Determinantal rings. (English) Zbl 0673.13006

Bruns, Winfried; Vetter, Udo

Lecture Notes in Mathematics, 1327. Berlin etc.: Springer-Verlag. vii, 236 p. (1988).

Investigations on determinantal ideals and varieties are an interesting part of commutative ring theory and algebraic geometry. The authors avoid geometric methods and develop a purely algebraic approach to determinantal rings using mainly the theory of algebras with straightening law (Hodge algebras) on posets of minors. This is done simultaneously with the treatment of the homogeneous coordinate rings of the Schubert varieties of Grassmannians (so called Schubert cycles), where the combinatorial structure is simpler. [Algebraically every determinantal ring may be considered as a dehomogenization of a Schubert cycle.]
The subjects of this book include results on height and grade, the Cohen- Macaulay property and normality of determinantal rings and Schubert cycles and the computation of their singular locus. Moreover the divisor class groups of Schubert cycles and determinantal rings over a normal ground ring of coefficients are considered. In section \(12\) the authors also discuss Hochster-Eagon’s proof of the perfection of determinantal ideals where principal radical systems instead of standard monomials are used. Finally Kähler differentials and representation-theoretical aspects of determinantal rings are described in a systematic way. The dominating example for illustrating properties of determinantal ideals is the ideal generated by the maximal minors.
The book is self-contained and includes most of the details needed for beginners.

Reviewer: M.Herrmann

MSC:

13C99	Theory of modules and ideals in commutative rings
13-02	Research exposition (monographs, survey articles) pertaining to commutative algebra
14M12	Determinantal varieties
14-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
14M15	Grassmannians, Schubert varieties, flag manifolds
13A15	Ideals and multiplicative ideal theory in commutative rings
14A05	Relevant commutative algebra

Keywords:

determinantal ideals; algebras with straightening law; Hodge algebras; Schubert cycles; divisor class groups