# The Centennial Celebration of Commutative Algebra Abstracts

## Abstracts

Fri 9:20–9:40a
Martine Picavet, Universite' Blaise Pascal,
"Factorization functions in weakly factorial quadratic orders"

A weakly factorial domain is such that every nonunit element is a product of
primary elements. Weakly factorial orders are characterized by means of
their group of units. If $R$ is a weakly factorial quadratic order, the
fundamental unit of the integral closure of $R$ determines the structure of
atoms of $R$, thereby allowing to compute the following factorization
functions on $R$. We denote respectively by $l(x)$ and $L(x)$ the inf and sup
of the lengths of factorizations of a nonzero nonunit $x\in R$ into a
product of irreducible elements and by $f(x)$ the number of nonassociated
irreducible factorizations of $x$ in $R$. Explicite formulas for $l(x)$ and
$L(x)$ are given from which the asymptotic behavior of these functions is
deduced. When $R$ is in addition a half-factorial domain, we give similar
results for $f(x)$.

Fri 9:50–10:10a
Sophie Frisch, Technische Universitaet Graz,
"Hilbert's Nullstellensatz for rings of integer-valued polynomials"

For a large class of Noetherian domains D (recently characterized
completely) the ring of integer-valued polynomials in several indeterminates
satisfies suitably phrased versions of Hilbert's Nullstellensatz. (The ring
Int(D^n) of integer-valued polynomials in n indeterminates over a domain D is
the ring of those polynomialswith coefficients in the quotient field of D
that map every n-tuple of elements of D into D.) This is one of several
instances where properties of the ring of polynomials over a field, which
fail for D[x_1,...,x_n] even for "nice" domains D, can and do hold for
Int(D^n).

Fri 9:50–10:10b
"Strong test ideals"

Let $R$ be a characteristic $p$ ring. A strong test ideal is an ideal $\tau$ such that $\tau I^* =\tau I$ for every ideal $I$ of $R$.  We show that if
$R$ is a complete local ring, the test ideal is a strong test ideal.  As an
application, we obtain a large class of tightly closed ideals.  We also
discuss a class of ideals $J$ in Gorenstein rings for which $J^*=J:\tau$,
where $\tau$ is the test ideal.

Fri 10:30–10:50a
Gabriel Picavet, Universite' Blaise Pascal,
"Generalized going-down"

Recently, Kang and Oh proved that a chain of prime ideals in a ring can be
lifted up to a valuation domain. We examine two  overlapped questions. A ring
morphism $A \to B$ is called a chain morphism if arbitrary chains of prime
ideals of $A$ can be lifted through $A \to B$. We give criteria for a ring
morphism to be a chain morphism. As a main result, we have that universally
subtrusive morphism (a special class of submersive morphisms like injective
integral morphisms, pure morphisms or lying-over  universally going-down
morphisms) are universally chain morphisms. We say that a chain of prime
ideals $X$ is local if it has a greatest element $U(X)$. Then a ring morphism
$A \to B$ has generalized going-down if an arbitrary local chain $X = \{P\sb i\}\sb{i\in I}\subset Spec(A)$  where $U(X)$ is dominated by a prime ideal
$Q$ of $B$ is dominated by a local chain $Y = \{Q\sb i\}\sb{i\in I}\subset Spec(B)$ such that $U(Y) = Q$. We give criteria for a ring morphism to be
generalized going-down. Going-down rings are deeply involved. The main result
is that universally going-down ring morphisms have generalized going-down.

Fri 10:30–10:50b
Chin-Yi (Jean) Chan, University of Utah,
"Euler characteristics and Euler forms"

Let $M$  and $N$ be finitely generated modules over a Noetherian
Cohen-Macaulay ring  $A$ such that $M$ and $N$ both have finite  projective
dimension, $\dim M + \dim N \leq \dim A$ and $M \otimes N$ has finite length.
We study the relation between Serre's intersection multiplicity  $\chi(M, N)$ and the Euler form $\xi(M,N)= \sum (-1)^i \textrm{length} \textrm{Ext}^i(M,N),$ which has been applied in the development of
noncommutative intersection theory. Over commutative rings, we study the
extent to which the relation $\chi(M,N) = (-1)^{\dim A - \dim M} \xi(M,N)$  will hold. We will introduce Roberts rings defined by K. Kurano. The
above relation is highly related to the vanishing theorem. The relation in
between them will also be discussed.

Fri 11:00–11:20a
Bruce Olberding, Northeast Louisiana University,
"Krull-Schmidt for ideals and modules over domains"

(Joint work with Pat Goeters)  A Noetherian domain has the torsion-free
Krull-Schmidt property (TFKS) if decompositions of finitely
generated torsion-free modules into indecomposables are unique up to
isomorphism.  A weaker property, UDI, asserts only uniqueness of
decompositions into summands isomorphic to ideals.  L. Levy and C.
Odenthal, in 1996, characterized the one-dimensional domains satisfying
TFKS.  Recently, we characterized the domains satisfying UDI.  As a local
phenomenon, neither TFKS nor UDI are unusually strong; globally, they force
the Picard group to be trivial.  Since this strongly limits geometric
interpretations of UDI and TFKS, we introduce weak versions of UDI and TFKS
that require only that the Picard group be torsion.  Using a result of R.
Wiegand on Picard groups of arithmetical curves, one can then give examples
of singular curves satsifying weak TFKS and weak UDI.  Similarly, work of
R. and S. Wiegand allows one to exhibit a number of examples of subrings of
algebraic number fields satisfying the weak and strong versions of TFKS and
UDI.  Finally, we sketch an approach to UDI and TFKS that proceeds from
general principles rather than Noetherian assumptions.

