All talks take place in Avery Hall

Note to speakers: The rooms have Windows XP machines with both Powerpoint

and Acrobat Reader installed. Probably the most convient way to bring your

talk electronically, is to have it on a USB flash drive, as all the computers

have a USB port for exactly this. In addition to the computers in the room,

there is also a VGA plug to connect laptops to the projection equipment in

the room. (If you have a Mac laptop, you will need to have one with a VGA out

or a connector that allows a VGA connection.) It is strongly recommended that

you have a copy of you talk on transparencies, just in case.

Ian Aberbach

Title: Some cases where the F-signature exists

Abstract: Let (R,m,k) be a local ring of positive prime characteristic

p and dimension d. For simplicity, assume that k is perfect. Then R

is known to be strongly F-regular if and only if lim inf_q a_q/q^d > 0,

where q = p^e is a varying power of p, and a_q is the number of

R-free summands of R^{1/q}. The F-signature of R is the limit of

a_q/q^d, if this limit exists. We give a new case where the limit

exists (the dimension of the non Q-Gorenstein locus is at most one).

Interestingly, this is the only case I am aware of where one can show

that the F-signature exists, but the method does not suffice to show

that weak and strong F-regularity coincide.

Back to Top

---

Baoping Jia

Title: Recent Progress in Valuation Theory

Abstract:

Back to Top

---

Joseph Brennan

Title: Apolarity and Partitions of Unity

Abstract: Apolar covariants of a binary n-tic of degree

greater than or equal to n are expressable as sums of powers of

the roots of the base n-tic. This talk explores the character of

the coeffiecents of such an expansion as functions in the roots of

the base n-tic.

Back to Top

---

C-Y. Jean Chan

Title: Chern Classes on the Product of Projective Spaces

Abstract: As an application of the Hirzebruch-Riemann-Roch theorem,

there exists a correspondence between the Chern polynomials and the Hilbert

polynomials of coherent sheaves on the projective space $\mathbb P^d$

over an algebraically closed field. This can be extended to a

product of two such spaces, $\mathbb P^{d_1} \times \mathbb P^{d_2}$.

In this discussion, each coherent sheaf is associated with a bigraded

module and the Hilbert polynomials under consideration are in two

variables.

The purpose of this work is to provide the realization of the

Riemann-Roch theorem from an algebraic point of view.

Back to Top

---

Lars Winther Christensen

Title: New formulas of the Auslander-Buchsbaum type

Abstract: Let M, N, and P be modules over a commutative

noetherian local ring R. New criteria for invertibility of natural

homomorphisms like

Hom(M,N) \otimes P --> Hom(M,N \otimes P)

give a fast passage to formulas of the Auslander-Buchsbaum type.

The proof exploits the strong computational properties of the complex

Hom(K^R,E), where K^R is a Koszul complex on a set of generators for

the maximal ideal of R, and E is the injective hull of the maximal

ideal.

This is joint work with Henrik Holm.

Back to Top

---

Anthony Crachiola

Title: Some cancellation results for affine domains

Abstract: Let $A$ and $B$ be integral domains which are finitely

generated over an algebraically closed field $k$ such that

$A[x_1,...,x_n] \cong B[x_1,...,x_n]$. If $A$ and $B$ have

transcendence degree 1 over $k$, then $A \cong B$. This is a

well known result from a 1972 paper by Shreeram Abhyankar,

Paul Eakin, and William Heinzer. I will discuss recent

efforts to study cancellation problems of this type by

algebraic means, in particular using actions of the additive

group $(k,+)$. These efforts include a new proof of the

Abhaynkar-Eakin-Heinzer theorem.

Back to Top

---

Steven Dale Cutkosky

Title: Toroidalization of Morphisms

Abstract: A morphism of nonsingular varieties is locally monomial if

it is locally formally isomorphic to a morphism of toric varieties. The

morphism is toroidal if the isomorphism respects fixed simple normal

crossings divisors on X and Y.

The toroidalization conjecture is that every dominant morphism

$f:X\rightarow Y$ of varieties over a field of characteristic zero

can be made into a toroidal morphism by sequences of blowups of

nonsingular subvarieties over $X$ and $Y$.

