Commutative Algebra and Algebraic Geometry
Lawrenceville, New Jersey
April 1718, 2004
List of Talks
Warning: If you are using a browser on a Unix machine,
you may have problems in visualizing the math characters in
the abstracts below.

Homological invariants of modules over local homomorphisms
by Luchezar AVRAMOV, University of Nebraska at Lincoln
Abstract:
Numerical invariants, such as Betti numbers and Bass numbers, will be defined
for a finitely generated module N over a local ring S,
in the the presence of a local homomorphism j:
R ® S.
Properties and applications of these invariants will be discussed.
This is joint work with Srikanth Iyengar and Claudia Miller.

The integral closure of monomial rings
by Joseph BRENNAN, North Dakota State University of
Abstract:
This talk will examine the question of computation of the integral closure of
subrings of polynomial rings generated by monomials. Particular attention will
be focused on the question of normality of Rees rings of ideals generated by
monomials.

BuchsbaumRim multiplicity formulas
by C.Y. Jean CHAN, University of Arkansas
Abstract:
Over a commutative Noetherian local ring A,
the BuchsbaumRim multiplicity e_{BR}(M),
defined for a submodule M of a free Amodule F of rank
r such that F/M has finite length, is a generalization of
the HilbertSamuel multiplicity of an mprimary ideal.
In a special case where r=2, there exist two mprimary ideals
I and J such that F/M @
I/J.
We have the following multiplicity formula for e_{BR}(M)
e_{BR}(M) =
e(J)  e(I) + e(Fitt_{0}(I/J)) 
e(Fitt_{0}(I/J')),
where J' is a minimal reduction of J and the above expression is
independent from the choices of minimal reductions. This also improves the
result of E. Jones in Computation of BuchsbaumRim multiplicities in
JPAA, vol. 162/1 (2001). We will discuss the extension of the above
multiplicity formula for modules of an arbitrary rank over a two dimensional
regular local ring. In such a case, e_{BR}(M)
can be expressed in terms of
the HilbertSamuel multiplicities of ideals in the same linkage class with the
Fitting ideals of F/M.
This leads a possibility of relating the BuchsbaumRim
multiplicity with the Chern classes of certain types of matrices.
This is joint work with JungChen Liu and Bernd Ulrich.

Maximal rank conjectures and algorithms for infinitesimal
neighborhoods
by Karen CHANDLER, University of Notre Dame
Abstract:
Alexander and Hirschowitz have shown, asymptotically, an upper bound of
the degree in which every general collection of infinitesimal neighbourhoods
of points of given order in a projective space displays maximal rank
with respect to ideal cohomology.
Namely, for each dimension n and multiplicity m there exists
d(n,m) so that the Hilbert function of such a scheme in each degree
d ³ d(n,m) has the expected value.
One would like to obtain concrete
values (or at least bounds) for the function d(n,m).
According to the SegreHartshorneHirshowitz, we should have d(2,m)=3m.
This has been shown to hold for each d £
20 by Ciliberto, Miranda, and Orecchia.
Indeed, we have conjectured that in general d(n,m) £ 3m. We shall describe algorithmic
methods on verifying this bound, as we have used, for example,
in the case of each d(n,3).

bgeneric modules
by Hara CHARALAMBOUS, SUNY at Albany
Abstract:
We will introduce and discuss the notion of bgeneric ideals and modules and
their free resolutions.

Asymptotic behavior of length of local cohomology
by Dale CUTKOSKY, University of Missouri at Columbia
Abstract:
Let k be a field of characteristic 0,
R=k[x_{1},..., x_{d}]
be a polynomial ring, and m its maximal
homogeneous ideal. Let I Ì R be a
homogeneous ideal in R. We show that
lim_{n ® ¥}
length(H^{0}_{m}(R/I^{n}))/n^{d} =
lim_{n ® ¥}
length(Ext^{d}_{R}(R/I^{n},R(d)))/n^{d}
always exists. This limit has been
shown to be e(I)/d! for mprimary ideals I
in a local Cohen Macaulay ring, where e(I) denotes the multiplicity
of I. But we find that this limit may not be rational in general.
We give an example for which the
limit is an irrational number thereby showing that the lengths of these
extention modules may not have polynomial growth.
This is joint work with Tai Ha, Hema Srinivasan, and Emanoil Theodorescu.

