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＜電子ブック＞
Determinants, Gröbner Bases and Cohomology / by Winfried Bruns, Aldo Conca, Claudiu Raicu, Matteo Varbaro
(Springer Monographs in Mathematics. ISSN:21969922)

版	1st ed. 2022.
出版者	(Cham : Springer International Publishing : Imprint: Springer)
出版年	2022
本文言語	英語
大きさ	XIII, 507 p. 21 illus : online resource
著者標目	*Bruns, Winfried author Conca, Aldo author Raicu, Claudiu author Varbaro, Matteo author SpringerLink (Online service)
件　名	LCSH:Algebraic geometry LCSH:Commutative algebra LCSH:Commutative rings LCSH:Algebra, Homological LCSH:Discrete mathematics FREE:Algebraic Geometry FREE:Commutative Rings and Algebras FREE:Category Theory, Homological Algebra FREE:Discrete Mathematics
一般注記	1 Gröbner bases, initial ideals and initial algebras -- 2 More on Gröbner deformations -- 3 Determinantal ideals and the straightening law -- 4 Gröbner bases of determinantal ideals -- 5 Universal Gröbner bases -- 6 Algebras defined by minors -- 7 F-singularities of determinantal rings -- 8 Castelnuovo–Mumford regularity -- 9 Grassmannians, flag varieties, Schur functors and cohomology -- 10 Asymptotic regularity for symbolic powers of determinantal ideals -- 11 Cohomology and regularity in characteristic zero This book offers an up-to-date, comprehensive account of determinantal rings and varieties, presenting a multitude of methods used in their study, with tools from combinatorics, algebra, representation theory and geometry. After a concise introduction to Gröbner and Sagbi bases, determinantal ideals are studied via the standard monomial theory and the straightening law. This opens the door for representation theoretic methods, such as the Robinson–Schensted–Knuth correspondence, which provide a description of the Gröbner bases of determinantal ideals, yielding homological and enumerative theorems on determinantal rings. Sagbi bases then lead to the introduction of toric methods. In positive characteristic, the Frobenius functor is used to study properties of singularities, such as F-regularity and F-rationality. Castelnuovo–Mumford regularity, an important complexity measure in commutative algebra and algebraic geometry, is introduced in the general setting of a Noetherian base ring and then applied to powers and products of ideals. The remainder of the book focuses on algebraic geometry, where general vanishing results for the cohomology of line bundles on flag varieties are presented and used to obtain asymptotic values of the regularity of symbolic powers of determinantal ideals. In characteristic zero, the Borel–Weil–Bott theorem provides sharper results for GL-invariant ideals. The book concludes with a computation of cohomology with support in determinantal ideals and a survey of their free resolutions. Determinants, Gröbner Bases and Cohomology provides a unique reference for the theory of determinantal ideals and varieties, as well as an introduction to the beautiful mathematics developed in their study. Accessible to graduate students with basic grounding in commutative algebra and algebraic geometry, it can be used alongside general texts to illustrate the theory with a particularly interesting and important class of varieties HTTP:URL=https://doi.org/10.1007/978-3-031-05480-8

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