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Record Nr. |
UNISA996503552303316 |
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Autore |
Bruns Winfried |
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Titolo |
Determinants, Gröbner bases and cohomology / / Winfried Bruns [and three others] |
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Pubbl/distr/stampa |
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Cham, Switzerland : , : Springer, , [2022] |
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©2022 |
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ISBN |
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Descrizione fisica |
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1 online resource (514 pages) |
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Collana |
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Springer monographs in mathematics |
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Disciplina |
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Soggetti |
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Algebra, Homological |
Algebraic geometry |
Commutative algebra |
Àlgebra homològica |
Geometria algebraica |
Àlgebra commutativa |
Llibres electrònics |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Nota di bibliografia |
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Includes bibliographical references and index. |
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Nota di contenuto |
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Intro -- Preface -- Contents -- 1 Gröbner Bases, Initial Ideals and Initial Algebras -- 1.1 Monomial Orders -- 1.2 Gröbner Bases -- 1.3 Sagbi Bases -- 1.4 Initial Subspaces and Gradings -- 1.5 Initial Subspaces for Weights. Homogenization -- 1.6 Deformation to the Initial Object -- 2 More on Gröbner Deformations -- 2.1 Minimal and Associated Primes -- 2.2 Connectedness -- 2.3 Optimal Gröbner Deformations -- 2.4 Square-Free Gröbner Deformations -- 3 Determinantal Ideals and the Straightening Law -- 3.1 Minors and Determinantal Ideals -- 3.2 Standard Bitableaux and the Straightening Law -- 3.3 Refinement of the Straightening Law -- 3.4 Determinantal Rings -- 3.5 Powers and Products of Determinantal Ideals -- 3.6 Representation Theory -- 4 Gröbner Bases of Determinantal Ideals -- 4.1 An Instructive Case: the Ideal of 2-Minors -- 4.2 The Robinson-Schensted-Knuth Correspondence -- 4.3 RSK and Gröbner Bases of Ideals -- 4.4 Cohen-Macaulayness and Hilbert Series of Determinantal Rings -- 4.5 Gröbner Bases via Secants -- 4.6 Multigraded Structures |
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and the Multidegree -- 5 Universal Gröbner Bases -- 5.1 Universal Gröbner Bases -- 5.2 Universal Gröbner Bases for Maximal Minors -- 5.3 Universal Gröbner Bases for 2-Minors -- 6 Algebras Defined by Minors -- 6.1 A Recap of Toric Algebra -- 6.2 Algebras Defined by Maximal Minors -- 6.3 Rees Algebras of Determinantal Ideals -- 6.4 The Algebra of Minors -- 6.5 Algebraic Relations between Minors -- 6.6 Exterior Powers of Linear Maps -- 6.7 Sagbi Deformations of Determinantal Rings -- 7 F-singularities of Determinantal Rings -- 7.1 F-purity -- 7.2 F-regularity -- 7.3 F-rationality and Gröbner Deformations -- 7.4 Equivariant Deformations -- 7.5 F-pure Thresholds -- 8 Castelnuovo-Mumford Regularity -- 8.1 Castelnuovo-Mumford Regularity over General Base Rings. |
8.2 Castelnuovo-Mumford Regularity and Multigraded Structures -- 8.3 Vanishing over Nonstandard Multigraded Rings -- 8.4 Regularity of Powers and Products of Ideals -- 8.5 Linear Powers and Linear Products -- 8.6 A Family of Determinantal Ideals with Linear Products -- 8.7 Maximal Minors of Matrices of Linear Forms and Linear Powers -- 9 Grassmannians, Flag Varieties, Schur Functors and Cohomology -- 9.1 Fundamental Multilinear Algebra Constructions -- 9.2 Affine and Projective Bundles, Grassmannians, Flag Varieties -- 9.3 Basic Calculus with Direct Images -- 9.4 The Kempf Vanishing Theorem -- 9.5 Characteristic-Free Vanishing for some Nondominant Weights -- 9.6 A Sharp Characteristic-Free Vanishing on Projective Bundles -- 9.7 Algebraic Groups and Representations -- 9.8 Schur Functors -- 9.9 Grothendieck Duality -- 9.10 Cohomology Vanishing on Grassmannians -- 10 Asymptotic Regularity for Symbolic Powers of Determinantal Ideals -- 10.1 Quotients of Ideals Defined by Shape -- 10.2 Filtrations Associated with Ideals Defined by Shape -- 10.3 An algorithm to Compute Z(Σ) -- 10.4 The Filtration for Symbolic Powers of Determinantal Ideals -- 10.5 Sheaves Defined by Shape and Their Quotients -- 10.6 A Geometric Realization of the Modules J(σ)l -- 10.7 Regularity of Symbolic Powers of Determinantal Ideals -- 10.8 Some Geometric Properties of Determinantal Varieties -- 11 Cohomology and Regularity in Characteristic Zero -- 11.1 The Borel-Weil-Bott Theorem -- 11.2 The Optimality of Theorem 9.5.1摥映數爠eflinkthm:charspsfreespsvanishing9.5.19 -- 11.3 Borel-Weil-Bott on Grassmannians -- 11.4 Effective Vanishing of Cohomology in Characteristic Zero -- 11.5 Cauchy's Formula Revisited -- 11.6 Explicit Description of `3́9`42`"̇613A``45`47`"603AExt Modules for J(σ)l -- 11.7 `3́9`42`"̇613A``45`47`"603AExt Modules for Symbolic Powers. |
11.8 The Basics on `3́9`42`"̇613A``45`47`"603AGL-Invariant Ideals -- 11.9 `3́9`42`"̇613A``45`47`"603AExt Modules for `3́9`42`"̇613A``45`47`"603AGL-Invariant Ideals -- 11.10 Local Cohomology with Determinantal Support -- 11.11 Syzygies of `3́9`42`"̇613A``45`47`"603AGL-Invariant Ideals -- Appendix List of Symbols -- References -- -- Index. |
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