The tracking number of an algebra
by Kia DALILI, Rutgers University
Abstract:
We introduce the technique of tracking numbers of graded algebras and modules.
It is a modified version of the first Chern class of its free resolution
relative to any of its standard Noether normalizations. Several estimations are
obtained which are used to bound the length of chains of algebras occurring in
the construction of the integral closure of a graded domain. Noteworthy is a
quadratic bound on the multiplicity for all chains of algebras that satisfy the
condition S_{2} of Serre.
This is joint work with Wolmer Vasconcelos.

A tight closure analogue of analytic spread
by Neil EPSTEIN, University of Kansas
Abstract:
Let R be an excellent commutative Noetherian local ring of positive
characteristic. Inspired by the classical work of Northcott and Rees (1954) on
minimal reductions and analytic spread, we first note that their proof of the
existence of minimal reductions provides most of a proof that minimal
*reductions exist. (A *reduction of an ideal I
is an ideal J such that J Í I
Í J^{*},
where the * indicates tight closure.) The
missing ingredient is a `Nakayama lemma for tight closure,' a useful new tool.
Then, using an `exchange lemma' style of proof, we show that all minimal
*reductions of an ideal I have the same minimal number
of generators, at least in the case where R is analytically irreducible
and its normalization has perfect residue field.
We call this common number the *spread of the
ideal, denoted ell^{*}(I), by analogy with analytic spread.

Monomial ideals via squarefree monomial ideals
by Sara FARIDI, Université du Québec à
Montréal
Abstract:
Polarization is an operation that transforms a monomial ideal
in a polynomial
ring into a squarefree monomial ideal in an extension of the original ring.
This talk focuses on what algebraic and combinatorial properties are preserved
under polarization. Using this operation, we extend combinatorial structures
defined on squarefree monomial ideals to all monomial ideals, and explore
possible ways to deduce properties of monomial ideals and their Rees rings
using this technique.

Baxtertype operators and shuffletype products
by Li GUO, Rutgers University, Newark
Abstract:
A Baxter operator is a linear operator P on an algebra A that satisfies the
relation P(x)P(y)=P(xP(y))+P(P(x)y)+t P(xy) for all x and y in A. Here t is
a constant. When t=0, we get the integration by parts formula.
The study
of Baxter operators originated in the work of G. Baxter in 1960 on fluctuation
theory, and the algebraic study of Baxter operators was started by Rota in
the 1970s. The study has experienced a renaissance in the last couple of years,
with applications in Hopf algebras, multiple theta values, umbral calculus,
dendriform algebras and quantum field theory. Much of these are related to
the shuffle like structure of the free Baxter algebras.
We will present the
basic algebraic theory of Baxter operators and several related operators and
algebras, with emphasis on the free algebras. We will also discuss some
applications mention above. This is joint work with Kurusch
EbrahimiFard, from IHES.

The general plane section of a curve in P^{3}
by Elisa GORLA, University of Notre Dame
Abstract:
We will discuss some necessary and sufficient conditions for a space curve to
be arithmetically CohenMacaulay, in terms of its general hyperplane section.
We obtain a characterization of the degree matrices that can occur for points
in the plane that are the general plane section of a non arithmetically Cohen
Macaulay curve of P^{3}.
We will see that almost all the degree matrices with
positive subdiagonal that occur for the general plane section of a non
arithmetically CohenMacaulay curve of P^{3},
arise also as degree matrices of
some smooth, integral, non arithmetically CohenMacaulay curve, and we will
describe the exceptions.