We give examples, and discuss the cases where the conjecture is true,

including our recent proof of the toroidalization conjecture for morphisms

of 3-folds.

Back to Top

---

Jesse Elliott

Title: Universal Properties of Integer-Valued Polynomial Rings

Abstract: Let $D$ be an infinite integral domain. We say that a

domain $A$ containing $D$ is a {\it polynomially dense extension of $D$}

if for all $f(X) \in K[X]$ with $f(D) \subset A$ one has $f(A) \subset A$,

where $K$ is the quotient field of $A$. For example, a polynomially dense

$\ZZ$-algebra is a {\it binomial domain}, that is, a domain $A$ of

characteristic zero such that the element $a(a-1)(a-2)\cdots(a-n+1)/n!$ of

$A \otimes_\ZZ \QQ$ lies in $A$ for every $a \in A$ and every positive

integer $n$. The ring $\Int(D^\XX)$ of integer-valued polynomials on

$D^\XX$ is the free polynomially dense extension of $D$ on the set $\XX$.

We say that a $D$-algebra $A$ is a {\it polynomially dense $D$-algebra} if

$A$ is a homomorphic image of a polynomially dense extension of $D$.

A $D$-algebra $A$ is a polynomially dense $D$-algebra if and only if it

is a homomorphic image of $\Int(D^\XX)$ for some set $\XX$. We study

various characterizations of polynomially dense $D$-algebras, with

particular attention to the case where $D$ is a Dedekind domain.

Back to Top

---

Alberto Facchini

Title: Local homomorphisms in noncommutative rings

Abstract: Let $R$ and $S$ be rings, not necessarily commutative.

A ring morphism $\varphi \colon R \to S$ is said to be {\em local} if,

for every $r\in R$, $r$ is invertible in $R$ whenever $\varphi (r)$ is

invertible in $S$. For instance, if $R$ is a ring and $I$ is a two-sided

ideal of $R$ contained in the Jacobson radical of $R$, the canonical

projection $R \to R/I$ is a local morphism. Conversely, the kernel of

every local morphism $R \to S$ is contained in the Jacobson radical

$J(R)$ of $R$.

In Algebraic Geometry and Commutative Algebra, local morphisms are

defined as the ring morphisms $\varphi\colon R\to S$, between local

commutative rings $(R,\mathcal{M})$ and $(S,\mathcal{N})$, for which

$\varphi(\mathcal{M})\subseteq \mathcal{N}$. This definition coincides

with ours in the case of $R$ and $S$ local.

In this spirit, Cohn considered local morphisms $R \to S$ when $R$, $S$

are not necessarily commutative and $S$ is a division ring. It is easily

seen that if a ring $R$ has a local morphism into a division ring, then

$R$ is a local ring.

Recall that a ring $R$ is called \emph{semilocal} if $R/J(R)$ is a

semisimple artinian ring. Under weak finiteness assumptions on an object

$A$ of a Grothendieck category $\Cal C$, the endomorphism ring

$\End_{\Cal C}(A)$ of $A$ is semilocal. We prove that these rings

$\End_{\Cal C}(A)$ are semilocal making use of suitable ring homomorphisms

which we show to be local morphisms. Most of the result obtained were

obtained with Dolors Herbera.

Back to Top

---

Robert Guralnick

Title: Mappings from the generic Riemann surface of genus g

Abstract: Let f: X --> Y be a rational map from the generic

Riemann surface of genus g of degree n (this notion can be

made precise). Zariski observed that Y must be the Riemann

sphere and the critical case is when f is indecomposable. He showed

that the monodromy group of such a cover could not be solvable

for g > 6 (for g < 7, any curve of genus g has a solvable map to

the Riemann sphere). His methods used some elementary group

theory. We will discuss extensions of this result using more serious

group theory.

Back to Top

---

Wolfgang Hassler

Title: Large indecomposable modules over non-Cohen-Macaulay rings

Abstract: This is joint work with R. Karr, L. Klingler and R. Wiegand.

