March 2021
Public Access Category: All |
5:30pm» Kazuho Ozeki, Yamaguchi University, Japan- Time:
- 5:30pm
- Description:
- Speaker:* Kazuho Ozeki, Yamaguchi University, Japan*
Date/Time: *5 March 2021, 5:30pm IST/ 12:00pm GMT/ 7:00am EST* (joining
time: 5:15 pm IST - 5:30
pm IST)
Google meet link: https://meet.google.com/mjh-prqn-rio
Title:* The reduction number of stretched ideals*
Abstract: The homological property of the associated graded ring of an
ideal is an important problem in commutative algebra and algebraic
geometry. In this paper we explore the structure of the associated graded
ring of stretched $\m$-primary ideals in the case where the reduction
number attains almost minimal value in a Cohen-Macaulay local ring
$(A,\m)$. As an application, we present complete descriptions of the
associated graded ring of stretched $\m$-primary ideals with small
reduction number.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar
- Time:
- 6:30pm
- Description:
- Speaker: Thomas Polstra, University of Virginia, Charlottesville, VA, USA
Date/Time: 12 March 2021, 6:30pm IST/ 1:00pm GMT / 8:00am EST (joining
time 6:15pm IST).
Google meet link: https://meet.google.com/xze-mbdb-qdb
Title: Strongly $F$-regular rings, maximal Cohen-Macaulay modules, and the
$F$-signature
Abstract: The singularities of a local prime characteristic ring are best
understood through the behavior of the Frobenius endomorphism. A
singularity class of central focus is the class of strongly $F$-regular
rings. Examples of strongly $F$-regular rings include normal affine toric
rings, direct summands of regular rings, and determinantal rings. Every
strongly $F$-regular ring enjoys the property of being a normal
Cohen-Macaulay domain. In particular, the study of finitely generated
maximal Cohen-Macaulay modules over such rings is a warranted venture. We
will demonstrate a surprising uniform behavior enjoyed by the category of
maximal Cohen-Macaulay modules over a strongly $F$-regular local ring.
Consequently, we can redrive Aberbach and Leuschke's theorem that the
$F$-signature of a strongly $F$-regular ring is positive in a novel and
elementary manner. Time permitting, we will present applications on the
structure of the divisor class group of a local strongly $F$-regular ring.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar
- Time:
- 6:30pm
- Description:
- Speaker:* Thomas Polstra, University of Virginia, Charlottesville, VA, USA*
Date/Time: *19 March 2021, 6:30pm IST/ 1:00pm GMT / 9:00am EDT* (joining
time 6:15pm IST).
Google meet link: https://meet.google.com/ujr-hykc-cjm
Title:* Prime characteristic singularities and the deformation problem*
Abstract: Let $P$ be a property of local rings (such as regular,
Gorenstein, or complete). We say that $P$ deforms if a local ring $R$
enjoys property $P$ provided there exists a nonzerodivisor $x$ such that
$R/xR$ is $P$. (For example, the properties of being regular or Gorenstein
deform, but the property of being complete does not deform). The
deformation problem, as it pertains to the prime characteristic singularity
classes of $F$-regular, $F$-rational, $F$-pure, and $F$-injective
singularities, has a rich history that dates to work of Fedder in the
1980's and remains an active research area. We will survey the history of
the deformation problem of these four prime characteristic singularity
classes and discuss a recent solution to the deformation of $F$-purity
problem in rings which are $\mathbb{Q}$-Gorenstein. This talk is based on a
collaboration with Austyn Simpson.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar
- Time:
- 6:30pm
- Description:
- Speaker: Jason McCullough, Iowa State University, Ames, IA, USA
Date/Time: 26 March 2021, 6:30pm IST/ 1:00pm GMT / 9:00am EDT (joining
time 6:15pm IST).
Google meet link: https://meet.google.com/qcy-jrqf-naa
Title: * The Eisenbud-Goto Conjecture*
Abstract: Let S be a polynomial ring over an algebraically closed field K.
There has been considerable research into effective upper bounds for the
Castelnuovo-Mumford regularity of graded ideals of S. Through work of
Bertram, Ein, Gruson, Kwak, Lazarsfeld, Peskine, and others, there are
several good bounds for the defining ideals of smooth projective varieties
in characteristic zero. However, for arbitrary ideals, the best upper bound
is doubly exponential (in terms of the number of variables and degrees of
generators), and this bound is asymptotically close to optimal due to
examples derived from the Mayr-Meyer construction. In 1984, Eisenbud and
Goto conjectured that the regularity of a nondegenerate prime ideal P was
at most deg(P) – codim(P) + 1, and proved this when S/P was Cohen-Macaulay
(even if P is not prime). In this talk I will explain the construction of
counterexamples to the Eisenbud-Goto Conjecture, joint work with Irena
Peeva, through the construction of Rees-Like algebras and a special
homogenization. While we show that there is no linear bound on regularity
in terms of the degree (or multiplicity) of P, we later showed that some
such bound exists. The latter part of this talk is joint work with Giulio
Caviglia, Marc Chardin, Irena Peeva, and Matteo Varbaro.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar