Ascend and Descend Conditions of Excellent Rings ( a work of Silvio Greco)
The "class" of Noetherian rings are not very well suited "class" to do rich geometry. Grothendieck introduced excellent rings as a "class" of rings which occur almost always in Geometry and Number Theory, and which is most likely to be the candidate suitable for geometrical purposes.
The aim of the talk is to define excellent rings and present the proof of their ascend and descend properties under module-finite maps which induce surjective maps of prime spectrums, which appeared in the following work:
"Two Theorems on Excellent Rings, Silvio Greco, Nagoya math. J., Vol. 60 (1976), 139 1.
Bounds on Waldschmidt constant
Given a finite set of points {p1,...,p_s} in $\mathbb{P}^N$ what is the minimal degree of a hypersurface that passes through these points with multiplicity at least m? Nagata raised the question to give counter-example to Hilbert's 14th problem. The minimum degree is in general very difficult to compute. In this talk, I will present various aspects of the lower bounds, some well known conjectural bounds, and some well-known containment conjectures which are used to determine better bounds. I also will present some techniques and some results regarding the lower bound. I will also present some results from our joint work with Eloísa Grifo, Huy Tà Hà, and Thái Thàn Nguyên
Base Change Along the Frobenius Endomorphism and the Gorenstein Property
Let R be a local ring of positive characteristic and X a complex with nonzero finitely generated homology and finite injective dimension. We prove that if derived base change of X via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then R is Gorenstein.
A Simple Study Of Colength
We define and study an invariant of any module over a local one dimensional analytically unramified Noetherian domain whose integral closure is a DVR. We shall prove a key property of this invarant and briefly venture into trace ideals and finally into reflexive ideals
Vector-bundles on punctured spectrum: a.k.a reflexive modules, locally free on punctured spectrum.
Projective modules over local rings are boring and "too nice" since they are just free. One way to get an interesting theory of "projective modules" in local Algebra is instead to look at "modules" that are "locally free on the punctured spectrum" i.e. to look at vector-bundles on punctured spectrum. Although this scenario looks non-affine at first glance, a celebrated Theorem of Horrocks says that these vector bundles on punctured spectrum are essentially reflexive modules, locally free on punctured spectrum when the ring has depth at least 2. After looking at the precise statement of Horrocks' theorem, we'll deduce some easy Corollary about triviality of such vector-bundles and will discuss when a local ring admits such non-trivial vector-bundles. Finally, we will discuss a cancellation problem of such vector-bundles from tensor product using local Cohomology, Krull-Schmidt property and other tools. This last part is still work in progress.
Dimension of syzygies of finite length modules
In this talk we will discuss the dimension of syzygies of finitely generated modules over a Noetherian local ring. We will mainly focus on modules of finite length and discuss a question raised by Alessandro De Stefani, Craig Huneke and Luis Núñez-Betancourt
Reducedness of Formally Unramified Algebras over fields.
It is well known that a finitely generated algebra over a field, which is also formally unramified over the same field is always reduced. This talk aims to discuss the reducedness of a general formally unramified algebra over a field. While it is not true that such algebras are necessarily reduced, we shall show in nicer cases formally unramified algebras are indeed reduced. In particular, we shall see that ‘nice’ local rings and graded rings which are formally unramified over a perfect field are always reduced. Then we shall produce examples to illustrate that these results cannot be extended further. This is based on a joint work with Karen Smith (https://arxiv.org/abs/2005.05833).
Finiteness Properties of Local Cohomology
Local cohomology modules are versatile tool that can be applied to problems as diverse as bounding the number of equations needed to cut out an algebraic set, measuring the singularities of a local ring, analyzing the syzygies of a projective embedding, and more. The most fundamental question about a local cohomology module is "when does it vanish?" Since local cohomology modules are rarely finitely generated, it can be very difficult to understand their supports. In this talk, I will review some fascinating open questions about the support and associated primes of local cohomology, and I will describe some recent work in the setting of complete intersection rings.