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

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*Valentina Barucci; David E. Dobbs; Marco Fontana*

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.

#### Table of Contents

# Table of Contents

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

- Contents vii8 free
- Abstract viii9 free
- Introduction ix10 free
- Chapter I. Maximality Properties in Numerical Semigroups 112 free
- 1. Symmetric semigroups and pseudo-symmetric semigroups 112
- 2. The Lipman semigroup of an ideal and semigroups of maximal embedding dimension 819
- 3. Arf semigroups 1425
- 4. Symmetric or pseudo-symmetric Arf semigroups 2031
- 5. Propinquity of symmetric Arf semigroups 2637
- 6. Arf semigroups and degree of singularity 3344

- Chapter II. Maximality Properties in One-Dimensional Analytically Irreducible Local Domains 3849
- 1. Gorenstein domains and Kunz domains 3849
- 2. Domains with maximal embedding dimension and Arf domains 4859
- 3. Arf Gorenstein domains and Arf Kunz domains 5768
- 4. Gorenstein overrings of a Gorenstein domain 6374
- 5. One-dimensional analytically irreducible local Arf domains and degree of singularity 6879

- References 7586