A local ring $(R,m)$ is called {\em Dedekind-like} provided $R$ is

one-dimensional and reduced, the integral closure $\bar R$ of $R$ generated

by at most $2$ elements as an $R$-module, and $m$ is the Jacobson radical of

$\bar R$. Let $(R,m,k)$ be Dedekind-like satisfying the following additional

condition: If ${\bar R}/m$ is a field, then it is a separable extension of

$k$. Only recently, L. Klingler and L. Levy have classified all finitely

generated indecomposable $R$-modules up to isomorphism. It follows from

their classification theorem that the multiplicities of all indecomposable

finitely generated $R$-modules are bounded by $4$.

Suppose now that $(R,m)$ is a one-dimensional local ring which is not

homomorphic image of a Dedekind-like ring. Then we prove that there is no

bound on the multiplicities of indecomposable finitely generated

$R$-modules.

In his talk, Lee Klingler will present our results assuming that $R$ is

Cohen-Macaulay, whereas I will deal with non-Cohen-Macaulay rings in my

talk.

Back to Top

---

Futoshi Hayasaka

Title: Modules of reduction number one

Abstract: I will talk about some recent results on the socle

modules of parameter modules, including a result on modules associated

to a certain good matrix. Let (A,m) be a Noetherian local ring and let

N be a parameter module in F=A^r. Let M=N:_F m be the socle module of N.

In this talk, I prove that the socle module M=N:_F m of a parameter module

N has a reduction number at most one and hence its Rees algebra R(M) is

Cohen-Macaulay, if the base ring A is Cohen-Macaulay of dimension two and

the rank of N is greater than or equal to two. This result gives numerous

examples of Cohen-Macaulay Rees algebras of modules, which are not

integrally closed.

Back to Top

---

William Heinzer

Title: Generic fiber rings of mixed polynomial/power series rings

Abstract: Joint with Christel Rotthaus and Sylvia Wiegand.

Let K be a field, m and n positive integers and X and Y

sets of m and n independent variables over K. If A is the

polynomial ring K[X] localized at (X), every prime ideal of K[[X]]

maximal with respect to intersecting A in (0) is of height m-2. If

B is K[[X]][Y] localized at (X, Y), and C is K[Y][[X]] localized

at (X, Y), then every prime ideal of K[[X, Y]] maximal with respect

to intersecting either B or C in (0) is of height m+n-2. Each

prime ideal of K[[X, Y]] that is maximal with respect to intersecting

K[[X]] in (0) is of height either n or m+n-2.

Back to Top

---

Kurt Herzinger

Title: Bricks and Perfect Bricks in Numerical Semigroups

Abstract: Let S be a numerical semigroup, I be a non-principal

relative ideal of S, and S - I the dual of I in S. Let \mu(I) represent

the size of the minimal generating set for I. We refer to the pair (S , I)

as a brick provided \mu(I)\mu(S - I) = \mu(I + (S - I)). We will survey

the past research that has been done on bricks and look at recent developments

related to this topic. We will also examine the role that bricks play in the

study of torsion in tensor products.

Back to Top

---

Craig Huneke

Title: A commutative history of Hom_R(M,M)

Back to Top

---

Daniel Katz

Title: Asymptotic sequences revisited

Abstract: Asymptotic sequences over an ideal were introduced by

Rees in order to improve an inequality of Burch concerning the

analytic spread of an ideal. Subsequently, they were studied in

their own right by Ratliff, McAdam and myself. In this talk I

will recount some of the history of asymptotic sequences and report

on recent work with Glenn Rice in which we introduce asymptotic

sequences over a module and use them to obtain an estimate for

the analytic spread of a module.