Normalization of modules
by Jooyoun HONG, Purdue University
Abstract:
Let R be a Noetherian normal domain and E a finitely generated torsionfree
Rmodule having a rank e. The Rees algebra of an Rmodule E is
defined as a subalgebra of the polynomial ring R[T_{1}, ...,
T_{e}] generated by all linear form a_{1}T_{1} + ...
+ a_{e}T_{e} in E. Our general goal is to
study the normalization of the Rees algebra of a module. First we deal with a
submodule E of free module R^{e} with finite colength.
In this case, we formulate relationships among the coefficients of
BuchsbaumRim polynomials of E and of the integral closure \overline{E}
by using the BriançonSkoda number of E. With this result we can measure
the length of the algorithmic procedures used to compute the integral
closure of the the Rees algebra of E. Next, we discuss bounds for degrees
of the generators of CohenMacaulay Rees algebra which is finite over
the Rees algebra of E. This is joint work with Bernd Ulrich and
Wolmer V. Vasconcelos.

Toralgebra operations and linkage
by Andrew KUSTIN, University of South Carolina
Abstract:
Let I be a perfect ideal in the local ring (R,k).
We conjecture that there exists a collection of operations
{h_{i}} on the Toralgebra
Tor_{·}^{R}(R/I,k),
such that I is in the linkage class of a complete intersection
if and only if
å im h_{i}= Tor_{2}.
Our best results are obtained when I is a grade four Gorenstein ideal.
This is joint work with Andreea N. Brezaie, Carrie E. Finch, Sara M.
Gabrielli, Derek J. Owens, Frank A. Sanacory, William H. Streyer,
Adela Vraciu, and Brooks D. Willet.

Pencils of forms and level Artinian algebras
by Anthony IARROBINO, Northeastern University
Abstract:
Consider a type two graded level algebra A, quotient of the polynomial
ring R=K[x_{1}, ..., x_{r}] in r variables,
defined by an inverse system R °
<
F,G>, F,G in D_{j}
where D = K_{DP}[X_{1}, ....,X_{r}] denotes the
ring of divided powers, upon which R acts by contraction.
We have A=R/Ann(F,G). For each
l in K, F_{l}
= F+ lG determines a socledegree
j Gorenstein algebra quotient of A, namely A_{l} = R/Ann(F_{l}). The family F_{l} is the pencil of
degreej forms in D determined by the twodimensional
vector space <
F,G>. We give lower bounds for the
Hilbert function H(A_{l}),
when l is generic, in
terms of the Hilbert functions H(R/Ann(F)) and H(R/Ann(G))
or in terms of the Hilbert function H(A). We give several applications
showing the impossibility of some candidate sequences to be Hilbert
functions for type two level Artinian algebras.
We also give an example of a compressed type two
level Artinian algebra not having a compressed Gorenstein
quotient of the same socle degree.

On the first infinitesimal neighborhood of a kconfiguration
by Juan MIGLIORE, University of Notre Dame
Abstract:
It is an open problem to classify the possible Hilbert functions of the first
infinitesimal neighborhood of a finite set of points in P^{2}
(fat points).
This is in contrast to the case of reduced points, where the Hilbert functions
are precisely the differentiable Osequences. All such Hilbert functions arise
as the Hilbert function of a set of points in a kconfiguration. Hence we study
sets Z of fat points supported on a kconfiguration X in P^{2}.
In certain cases, the minimal free resolution of I_{Z}
is uniquely determined by the type
of the kconfiguration (and not on the actual positioning of the points). In
other cases the minimal free resolution is not uniquely determined, but the
Hilbert function is. And in yet other cases not even the Hilbert function is
uniquely determined. The trick is to determine when Z can be realized as the
result of a sequence of basic double links. In the other direction: given any
differentiable Osequence H, we can produce a reduced set of points X with
Hilbert function H, such that we can explicitly give the minimal free
resolution (and hence the Hilbert function) of the set of fat points Z
supported on X. We also give an example of two sets of fat points with the same
Hilbert function, but whose supports have different Hilbert functions.
This joint work with Anthony V. Geramita and Lousindi R. Sabourin.

