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Homological invariants of modules over contracting endomorphisms
Luchezar Avramov, University of Nebraska
Abstract:
If φ: R → R is a contracting endomorphism of a local
ring R, then, viewed through φ, every non-zero finite
R-module M displays the homological behavior of the
residue field of R. The special case when R has positive
characteristic, φ is the Frobenius endomorphism, and
M=R contains a broad generalization of Kunz's criterion
for regularity. The talk is based on joint work with Mel Hochster,
Srikanth Iyengar, and Yongwei Yao.
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The Lex-Plus-Power inequality for local cohomology modules
Giulio Caviglia, Purdue University
Abstract:
This is joint work with E. Sbarra.
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Cohomology and Betti numbers of graded modules
Marc Chardin, CNRS/Université Pierre et Marie Curie, France
Abstract:
In the first part of this lecture, I will present joint results with
Nicolas Botbol related to the extension to general grading of the notion
of Castelnuovo-Mumford regularity. These results are very much connected
to previous ones on this matter obtained by several authors.
In a second part, I will present results with Tai Ha and Amir Bagheri
on Betti numbers and local cohomology of powers of graded ideals.
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Generating sequences and semigroups of valuations
Dale Cutkosky, University of Missouri
Abstract:
We discuss the general question of when a semigroup is the semigroup
of a valuation dominating a noetherian local domain, giving
some surprising examples and a necessary condition. We give a
necessary and sufficient condition for the pair of a semigroup S
and a field extension L/k to be the semigroup and residue field
of a valuation dominating a regular local ring R of dimension
two with residue field k, generalizing the
theorem of Spivakovsky for the case when there is no residue field
extension. We show how these methods can be used to
detect the defect type of a field extension; the defect
occurring in a finite projection is an essential obstruction to
extending resolution of singularities to positive characteristic.
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Intersection multiplicity on blow-ups
Sankar Dutta, University of Illinois at Urbana-Champaign
Abstract:
We propose a conjecture on vanishing and non-negativity of intersection
multiplicity on the blow-up of a regular local ring at its closed point.
Proofs of vanishing, several special cases of non-negativity and a
necessary and sufficient condition for non-negativity will be presented
in this talk. The connection of this conjecture with Serre's conjecture
on intersection multiplicity and Peskin & Szpiro's work will
also be discussed.
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Asymptotic syzygies of algebraic varieties
Lawrence Ein, University of Illinois at Chicago
Abstract:
This is joint work with Rob Lazarsfeld. We discuss the shape of the
minimal resolution of a smooth projective variety, when it is embedded by
the complete linear system of sufficiently positive line bundles.
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Almost Gorenstein rings
Shiro Goto, Meiji University, Japan
Abstract: author's pdf file.
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Stable Ulrich bundles
Robin Hartshorne, University of California at Berkeley
Abstract:
(joint work with Marta Casanellas and with the assistance of
F.-O. Schreyer and F. Geiss) Many years ago Bernd Ulrich observed that the
number of generators of a Cohen-Macaulay module over a local ring is
bounded by the product of the rank of the module and the multiplicity of
the ring. He initiated the study of modules for which this maximum occurs,
now called Ulrich modules. The corresponding notion in algebraic geometry
is that of an Ulrich bundle on a variety in projective space, meaning that
the corresponding module over the local ring of the cone over that variety
is an Ulrich module. In this paper we prove the existence of stable (hence
indecomposable) Ulrich bundles of all ranks on cubic surfaces in
P3 and
on general cubic threefolds in P4. For bundles on the cubic
surface, this
requires a careful analysis of the possible curves on the surface that can
be the first Chern class of an Ulrich bundle; for bundles on the cubic
threefold, we need certain curves of degrees 5 and 12 whose existence (up
to now) is only known via computations on Macaulay2 provided by Schreyer
and Geiss.
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Ideals generated by quadratic polynomials
Melvin Hochster, University of Michigan
Abstract:
Stillman has conjectured that given n forms of degree d
in a polynomial ring over a field, there is a bound on the
projective dimension of the ideal they generate that depends
on n and d but not on the number of variables. The talk
will survey what is known, and then discuss joint work of
Tigran Ananyan and the speaker that solves the problem
when d = 2. The bound is asymptotic to 2(n2n).
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Back to the future: multiplicities, integral closures, and
positive characteristic
Craig Huneke, University of Kansas
Abstract:
This talk will discuss recent joint work with Olgur Celikbas,
Hailong Dao, Yi Zhang, and myself, concerning bounding Hilbert-Kunz
multiplicities. A somewhat surprising point in the proofs is the
appearance of the integral closure of ideals, something that takes
me back to Purdue nearly 30 years ago.
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Complete ideals in two-dimensional regular local rings
Mee-Kyoung Kim, Sungkyunkwan University, Korea
Abstract: author's pdf file.
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Linearly presented grade 3 Gorenstein ideals in k[x,y,z]
Andrew Kustin, University of South Carolina
Abstract:
We exhibit a monomial presentation of such ideals.
This is joint work with Sabine El Khoury.
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Duality, residues, fundamental class
Joseph Lipman, Purdue University
Abstract:
This talk will be about a formalism for duality, à la Grothendieck,
which unifies the local theory---that is, for complete local rings or,
more generally, topological rings---and the global theory---that is,
for formal schemes. The formalism will be illustrated by a brief
discussion of how local residue maps can be pasted together into a global
map, the fundamental class. The whole topic is full of intriguing
relationships, most of which have been understood for some time;
but no definitive exposition has yet appeared.
As this is a conference on commutative algebra, the talk will keep
mainly to the context of commutative rings.