Back to Top

---

Lee Klingler

Title: Large Indecomposable Modules over Local Cohen-Macaulay Rings

Abstract: We determine which local Cohen-Macaulay rings have ``large''

indecomposable finitely generated modules. In earlier joint work of

mine with L. Levy, we showed that indecomposable finitely generated

modules over local Dedekind-like rings have torsion-free rank at most

two. In current joint work with W. Hassler, R. Karr, and R. Wiegand,

we show that any one-dimensional local Cohen-Macaulay ring which is not

a homomorphic image of a local Dedekind-like ring has indecomposable

finitely generated modules of arbitrary (and hence arbitrarily large)

torsion-free rank. For non-Cohen-Macaulay rings and rings of higher

dimension, multiplicity (rather than torsion-free rank) is an

appropriate measure of size; we show that any local ring which is not a

homomorphic image of a Dedekind-like ring has indecomposable finitely

generated modules of arbitrarily large multiplicity.

Back to Top

---

Larry Levy

Title: Direct-sum behavior of modules over Dedekind-like rings

Abstract: The structure and direct-sum behavior of finitely

generated modules over Dedekind-like rings --- a generalization of

Dedekind domains --- has recently been described in a memoir by Klingler

and Levy. (A preprint of this is available on my web page

www.math.wisc.edu/~levy)

One of the interesting features, in the case of Dedekind domains, is

that direct-sum behavior is completely determined by local data and a

group-theoretic invariant (the ideal class group); and a consequence of

this is that direct-sum cancellation holds.

For the more general Dedekind-like rings, direct-sum cancellation fails

quite often. Nevertheless, all direct-sum behavior is still controlled by

local data, and a group-theoretic invarant which we call the "web of class

groups". This is in marked contrast to situations encountered in K-theory,

where either the situation is "stable" and these two types of invariants

work, or else the situation is "unstable" and chaos reins.

Most of this talk will deal with this web of class groups.

Back to Top

---

Tom Lucas

Title: Classifying Prime Ideals

Abstract: Here are five types of primes in integral domains:

divisorial [P=(D:(D:P))], idempotent, branched [proper P-primary ideals

exist], sharp [D_P does not contain the intersection \cap D_M where the

M range over the maximal ideals that do not contain P], antesharp

[each maximal ideal of (P:P) that contains P, contracts to P]. One of

the main questions to be addressed is which 5-tuples L(P)=(V,W,X,Y,Z) can

occur where V=div./not div., W=idem./not idem., X=bran./unbran.,

Y=sharp/not sharp and Z=ashrp/notashrp. Using pullbacks and the Nagata

ring D(x), examples of divisiorial primes can be built from non-divisorial

ones without changing the other four components in L(P) when P is not

divisorial prime.

Back to Top

---

Naoyuki Matsuoka

Title: Ratliff-Rush closures of certain two-dimensional monomial ideals and Buchsbaumness of their Rees algebras

Abstract: Let $(A,m)$ be a regular local ring of dimension 2 and

let $I$ be an $m$-primary ideal in $A$. In my talk I will discuss the

Buchsbaumness of the Rees algebra $R(I)=\bigoplus_{n \geq 0}I^n$, the

associated graded ring $G(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$, and the

extended Rees algebra $R'(I)=\bigoplus_{n \in Z}I^n$ of $I$. Some effective

characterization for $R(I)$ to be a Buchsbaum ring shall be given in terms

of the ideal $I$ in the case where $\overline{I} = \widetilde{I}$.

Eventually I will show that $R(I)$ is a Buchsbaum ring if and only if so

is $G(I)$. Monomial ideals in the polynomial ring $A=k[x,y]$ over a field

$k$ are studied and numerous examples of ideals $I$ for which the Rees

algebras $R(I)$ are Buchsbaum shall be given among them.

Back to Top

---

Gabriel Picavet

Title: Some remarks about flat epimorphisms

Abstract: We show the ubiquity of flat epimorphisms in the theory

of schemes. This remark allows us to build for instance the integral

closure of a scheme as a scheme instead of as a quasi-coherent algebra. We

introduce standard flat epimorphisms. Flat epimorphisms can be

characterized with the help of standard flat epimorphisms and we give a

computable criterion for standard flat epimorphisms.

Back to Top

---

Tony Joseph Puthenpurakal

Title: Depth of Higher Associated graded modules

Abstract: Let $(A,\m)$ be a Noetherian local ring with $\depth A \geq 2$.