A normal form for space curves in a double plane
by Uwe NAGEL, University of Kentucky
Abstract:
We discuss space curves that are contained in some double plane, i.e. in a
quadric defined by the square of a linear form. Hartshorne and Schlesinger have
studied such curves from a geometric point of view. Our goal is to explicitly
relate the geometric and algebraic properties of these curves. We show in
particular that the minimal generators of the homogeneous ideal of such a curve
can be written in a very specific form. As applications we describe the minimal
free resolution and characterize the possible HartshorneRao modules of curves
in a double plane as well as the minimal curves in their even Liaison classes.
This is joint work with Nadia Chiarli and Silvio Greco.

Applications of the purity of nonstandard Frobenius
by Hans SCHOUTENS, College of Technology, CUNY
Abstract:
Originating in the work of HochsterRoberts on the CohenMacaulyness of rings
of invariants, and of MehtaRamanathan on vanishing theorems for Schubert
varieties, the purity of the Frobenius has proven to be a powerful tool in
singularity theory in positive characteristic. For instance, Smith, Hara et
al., have given a characterization of logterminal singularities in terms of
the purity of Frobenius. Smith also uses it to obtain KawamataViehweg
vanishing in positive characteristic. To make use of this tool in zero
characteristic, one uses reduction mod p.
However, in this reduction process,
some information gets lost (especially when dealing with quotients), so that
the results in zero characteristic remained unsatisfactory. This led Smith to
conjecture that any GIT quotient Y
of a Fano variety admits KawamataViehweg vanishing: for a numerically
effective line bundle L on Y, its higher sheaf
cohomology is zero, and, if L is moreover big, then
H^{i}(Y,L^{1}) vanishes
for i < dim Y.
I will indicate a proof of this conjecture making direct use of
a characteristic zero version of Frobenius. This is achieved by constructing a
faithfully flat cover Y_{¥}
® Y where
Y_{¥} is obtained from an
ultraproduct of projective varieties in positive characteristic.

Asymptotic limits of local cohomology modules
by Hema SRINIVASAN, University of Missouri at Columbia
Abstract:
For a standard graded ring R of dimension d and a homogenous ideal I of R, we
investigage the asymptotic growth of the length of Ext^{d}
(R/I^{n}, R) with
respect to n. We show that in characteristic zero, when R is a normal domain of
dimension d>1 and I is a proper homogenous ideal then this grows
asymptotically as cn^{d} for some constant c.
This number c may not always be
rational and hence the lengths of H^{0}_{m}(R/I^{n})
may not be a polynomial in n in general. This is joint work with
Dale Cutkosky, Tai Ha, and Emanoil Theodorescu.

On Extensions of modules
by Janet STRIULI, University of Kansas
Abstract:
In this talk we study closely Yoneda's correspondence between short exact
sequences and the Ext^{1} group. We prove a main theorem which gives
conditions on the splitting of a short exact sequence after taking the tensor
product with R/I, for any ideal I of R.
As an application we prove a generalization of Miyata's Theorem.
We introduce the notion of sparse module and we show that
Ext_{R}^{1}(M,N)
is a sparse module provided that there are
finitely many isomorphism classes of maximal CohenMacaulay modules having
multiplicity the sum of the multiplicities of M and N.
We prove that sparse modules are Artinian.

Almost Gorenstein rings
by Adela VRACIU, University of South Carolina
Abstract:
There are (at least) three equivalent characterizations for the Gorenstein
property of an Artinian ring R:
 for every ideal I, 0:(0:I)=I
 the canonical module w is isomorphic
to R
 the ideal (0) is irreducible.
Each of these properties can be generalized in a natural way,
giving rise to classes of rings that can be considered close to being
Gorenstein. Concrete instances where these rings arise naturally are as
specializations of rings of countable CM type modulo parameter ideals, or as
quotients of Gorenstein Artinian rings by their socle. We give a variety of
examples of such rings, and we explore the question of whether our
generalizations of the three properties listed above are equivalent.
This joint work with Craig Huneke.
