Determinants, Groebner Bases and Cohomology

The book covers determinantal rings and varieties,including methods from combinatorics,algebra,representation theory,and geometry. It studies determinantal ideals via monomial theory and the straightening law,introduces toric methods,and discusses singularities and regularity in positive characteristic. In characteristic zero,it presents sharper results for GL-invariant ideals and concludes with a computation of cohomology with support in determinantal ideals and a survey of their free resolutions.

Format: Hardback
Length: 507 pages
Publication date: 03 December 2022
Publisher: Springer International Publishing AG

algebra,the book offers a comprehensive account of the subject,covering topics such as Gröbner and Sagbi bases,determinantal ideals,representation theory,and algebraic geometry.

After a concise introduction to Gröbner and Sagbi bases,the book delves into the study of determinantal ideals through 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 and yield homological and enumerative theorems on determinantal rings.

Sagbi bases then lead to the introduction of toric methods,which are used to study properties of singularities in positive characteristic. The Frobenius functor is employed 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 is an essential resource for researchers and students in commutative algebra,algebraic geometry,and number theory.

Weight: 945g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031054792
Edition number: 1st ed. 2022