LEADER 05868nam 2200577 450 001 996503552303316 005 20230512152002.0 010 $a3-031-05480-6 035 $a(MiAaPQ)EBC7151651 035 $a(Au-PeEL)EBL7151651 035 $a(CKB)25554151000041 035 $a(EXLCZ)9925554151000041 100 $a20230404d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aDeterminants, Gro?bner bases and cohomology /$fWinfried Bruns [and three others] 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (514 pages) 225 1 $aSpringer monographs in mathematics 311 08$aPrint version: Bruns, Winfried Determinants, Gröbner Bases and Cohomology Cham : Springer International Publishing AG,c2023 9783031054792 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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 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. 327 $a8.2 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(?) -- 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(?)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????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 `3?9`42`"?613A``45`47`"603AExt Modules for J(?)l -- 11.7 `3?9`42`"?613A``45`47`"603AExt Modules for Symbolic Powers. 327 $a11.8 The Basics on `3?9`42`"?613A``45`47`"603AGL-Invariant Ideals -- 11.9 `3?9`42`"?613A``45`47`"603AExt Modules for `3?9`42`"?613A``45`47`"603AGL-Invariant Ideals -- 11.10 Local Cohomology with Determinantal Support -- 11.11 Syzygies of `3?9`42`"?613A``45`47`"603AGL-Invariant Ideals -- Appendix List of Symbols -- References -- -- Index. 410 0$aSpringer monographs in mathematics. 606 $aAlgebra, Homological 606 $aAlgebraic geometry 606 $aCommutative algebra 606 $aÀlgebra homològica$2thub 606 $aGeometria algebraica$2thub 606 $aÀlgebra commutativa$2thub 608 $aLlibres electrònics$2thub 615 0$aAlgebra, Homological. 615 0$aAlgebraic geometry. 615 0$aCommutative algebra. 615 7$aÀlgebra homològica 615 7$aGeometria algebraica 615 7$aÀlgebra commutativa 676 $a512.55 700 $aBruns$b Winfried$056675 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996503552303316 996 $aDeterminants, Gro?bner bases and cohomology$93084265 997 $aUNISA