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Determinants, Groebner Bases and Cohomology (Hardcover, 1st ed. 2022)
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Determinants, Groebner Bases and Cohomology (Hardcover, 1st ed. 2022)
Series: Springer Monographs in Mathematics
Expected to ship within 10 - 15 working days
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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 Groebner 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 Groebner 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, Groebner 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.
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