國立虎尾科技大學 |

Determinants, Gröbner Bases and Cohomology

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Determinants, Gröbner Bases and Cohomology/ by Winfried Bruns, Aldo Conca, Claudiu Raicu, Matteo Varbaro.
作者:	Bruns, Winfried.
其他作者:	Varbaro, Matteo.
面頁冊數:	XIII, 507 p. 21 illus.online resource. :
Contained By:	Springer Nature eBook
標題:	Discrete Mathematics. -
電子資源:	https://doi.org/10.1007/978-3-031-05480-8
ISBN:	9783031054808

Determinants, Gröbner Bases and Cohomology
Bruns, Winfried.

Determinants, Gröbner Bases and Cohomology[electronic resource] /by Winfried Bruns, Aldo Conca, Claudiu Raicu, Matteo Varbaro. - 1st ed. 2022. - XIII, 507 p. 21 illus.online resource. - Springer Monographs in Mathematics,2196-9922. - Springer Monographs in Mathematics,.

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.

ISBN: 9783031054808

Standard No.: 10.1007/978-3-031-05480-8doiSubjects--Topical Terms:

796600
Discrete Mathematics.

LC Class. No.: QA564-609

Dewey Class. No.: 516.35

Determinants, Gröbner Bases and Cohomology
LDR:03946nam a22004095i 4500 001 1086157
003 DE-He213
005 20221202115609.0
007 cr nn 008mamaa
008 221228s2022 sz | s |||| 0|eng d
020 $a 9783031054808 $9 978-3-031-05480-8
024 7 $a 10.1007/978-3-031-05480-8 $2 doi
035 $a 978-3-031-05480-8
050 4 $a QA564-609
072 7 $a PBMW $2 bicssc
072 7 $a MAT012010 $2 bisacsh
072 7 $a PBMW $2 thema
082 0 4 $a 516.35 $2 23
100 1 $a Bruns, Winfried. $e author. $0 (orcid)0000-0002-7081-2261 $1 https://orcid.org/0000-0002-7081-2261 $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1392854
245 1 0 $a Determinants, Gröbner Bases and Cohomology $h [electronic resource] / $c by Winfried Bruns, Aldo Conca, Claudiu Raicu, Matteo Varbaro.
250 $a 1st ed. 2022.
264 1 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2022.
300 $a XIII, 507 p. 21 illus. $b online resource.
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
347 $a text file $b PDF $2 rda
490 1 $a Springer Monographs in Mathematics, $x 2196-9922
505 0 $a 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.
520 $a 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.
650 2 4 $a Discrete Mathematics. $3 796600
650 2 4 $a Category Theory, Homological Algebra. $3 678397
650 2 4 $a Commutative Rings and Algebras. $3 672054
650 1 4 $a Algebraic Geometry. $3 670184
650 0 $a Discrete mathematics. $3 1254302
650 0 $a Algebra, Homological. $3 672614
650 0 $a Commutative rings. $3 672474
650 0 $a Commutative algebra. $3 672047
650 0 $a Algebraic geometry. $3 1255324
700 1 $a Varbaro, Matteo. $e author. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1362752
700 1 $a Raicu, Claudiu. $e author. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1362751
700 1 $a Conca, Aldo. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1197533
710 2 $a SpringerLink (Online service) $3 593884
773 0 $t Springer Nature eBook
776 0 8 $i Printed edition: $z 9783031054792
776 0 8 $i Printed edition: $z 9783031054815
776 0 8 $i Printed edition: $z 9783031054822
830 0 $a Springer Monographs in Mathematics, $x 1439-7382 $3 1254272
856 4 0 $u https://doi.org/10.1007/978-3-031-05480-8
912 $a ZDB-2-SMA
912 $a ZDB-2-SXMS
950 $a Mathematics and Statistics (SpringerNature-11649)
950 $a Mathematics and Statistics (R0) (SpringerNature-43713)