We give a necessary and sufficient condition for $\depth G_{\m^n}(A) \geq 2$

for all $n \gg 0$.

Back to Top

---

Christel Rotthaus

Title: Open loci in graded modules

Abstract: Let $A=\oplus_{i\in \Bbb N}A_i$ be an excellent homogeneous

Noetherian graded ring and let $M=\oplus_{n\in \Bbb Z}M_n$ be a finitely

generated graded $A$-module. We consider $M$ as a module over $A_0$ and

show that the $(S_k)$-loci of $M$ are open in $\text{Spec}(A_0)$. In

particular, the Cohen-Macaulay locus

$U^0_{CM}=\{\frak p\in \text{Spec}(A_0) \mid M_{\frak p}\, \text{is Cohen-Macaulay}\}$

is an open subset of $\text{Spec}(A_0)$. We also show that the $(S_k)$-loci

on the homogeneous parts $M_n$ of $M$ are eventually stable. As an

application we obtain that for a finitely generated Cohen-Macaulay module

$M$ over an excellent ring $A$ and for an ideal $I\subseteq A$ which is not

contained in any minimal prime of $M$ the $(S_k)$-loci for the modules

$M/I^nM$ are eventually stable. (This is joint work with Liana Sega.)

Back to Top

---

Hideto Sakurai

Title: Index of reducibility of parameter ideals and the Cohen-Macaulay types for modules possessing finite local cohomology modules

Abstract: This is a joint work with Shiro Goto.

Let $A$ be a Noetherian local ring with the maximal ideal $\frak{m}$,

and $M$ be a finitely generated $A$-module with $d=\dim M$.

This talk is aimed at exploring the index $\ell_A((QM:\frak{m})/QM)$

of reducibility of parameter ideals $Q$ for $M$

and the Cohen-Macaulay type of $M$.

In the most part of this talk we shall explore it

when $M$ has finite local cohomology modules (shortly FLC),

that is the $i$-th local cohomology module of $M$ with respect

to $\frak{m}$ is finitely generated for all integers $i \ne d$.

Firstly we shall determine the supremum of the index of

reducibility of standard parameter ideals for $M$ possessing FLC,

and denote it by $\roman{s}(M)$.

The supremum $\roman{s}(M)$ is expressed in terms of the lengths of

the socles of local cohomology modules of $M$.

Also, the problem of when the equality

$\ell((QM:\frak{m})/QM)=\roman{s}(M)$ holds true is explored.

Moreover we shall give that the invariant $\roman{s}(A)$ has

relationships to reduction numbers of certain ideals when $A$ has FLC.

Back to Top

---

Victoria Sapko

Title: Associated Graded Rings of Complete Intersection Numerical Semigroup Rings

Abstract: In this presentation we examine the associated graded

ring of $R=k[t^a,t^b,t^c]_m$, where $m$ is the homogeneous maximal ideal.

If the associated graded ring of $R$ is Buchsbaum, we show that the

associated graded ring of $R$ is Cohen-Macaulay in two cases: when $R$

is a complete intersection and when $a$ and $b$ are not relatively prime.

Back to Top

---

Sean Sather-Wagstaff

Title: Ascent and descent of semidualizing modules under completion

Abstract: Semidualizing modules provide interesting duality theories

over local rings, for instance, Grothendieck duality over a Cohen-Macaulay

ring. In this talk, we will discuss the behavior of semidualizing modules

under completion. It is known that this assignment C \mapsto \hat{C} is

injective, but it is not in general surjective. We will present an

example, based on a theorem of R. Heitmann and motivated by work of C.

Rotthaus, D. Weston, and R. Wiegand, demonstrating how badly the

surjectivity of this assignment can fail. Also, we will discuss criteria

that guarantee the surjectivity of this assignment. This is joint work

with L.W. Christensen.

Back to Top

---

Markus Schmidmeier

Title: Subgroups of Abelian Groups vs. Submodules of k[T]-Modules

Abstract: Let R be a commutative uniserial ring, for example

R = Z/(p^n) or R = k[T]/(T^n) where k = Z/(p) . We are interested in

all possible pairs (B; A) where B is a finitely generated R -module and A

a submodule of B.

