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.