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Holonomic D-modules, a characteristic-free approach
Gennady Lyubeznik, University of Minnessota
Abstract:
Holonomic D-modules have been studied in characteristic zero for several
decades now. In characterisitc p>0 there has been no comparable theory.
Recently V. Bavula has given a characteristic-free definition of holonomicity
and shown that in characteristic zero it coincides with the usual notion.
We will discuss some developments resulting from this approach.
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A tour of the weak and strong Lefschetz properties
Juan Migliore, University of Notre Dame
Abstract:
In the first part of the talk we will give an overview of open
problems and known results on the Weak and Strong Lefschetz
properties. We will emphasize the many different approaches and
tools that have been used to study it, behavior in positive
characteristic, and connections that have been made with seemingly
unrelated problems. In the second part of the talk we will describe
recent work, done jointly with Rosa Miró-Roig and Uwe Nagel, on
ideals generated by powers of general linear forms, and their
behavior with respect to the Weak Lefschetz property.
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Multiplier ideals, test ideals, and ordinary varieties
Mircea Mustaţă, University of Michigan
Abstract:
Multiplier ideals are invariants of singularities defined
via divisorial valuations, while test ideals are characteristic p
versions defined via the Frobenius morphism. I will discuss a
conjecture relating the multiplier ideals to the test ideals via
reduction mod p, and the relation of this conjecture to the ordinarity
of the reductions to prime characteristic of a smooth projective
variety. This is based on joint work with V. Srinivas.
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Resolutions over complete intersections
Irena Peeva, Cornell University
Abstract:
This talk is on the asymptotic structure of minimal free
resolutions over a graded or local complete intersection.
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An effective approach to the study of the local complete intersections of height two
Maria Evelina Rossi, Università di Genova, Italy
Abstract:
Motivated by some papers of S. Goto, W. Heinzer, M.K. Kim and B. Ulrich,
in this talk we present a concrete approach to the study of numerical invariants
attached to a local complete intersection of height two. The underlying philosophy
is that all basic non-algorithmic facts concerning Groebner bases translate into valid
results on standard bases in rings of power series.
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Test ideals via alterations
Karl Schwede, Penn State University
Abstract: Given a ring R of characteristic p > 0, one can associate a
test ideal which reflects subtle properties of the singularities of R.
Over the last 15 years, a great deal of interest in the test ideal has
been with regards to its links with multiplier ideals, similar measures of
singularities often defined using complex analytic techniques. In order
to compute these multiplier ideals one uses a resolution of singularities
in order to change coordinates. In this talk we describe how to use
alterations (partial replacements for resolutions of singularities in
characteristic p > 0) to give a uniform description of an ideal which
coincides with the test ideal in characteristic p > 0, the multiplier
ideal in characteristic zero, and suggests a number of tantalizing
questions related to the direct summand conjecture in mixed characterstic.
This is joint work with Manuel Blickle and Kevin Tucker.
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F-pure thresholds of hypersurfaces
Anurag Singh, University of Utah
Abstract:
The F-pure threshold is a characteristic p analogue of
characteristic zero log canonical thresholds. We will discuss the
calculation of F-pure thresholds for supersingular Calabi-Yau
hypersurfaces.
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F-splitting dimension equals the dimension of the F-splitting prime
Kevin Tucker, University of Utah/Princeton University
Abstract:
When working in equal characteristic p > 0, splittings of the Frobenius
endomorphism (and its iterates) have long been used to study singularities.
To that end, the F-splitting dimension of a local ring gives the asymptotic
growth of the number of such splittings. When this invariant was introduced
by I. Aberbach and F. Enescu, they further questioned whether it was equal to the
dimension of a certain naturally defined prime ideal (the F-splitting prime).
In this talk, I will describe joint work in progress with M. Blickle and K. Schwede
showing that this question has a positive answer. The main ingredient in the proof
is the construction of generalized F-signature (following the generalizations of tight
closure and test ideals to incorporate ideal and divisor pairs). Time permitting,
we hope to mention some other applications and examples of this beautiful and still
developing theory.
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Comparing powers and symbolic powers of ideals
Javid Validashti, University of Kansas
Abstract:
Comparing the ordinary and symbolic powers of ideals and giving criteria for equality
are subjects of interest in both commutative algebra and algebraic geometry.
We show that the symbolic topology defined by a prime ideal is uniformly linearly
equivalent to the adic topology for a large class of isolated singularities.
We also consider the following question posed by Huneke on the equality.
In a regular local ring, if the ordinary and symbolic powers of a prime ideal
are the same up to its height, then are they the same for all powers?
We provide supporting evidence for ideals defining classes of monomial curves
and for prime ideals defining rings with low multiplicity.
This talk is based on joint works with Aline Hosry, Youngsu Kim, Craig Huneke and Dan Katz.
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The homology of parameter ideals
Wolmer Vasconcelos, Rutgers University
Abstract: author's pdf file.
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Certain jet schemes have rational singularities
Kei-ichi Watanabe, Nihon University, Japan
Abstract: author's pdf file.
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Power series over Noetherian rings
Sylvia Wiegand, University of Nebraska
Abstract:
This is joint work with William Heinzer and Christel Rotthaus.
We show how to create exotic Noetherian and non-Noetherian rings using power
series over well-understood Noetherian commutative integral domains.
This construction synthesizes a classical technique used by Akizuki in
the nineteen thirties and by Nagata in the nineteen fifties.
Usually we simply intersect an appropriate field with the power series ring
over the base Noetherian domain. We have been able to show that in
certain circumstances, such an intersection is computable as a directed
union, and the Noetherian property for the associated directed union
is equivalent to a flatness condition. This flatness criterion simplifies
the analysis of several classical examples and yields new examples.
We also describe some examples produced using the technique.
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