Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains.

Barucci, Valentina:

ISBN 10: 0821805444 / ISBN 13: 9780821805442

Published by Providence (R.I.), American Mathematical Society, 1997

About this Item

x, 78 p. Ex-library with stamp and library-signature. GOOD condition, some traces of use. Ehem. Bibliotheksexemplar mit Signatur und Stempel. GUTER Zustand, ein paar Gebrauchsspuren. C-05520 9780821805442 Sprache: Englisch Gewicht in Gramm: 550.

Seller Inventory # 2491774

Bibliographic Details

• Title: Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains.
• Publisher: Providence (R.I.), American Mathematical Society
• Publication Date: 1997
• Language: English
• ISBN 10: 0821805444
• ISBN 13: 9780821805442
• Binding: Softcover
• Weight: 550 grams
• Seller catalogs:
  • SA MATHEMATIK

About this title

Synopsis

If $k$ is a field, $T$ an analytic indeterminate over $k$, and $n_1, \ldots , n_h$ are natural numbers, then the semigroup ring $A = k[[T^{n_1}, \ldots , T^{n_h}]]$ is a Noetherian local one-dimensional domain whose integral closure, $k[[T]]$, is a finitely generated $A$-module. There is clearly a close connection between $A$ and the numerical semigroup generated by $n_1, \ldots , n_h$. More generally, let $A$ be a Noetherian local domain which is analytically irreducible and one-dimensional (equivalently, whose integral closure $V$ is a DVR and a finitely generated $A$-module).

As noted by Kunz in 1970, some algebraic properties of $A$ such as "Gorenstein" can be characterized by using the numerical semigroup of $A$ (i.e., the subset of $N$ consisting of all the images of nonzero elements of $A$ under the valuation associated to $V$ ). This book's main purpose is to deepen the semigroup-theoretic approach in studying rings A of the above kind, thereby enlarging the class of applications well beyond semigroup rings. For this reason, Chapter I is devoted to introducing several new semigroup-theoretic properties which are analogous to various classical ring-theoretic concepts. Then, in Chapter II, the earlier material is applied in systematically studying rings $A$ of the above type.

As the authors examine the connections between semigroup-theoretic properties and the correspondingly named ring-theoretic properties, there are some perfect characterizations (symmetric $\Leftrightarrow$ Gorenstein; pseudo-symmetric $\Leftrightarrow$ Kunz, a new class of domains of Cohen-Macaulay type 2). However, some of the semigroup properties (such as "Arf" and "maximal embedding dimension") do not, by themselves, characterize the corresponding ring properties. To forge such characterizations, one also needs to compare the semigroup- and ring-theoretic notions of "type". For this reason, the book introduces and extensively uses "type sequences" in both the semigroup and the ring contexts.