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**Determinantal rings.**
*(English)*
Zbl 1079.14533

Monografías de Matemática (Rio de Janeiro) 45. Rio de Janeiro: Instituto de Matemática Pura e Aplicada (IMPA). viii, 236 p. (1988).

Let \(A\) be a commutative ring and let \(X=(X_{ij})\) be an \(m\times n\) matrix whose entries are indeterminates over \(A\). Let \(A[X]\) denote the polynomial ring \(A[X_{ij}]\), \(i=1,\cdots,m\), \(j=1,\cdots,n\), and let \(I_t(X)\) be the ideal of \(A[X]\) generated by all \(t\times t\) minors of \(X\). Ideals of the form \(I_t(X)\) are called determinantal ideals and the factor ring \(R_t(X)=A[X]/I_t(X)\) is called the determinantal ring associated to \(I_t(X)\). If \(A\) is a field then the affine variety determined by \(I_t(X)\) is called the determinantal variety.

The work under review is the first book in 50 years devoted entirely to determinantal rings and varieties. T. G. Room’s book “Geometry of determinantal loci” [Cambridge Univ. Press, Cambridge (1938; Zbl 0020.05402, JFM 64.0693.04)] is a source of classical results in this field. The subject has attracted many algebraists, geometers and combinatorialists over the last 30 years and the book presents a systematic treatment of the theory which took shape in the last few decades. The authors’ approach is algebraic and is based on the theory of algebras with straightening law. However, principal radical systems and the relationship with invariant theory and representation theory are also discussed in detail. The main topics not covered in the book include geometric methods, finite free resolutions of determinantal ideals (except those of maximal minors) and ideals generated by minors of symmetric matrices and Pfaffians of skew-symmetric matrices. However, the rather extensive bibliography also contains references to topics concerning determinantal ideals which are not discussed in the book.

The list of topics covered in the consecutive chapters is as follows: 1. Preliminaries; 2. Ideals of maximal minors; 3. Generically perfect ideals; 4. Algebras with straightening law on posets of minors; 5. The structure of an algebra with straightening law; 6. Integrity and normality. The singular locus; 7. Generic points and invariant theory; 8. The divisor class group and the canonical class; 9. Powers of ideals of maximal minors; 10. Primary decomposition; 11. Representation theory; 12. Principal radical systems; 13. Generic modules; 14. The module of Kähler differentials; 15. Derivations and rigidity. In an appendix the authors discuss topics in commutative algebra for which they were not able to find an adequate reference in standard textbooks.

The work under review is the first book in 50 years devoted entirely to determinantal rings and varieties. T. G. Room’s book “Geometry of determinantal loci” [Cambridge Univ. Press, Cambridge (1938; Zbl 0020.05402, JFM 64.0693.04)] is a source of classical results in this field. The subject has attracted many algebraists, geometers and combinatorialists over the last 30 years and the book presents a systematic treatment of the theory which took shape in the last few decades. The authors’ approach is algebraic and is based on the theory of algebras with straightening law. However, principal radical systems and the relationship with invariant theory and representation theory are also discussed in detail. The main topics not covered in the book include geometric methods, finite free resolutions of determinantal ideals (except those of maximal minors) and ideals generated by minors of symmetric matrices and Pfaffians of skew-symmetric matrices. However, the rather extensive bibliography also contains references to topics concerning determinantal ideals which are not discussed in the book.

The list of topics covered in the consecutive chapters is as follows: 1. Preliminaries; 2. Ideals of maximal minors; 3. Generically perfect ideals; 4. Algebras with straightening law on posets of minors; 5. The structure of an algebra with straightening law; 6. Integrity and normality. The singular locus; 7. Generic points and invariant theory; 8. The divisor class group and the canonical class; 9. Powers of ideals of maximal minors; 10. Primary decomposition; 11. Representation theory; 12. Principal radical systems; 13. Generic modules; 14. The module of Kähler differentials; 15. Derivations and rigidity. In an appendix the authors discuss topics in commutative algebra for which they were not able to find an adequate reference in standard textbooks.

Reviewer: Tadeusz Józefiak (MR0986492)