideXlab platform.
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the geometry of syzygies a second course in Commutative Algebra and Algebraic geometry
2004Co-Authors: David EisenbudAbstract:Preface: Algebra and Geometry * Free Resolutions and Hilbert Functions * First Examples of Free Resolutions * Points in P^2 * Castelnuovo-Mumford Regularity * The Regularity of Projective Curves * Linear Series and 1-Generic Matrices * Linear Complexes and the Linear Syzygy Theorem * Curves of High Degree * Clifford Index and Canonical Embedding * Appendix 1: Introduction to Local Cohomology * Appendix 2: A Jog Through Commutative Algebra * References * Index
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Commutative Algebra Algebraic geometry and computational methods
1999Co-Authors: David Eisenbud, Computational MethodsAbstract:About Wolfgang Vogel: papers contributed each by David Eisenbud, Hubert Flenner, Jurgen Stuckrad, and Ngo Viet Trung. Survey Articles (8 contributions): some archetypal papers include: Gorenstein Artin Algebras, additive decompositions of forms and the pure Hilbert scheme (AA Iarrobino). On subintegrability (B Singh). Appreciating Applonomius, 2000 years later (W Vogel). Graded local cohomology and some applications (RY Sharp). Recovery of vanishing cycles by log geometry: Case of variables (S Usui), and other papers. Research Articles (9 contributions): archetypal papers include Rectangular simplicial semigroups (W Burns & J Gubeladze). Cohen-Macaulay fiber cones (C D'Cruz et al). Homological aspects of equivariant modules: Matijevic-Roberta and Buchsbaum-Rim (M Hashimoto), and other papers.
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computational methods in Commutative Algebra and Algebraic geometry
1997Co-Authors: Wolmer V Vasconcelos, David Eisenbud, Daniel R Grayson, Mike Stillman, Juurgen HerzogAbstract:This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of Commutative Algebra and Algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation. Although intended for advanced students and researchers with interests both in Algebra and computation, many parts may be read by anyone with a basic abstract Algebra course.
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Commutative Algebra
Graduate Texts in Mathematics, 1995Co-Authors: David EisenbudAbstract:Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards Algebraic geometry. The author presents a comprehensive view of Commutative Algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in Commutative Algebra and Algebraic geometry that flow from it. Applications of the theory and even suggestions for computer Algebra projects are included. This book will appeal to readers from beginners to advanced students of Commutative Algebra or Algebraic geometry. To help beginners, the essential ideals from Algebraic geometry are treated from scratch. Appendices on homological Algebra, multilinear Algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text
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Commutative Algebra with a view toward Algebraic geometry
1995Co-Authors: David EisenbudAbstract:Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards Algebraic geometry. The author presents a comprehensive view of Commutative Algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in Commutative Algebra and Algebraic geometry that flow from it. Applications of the theory and even suggestions for computer Algebra projects are included. This book will appeal to readers from beginners to advanced students of Commutative Algebra or Algebraic geometry. To help beginners, the essential ideals from Algebraic geometry are treated from scratch. Appendices on homological Algebra, multilinear Algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text.
Gennady Lyubeznik - One of the best experts on this subject based on the ideXlab platform.
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finiteness properties of local cohomology modules an application of d modules to Commutative Algebra
Inventiones Mathematicae, 1993Co-Authors: Gennady LyubeznikAbstract:The main goal of this paper is to establish finiteness properties of local cohomology modules in characteristic 0 that would be analogous to those proven by C. Huneke and R. Sharp in characteristicp>0. Our method, based on the theory of AlgebraicD-modules, seems to be the first application ofD-modules to Commutative Algebra.
Benjamin Cooper - One of the best experts on this subject based on the ideXlab platform.
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Topological Field Theories and Harrison Homology
Letters in Mathematical Physics, 2013Co-Authors: Benjamin CooperAbstract:Tools and arguments developed by Kevin Costello are adapted to families of “Outer Spaces” or spaces of graphs. This allows us to prove a version of Deligne’s conjecture: the Harrison homology associated with a homotopy Commutative Algebra is naturally a module over a cobordism category of 3-manifolds.
Qiuhui Mo - One of the best experts on this subject based on the ideXlab platform.
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Embedding Theorems for L-Algebras and Anti-Commutative Algebras
Algebra Colloquium, 2019Co-Authors: Qiuhui MoAbstract:Bokut, Chen and Huang proved that every countably generated L-Algebra over a countable field can be embedded into a simple two-generated L-Algebra. In this paper, we prove that every countably generated L-Algebra can be embedded into a simple two-generated L-Algebra. We also prove that every anti-Commutative Algebra can be embedded into a simple anti-Commutative Algebra, and that every countably generated anti-Commutative Algebra can be embedded into a simple two-generated anti-Commutative Algebra. Finally, we prove that every anti-Commutative Algebra can be embedded into its universal enveloping non-associative Algebra.