Fri 11:00–11:20b
Ruth Michler, University of North Texas,
"Groebner bases for symmetric quotients"

By using so-called orbit notation (which will be defined in the talk) it
will be possible to explicitly compute Groebner bases for ideals that are
invariant under the action of S_n.  Explicit examples motivated by the study
of isolated singularities will be given.

Sat 11:00–11:20a
Tom Lucas, University of North Carolina-Charlotte,
"Degree of sharpness for a Pruefer domain"

For a prime  ideal $P$ of a Pr\"ufer domain $R$, say that $P$ is a
$\#$-prime (read as \lq\lq sharp prime") if there is a finitely generated
ideal which is contained in $P$ and in no  maximal ideal that does not
contain $P$.  Consider the ring $R_1= \bigcap \{R_{M_\alpha} \,|\, M_\alpha\in \flat Max(R)\}$ where $\flat Max(R)$ is the set of maximal
ideals of $R$ which are not $\#$-primes.    Each maximal ideal in
$Max(R)\setminus \flat Max(R)$  blows up in $R_1$, and each maximal ideal in
$\flat Max(R)$ extends to a maximal ideal of $R_1$. If $R\ne R_1$, then it
may be that some maximal ideals in $\flat Max(R)$ extend to $\#$-prime
ideals in $R_1$.    Recursively form the ring $R_n$ by intersecting over the
localizations at the maximal ideals of $R_{n-1}$ which are in $\flat Max(R_{n-1})$.  We say that $R=R_0$ has $\#$-degree $n$ if $R_n\ne R_{n-1}$
and each maximal ideal of $R_n$ is a $\#$-prime (of $R_n$).     A simple
algorithm  will be given to show how to construct almost Dedekind domains of
any given finite $\#$-degree. Another simple algorithm will be given to show
how to construct an almost Dedekind domain with no $\#$-primes. Such a domain
is declared to be a {\it dull domain} (\lq\lq dull" because the word \lq\lq
flat" seems to have been taken).  By blending the two algorithms we can
produce an almost Dedekind domain $R$ for which  $R$ through $R_{n-1}$ have
$\#$-primes but $R_n$ is a dull domain.

Sat 11:00–11:20b
Sara Faridi, University of Michigan,
"The multiple closure of a set of ideals"

We introduce the multiple closure of a set of ideals. We show that under
mild conditions on the ring, multiple closure agrees with tight integral
closure. This translates the tight integral closure of a set of ideals in a
ring into the tight closure of one ideal in a larger ring.  Using this
observation, we settle several questions on the properties of
tight integral closure.

Sat 11:30–11:50b
Sean Sather-Wagstaff, University of Utah,
"A dimension inequality for Cohen-Macaulay modules"

Motivated by the work of Gabber, Kurano and Roberts on Serre'spositivity
conjecture, we consider the following conjecture.
Conjecture:  Assume that $(R,m)$ is a quasi-unmixed local ring with
prime ideals $p$ and $q$ such that $p+q$ is $m$-primary.  Assume further
that$M$ is a Cohen-Macaulay $R$-module such that $p$ and $q$ are in the
support of $M$ and $e(M_p)=e(M)$.  Then $\dim(R/p) + \dim(R/q) \leq \dim(M)$.
I will give a brief history of the genesis of this conjecture and a
sketch of the verification in the case where $R$ is excellent and $R/\ann(M)$
contains a field.

Sat 4:30–4:50
Mark Johnson, University of Arkansas,
"Equidimensional symmetric algebras and residual intersections"

For a finitely generated module M, over a universally catenary local ring,
whose symmetric algebra is equidimensional, the ideals generated by the rows
of a minimal presentation n x m matrix are shown to have height at most n -
rank(M).  Moreover, in the extremal case, they are Cohen-Macaulay ideals if
the symmetric algebra is Cohen-Macaulay.  Some applications are given to
residual intersections of ideals.

Sat 5:00–5:20
Susan Morey, Southwest Texas State University,
"Primes associated to powers of ideals of graphs"

If R is a Noetherian ring and I is an ideal of R, then Brodmann showed that
the sets of associated primes Ass(R/I^n) stabilize for n large. In general it
is not known where (for which n) the sets stabilize or which primes are in
the stable set.  Suppose R is a polynomial ring and I is the edge ideal of a
graph G. If G is bipartite, then it has been shown by Simis, Vasconcelos and
Villarreal that Ass(R/I^n)=Min(R/I) for all n. So assume G contains at least
one odd cycle.  We will give a method of constructing the primes that are in
the stable set of I and we will give an upper bound for where the stable set
occurs.

Sat 5:30–5:50
Evan Houston, University of North Carolina-Charlotte,
"On integral domains whose overrings are Kaplansky ideal transforms"

Let $R$ be a domain with quotient field $K$.  For an ideal $I$ of $R$, call
the set $\varOmega_{R}(I) =\{x\in K \mid \text{ for each } a \in I \text{ there is an integer } n \geq 1 \text{ such that } xa^n \in R\}$ the
\emph{Kaplansky ideal transform of $I$}.  For finitely generated ideals, this
agrees with the familiar Nagata transform.  We attempt to classify those
domains each of
whose overrings is a Kaplansky transform; we dub such rings
$\varOmega$-domains.  Among other things, we prove:
\textbf{Theorem.} A Pr\"ufer domain $R$ is a $\varOmega$-domain if
and only if each nonzero prime ideal $P$ of $R$ for which $\{Q \in \text{Spec}(R) \mid Q \subseteq P\}$ is not open in $\text{Spec}(R)$ is the
union of the prime ideals properly contained in $P$.