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Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains
     

Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains

by Valentina Barucci, David E. Dobbs, Marco Fontana
 

ISBN-10: 0821805444

ISBN-13: 9780821805442

Pub. Date: 02/12/1997

Publisher: American Mathematical Society

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

Overview

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.

Product Details

ISBN-13:
9780821805442
Publisher:
American Mathematical Society
Publication date:
02/12/1997
Series:
Memoirs of the American Mathematical Society Series , #125
Pages:
78

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