It is known since Reinhold Baer that the lattice of submodules reflects

the structure of the underlying ring, in particular the addition in R can

be recovered from the lattice of submodules of the direct sum of three

copies of R.

In my talk however, I will discuss two results where the classification

of the pairs (B; A) is independent of the additive structure of the ring R.

This is the case if there are only finitely many indecomposable pairs

(B; A), up to isomorphism, or if the isomorphism types of B and A are kept

fixed.

Back to Top

---

Liana Sega

Title: Asymmetric vanishing of Ext over Gorenstein rings

Abstract: We construct a class of Gorenstein local rings $R$ which

admit minimal complete $R$-free resolutions $\bd C$ such that the sequence

$\{\rank_R C_i\}$ is constant for $i< 0$, and grows exponentially for all

$i>0$. Consequently, over these rings we show there exist finitely

generated $R$-modules $M$ and $N$ such that $\Ext^i_R(M,N)=0$ for all

$i> 0$, but $\Ext^i_R(N,M)\ne 0$ for all $i>0$.

Back to Top

---

Brent Strunk

Title: Hilbert Functions and Castelnuovo-Mumford Regularity

Abstract: Suppose G is a standard graded ring over an infinite field.

From the minimal graded free resolution of G, it is possible to derive

several invariants, among them the multiplicity, the Castelnuovo Mumford

regularity, the Hilbert series, and the postulation number. I discuss a

sharp lower bound for the regularity of G in terms of the postulation

number, depth, and dimension. I also present a class of examples in

dimension 1 where the postulation number is 0 and the regularity of G can

take on any value between 1 and the embedding codimension of G.

Back to Top

---

Jan Strooker

Title: Stiffness: what does it mean?

Abstract: Auslander-Buchweitz theory furnishes a link between certain

Homological Conjectures, notably Hochster's Canonical Element Conjecture,

and a linear algebra property which we call "stiffness". This is akin to

the useful Buchsbaum-Eisenbud criterion from their classical "What makes

a complex exact?". We skim over this connexion, developed over the years

with Anne-Marie Simon, focusing on intriguing open questions.

Back to Top

---

Irena Swanson

Title: Adjoints of ideals

Abstract: This is joint work with Reinhold Huebl

Adjoints of ideals are defined by taking intersection of ideals

in an infinite family. At least for each zero-dimensional ideal,

a finite subset suffices. This finite subset depends on the ideal,

however, it need not be known ahead of time. We prove several cases

when the finite subset is obtained from the Rees valuations of the ideal.

The motivation for our work came from the work of Ein, Lazarsfeld and

Smith,and Hochster and Huneke. They proved that in a regular ring R

containing a field, for any prime ideal P, $P^{n\, ht\, P} \subseteq P^n$

for all positive integers n. The proofs are based on the theory of

multiplier ideals and tight closure, respectively. The question remains

if the same type of result holds in an arbitrary regular ring. One way to

tackle this is via adjoints of ideals.

Back to Top

---

Ryo Takahashi

Title: Certain direct summands of syzygies of the residue field

Abstract: Let $R$ be a commutative Noetherian local ring with

residue field $k$. In this talk, we investigate direct summands

of the syzygies of $k$. We prove that $R$ is regular if and

only if some syzygy of $k$ has a semidualizing summand. After

that, we consider whether $R$ is Gorenstein if and only if some

syzygy of $k$ has a G-projective summand.

Back to Top

---

Diana White

Title: An Euler characteristic for modules of finite G-dimension

Abstract: Let R be a local ring and M a finitely generated R-module.

When M has finite projective dimension, the Euler characteristic is defined

as the alternating sum of the Betti numbers.

Based on work of Avramov and Martsinkovsky, we generalize the notion of the

Euler characteristic to a G-Euler characteristic, defined for modules of

finite G-dimension. We show which basics facts about the Euler characteristic

carry over to the G-Euler characteristic, and offer examples to demonstrate

differences. We also present some unexpected results that arise from this

invariant.

Back to Top

Back to the conference homepage