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Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains
 
Valentina Barucci Universita di Roma, “La Sapienza”, Rome, Italy
David E. Dobbs University of Tennessee, Knoxville, Knoxville, TN
Marco Fontana Terza Universita di Roma, Rome, Italy
Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains
eBook ISBN:  978-1-4704-0183-2
Product Code:  MEMO/125/598.E
List Price: $45.00
MAA Member Price: $40.50
AMS Member Price: $27.00
Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains
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Maximality Properties in Numerical Semigroups and Applications to One-Dimensional Analytically Irreducible Local Domains
Valentina Barucci Universita di Roma, “La Sapienza”, Rome, Italy
David E. Dobbs University of Tennessee, Knoxville, Knoxville, TN
Marco Fontana Terza Universita di Roma, Rome, Italy
eBook ISBN:  978-1-4704-0183-2
Product Code:  MEMO/125/598.E
List Price: $45.00
MAA Member Price: $40.50
AMS Member Price: $27.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1251997; 78 pp
    MSC: 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 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.

    Readership

    Advanced 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 one-dimensional analytically irreducible local domains
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1251997; 78 pp
MSC: 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 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.

Readership

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 one-dimensional analytically irreducible local domains
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.