Regular Sequences and Resultants
The main purpose of this monograph is to investigate eliminations in weighted projective spaces over commutative noetherian rings. Among the important algebraic tools introduced and generalised, regular sequences and complete intersections play the key role in the theory developed by the authors.
The book has four chapters. Chapter I contains the general concept of Kronecker extensions of a ring and its modules, and presents an investigation of numerical monoids that turn up in a natural way as monoids of positive weights of indeterminates in polynomial algebras. Chapter II deals with regular sequences and complete intersections. The authors present special (but for elimination theory essential) cases of regular sequences, which are characterised by combinatorial means (sequences of generic polynomials and generic Laurent polynomials). Chapter III is a study of the main case of elimination with respect to projective spaces. In particular, the structure of elimination ideals of ground rings of algebras is investigated. It is proved (for regular sequences) that the elimination ideal over an integrally closed noetherian domain is a divisible ideal, and that the elimination ideal over a factorial noetherian ring is principal. The concept of resultant ideals for a regular sequence of homogeneous polynomials is introduced in the Chapter IV, where the construction and basic properties of resultants (canonical generators of resultant ideals) are presented.
This monograph is written carefully and lucidly in a fresh mathematical style. All topics are arranged very clearly. Each section contains a supplement in which additional details and examples are presented. The book is suitable as a useful reference for researchers with an interest in commutative algebra, and for those interested in applications in applied and computational algebra and in algebraic geometry.