eBook ISBN:  9781470401832 
Product Code:  MEMO/125/598.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $27.00 
eBook ISBN:  9781470401832 
Product Code:  MEMO/125/598.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $27.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 125; 1997; 78 ppMSC: Primary 13; 20
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 onedimensional 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 onedimensional (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 semigrouptheoretic 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 semigrouptheoretic properties which are analogous to various classical ringtheoretic 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 semigrouptheoretic properties and the correspondingly named ringtheoretic properties, there are some perfect characterizations (symmetric \(\Leftrightarrow\) Gorenstein; pseudosymmetric \(\Leftrightarrow\) Kunz, a new class of domains of CohenMacaulay 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 ringtheoretic notions of “type”. For this reason, the book introduces and extensively uses “type sequences” in both the semigroup and the ring contexts.
ReadershipAdvanced graduate students, research mathematicians, algebraists, commutative ring theorists, algebraic geometers, and semigroup theorists.

Table of Contents

Chapters

I. Maximality properties in numerical semigroups

II. Maximality properties in onedimensional analytically irreducible local domains


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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 onedimensional 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 onedimensional (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 semigrouptheoretic 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 semigrouptheoretic properties which are analogous to various classical ringtheoretic 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 semigrouptheoretic properties and the correspondingly named ringtheoretic properties, there are some perfect characterizations (symmetric \(\Leftrightarrow\) Gorenstein; pseudosymmetric \(\Leftrightarrow\) Kunz, a new class of domains of CohenMacaulay 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 ringtheoretic notions of “type”. For this reason, the book introduces and extensively uses “type sequences” in both the semigroup and the ring contexts.
Advanced graduate students, research mathematicians, algebraists, commutative ring theorists, algebraic geometers, and semigroup theorists.

Chapters

I. Maximality properties in numerical semigroups

II. Maximality properties in onedimensional analytically irreducible local domains