Regular sequences and resultants (Q2724093)

Elimination theory is a classical topic in commutative algebra and algebraic geometry. In recent time it became important in context to computational algebra and algebraic geometry. A view towards resultants from a geometric point of view is given by \textit{J. P. Jouanolou} [Adv. Math. 126, No. 2, 119-250 (1997; Zbl 0882.13008); 114, No. 1, 1-174 (1995; Zbl 0882.13007); 90, No. 2, 117-263 (1991; Zbl 0747.13007); 37, 212-238 (1980; Zbl 0527.13005)] and in the book by \textit{I. M. Gelfand, M. M. Kapranov} and \textit{A. N. Zelevinsky} [``Discriminants, resultants and multidimensional determinants'' (Boston 1994; Zbl 0827.14036)]. The main topic of the paper under review is the investigation of aspects of elimination by treating regular sequences and resultants, the authors present in particular elimination theory in weighted projective spaces over arbitrary Noetherian base rings. NEWLINENEWLINENEWLINEThe book is divided into four chapters. Chapter I, `Preliminaries', deals with the concept of Kronecker extensions of a ring and its modules and the study of numerical monoids, i.e. submonoids of the non-negative integers \(\mathbb N\) containing almost all elements of \(\mathbb N.\) Note that a weighted polynomial ring over a commutative ring leads to questions about the monoid generated by the weights. Chapter II, `Regular sequences', contains the basic treatment on regular sequences and the concept of (relative) complete intersections and locally complete intersections. A crucial point is the study of generic polynomials. It turns out that a sequence of such polynomials is a regular sequence with a certain additional property of the primary decomposition of the ideal generated by them, described by combinatorial data. Chapter III, `Elimination', presents the main part of elimination theory (with respect to projective spaces). Among others it is shown that the generic elimination ideal is principal. In the case of an integrally closed base ring it follows that the ideal of elimination is divisorial. Here an extended version of duality of graded complete intersection is developed. Chapter IV, `Resultants', is devoted to the study of the resultant for a regular sequence \(F_0,\ldots,F_n\) of homogeneous polynomials in the polynomial algebra \(A[T_0,\ldots,T_n]\) over an integrally closed domain \(A.\) It follows that it is a divisorial ideal (like the elimination ideal) with a more functorial property. Using the Koszul resolution there is a construction of a canonical generator of the resultant ideal. In the classical case this was done in the work of \textit{A. Hurwitz} [Ann. Mat. Pure Appl., III. Ser. 20, 113-151 (1913; JFM 44.0142.02)], the authors' starting point for their generalization for arbitrary Noetherian ground rings. Supplements following each chapter provide extra details and insightful examples. NEWLINENEWLINENEWLINEThe exposition provides a complement to sparse elimination theory. In details the difficulties of working over general base rings are carefully investigated. This is essential for many applications, including arithmetic geometry.