Umfang:
1 Online-Ressource(XIII, 507 p. 21 illus.)
Ausgabe:
1st ed. 2022.
ISBN:
9783031054808
Serie:
Springer Monographs in Mathematics
Inhalt:
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.
Inhalt:
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.
Weitere Ausg.:
ISBN 9783031054792
Weitere Ausg.:
ISBN 9783031054815
Weitere Ausg.:
ISBN 9783031054822
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe ISBN 9783031054792
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe ISBN 9783031054815
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe ISBN 9783031054822
Weitere Ausg.:
Erscheint auch als Druck-Ausgabe Bruns, Winfried, 1946 - Determinants, Gröbner bases and cohomology Cham, Switzerland : Springer Nature, 2022 ISBN 9783031054792
Weitere Ausg.:
ISBN 9783031054815
Weitere Ausg.:
ISBN 9783031054822
Sprache:
Englisch
DOI:
10.1007/978-3-031-05480-8
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