The commutative algebra group of the Institute for Research in Fundamental Sciences (IPM) is organizing the 16th seminar on Commutative Algebra and Related Topics. The goal is to bring together people who are working on commutative algebra and related fields and to introduce recent developments to the PhD students and young researchers.

In addition, a workshop will be held on January 13 and 14, 2020. The aim is to acquaint the young researchers with the new trends in the subject.Your contribution is highly appreciated.

The speakers of the workshop are as follows:

• Hamid Hassanzadeh (Federal University of Rio de Janeiro, Brazil)
• Doan Trung Cuong (Institute of Mathematics, Hanoi, Vietnam)
• Nguyen Tu Cuong (Institute of Mathematics, Hanoi, Vietnam)
• Sverre Olaf Smalo (Norwegian University of Science and Technology, Norway)
.

#### Organizers:

• Abdolnaser Bahlekeh (Gonbad Kavous University)
• Kamran Divaani-Aazar (IPM and Alzahra University)
• Zahra Nazemian (IPM)

# Seminar (January 15 - 16, 2020)

## Invited Speakers

Abstract: In this talk, we study Cohen-Macaulay Auslander algebras using certain intermediate extension functors. In particular, it will be shown that two Gorenstein algebras of G-dimension one that are of finite Cohen-Macaulay type are Morita equivalent if and only if their Cohen-Macaulay Auslander algebras are Morita equivalent. The talk is based on a joint work with Rasool Hafezi.
Abstract: Cohomological degrees (or extended degrees) were introduced by Doering-Gunston-Vasconcelos twenty years ago as a measure for the complexity of algebraic structures of finitely generated modules over a Noetherian ring. These degrees plays the role of multiplicities in the case of Cohen-Macaulay modules. In particular, Doering-Gunston-Vasconcelos obtained several upper bounds for number of generators, Castelnuovo-Mumford regularity, Betti numbers, etc in terms of a cohomological degree. On the other hand, Vasconcelos and Gunston introduced respectively the homological degree $\mathrm{hdeg}$ and the extremal degree $\mathrm{bdeg}$ as two examples of cohomological degrees. Recently, NT Cuong-PH Quy has introduced the notion of unmixed degree and showed that it is also a cohomological degree. This is the third example of cohomological degree. In this talk, we will discuss a construction that gives rise to an infinite family of cohomological degrees. The construction relies on special properties of annihilators of local cohomology modules. This is a joint work with Pham Hong Nam.
Abstract: Residual intersection is a generalization of linkage. It provides geometric and algebraic classifications of ideals. The definition is the following: In a (Cohen-Macaulay) local ring R, an ideal J is an s-residual intersections of an ideal I if there is an s-generated ideal a such that J=a: I and codim(J) is at least s. Although this is a fundamental construction in Algebra and Geometry, very few are known about the structure of J (generators and other Betti numbers), except for some particular cases. In this talk, we show how one can determine the generators of J by using the DG-algebra structure of the Koszul complex of I. The results cover most of the classical works in this direction which include some works of Peskine-Szpiro, Huneke-Ulrich, Kustin, Bruns among others. The talk bases on joint work with Vinicius Bouca.
Abstract: Hankel determinantal rings, i.e., rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves. We study cyclic covers of Hankel determinantal rings in broad generality, and use these to prove that certain Hankel determinantal rings are F-regular. Based on a joint work with Aldo Conca, Anurag K. Singh and Matteo Varbaro.
Abstract: I will present a couple of counterexamples I have made to some homological questions raised by H. Bass and M. Auslander.

## List of Participants

The list of participants is available by clicking here.

## Registration

To register for the seminar, please fill out the Registration Form. Your request will be considered by the organizing committee. Upon acceptance you will receive an email. After the acceptance, you may complete the registration by doing the payment process.

Registration fee for the seminar:

1) Registration fee for Iranian participants:

2) Registration fee for international participants:
The registration fee is 150 Euro. The registration fee for international participants will be due in cash at the time of registration on the first day of the meeting. Please note that standard credit cards; e.g., Visa, Master or AmEXP, cannot be used in Iran.

Registration fee includes: Documentation package, participation in the lectures, lunches coupons, and coffee breaks during the meeting.
Residence fee at IPM guest house is 25 Euro per night.

#### Registration Form

Name:
Surname:
Employment:
(only for Students):
Other explain here:
Affiliation:
Email:
Phone:
Accommodation:
From: To:

## Call for Papers

Papers will be accepted for presentation at the seminar subject to approval by the Organizers. Please send submissions (extended abstract or full paper) electronically (in PDF and Tex formats) to ipmcommalg@ipm.ir with the subject: "16th_commalg_abstract".

## Important Dates

Deadline for registration and paper submission: December 22, 2019 (اول دی ماه 1398)
Decision on acceptance of attendance and talks: December 29, 2019 (هشتم دی ماه 1398)
Deadline for registration confirmation: January 2, 2020 (دوازدهم دی ماه 1398)

# Workshop (January 13 - 14, 2020)

All commutative algebraists are welcome to participate in this event. Please note that attending in the workshop is free of charge and there is no need for registration.

## Lecturers

Abstract: The Koszul complex is one of the most fundamental objects in commutative algebra, historically and technically. The complex generalized to Eagon-Northcott complex to provide a resolution for determinantal ideals and to Buchsbaum-Rim complex to provide a resolution for any module. Both of these generalizations were unified by Buchasbaum and Eisenbud to construct what we know as the Buchsbaum-Eisenbud family of complexes. In this talk, after introducing this family, we show that indeed the Buchsbaum-Eisenbud family of complexes can be constructed as graded strands of a Koszul-Cech spectral sequence. The new construction helps to understand the family better. One of the applications of this result is in the determination of the generators of the residual intersection. A topic which I will expand in my next talk in the seminar.
Abstract: Abstract of Talk 1: In this talk we will discuss some bounds for the maximal dimension of formal fibers of a Noetherian local ring. We will first review the works of Matsumura, Rotthaus and others. Then we present some recent results obtained by ourselves.

Abstract of Talk 2: I would like to focus on Noetherian local rings whose formal fibers are of zero-dimensional. Some characterizations of these rings are discussed. Using these characterization, we show that certain local rings of a one-dimensional scheme over a one-dimensional complete local ring have all formal fibers of zero-dimensional. Then we present an application of this result to the theory of quadratic forms.

Abstract of Talk 3: Relations between singularities of the formal fibers of a local ring and of the ring itself was discovered by Grothendieck and played an important role in the theory of excellent rings. In this talk, I will discuss relations between the Cohen-Macaulayness of the formal fibers of a finitely generated module $M$ over a Noetherian local ring with the existence of a Cohen-Macaulay Rees module of $M$.
Abstract: The notion of systems of parameters in a local ring was first introduced by the author [1] in order to study the problem of Macaulayfication posed by G. Faltings. We proved in [2] that the existence of a p-standard system of parameters for a local ring R is equivalent to R is a quotient of a Cohen-Macaulay local ring. Therefore, the p-standard system of parameters is an useful for studying the structure of finitely generated R-modules provided R is a quotient of a Cohen-Macaulay local ring. Next, we present in this talk a short survey on applications of p-standard systems of parameters to solving problems:
- The existence of a Macaulayfication of a local ring which is a quotient of a Cohen-Macaulay ring [2].
- The stability of index of reducibility of parameter ideals [3].
- To define a new extended degree in the sense of W. Vasconselos called the unmixed degree for a finitely generated module [4].

References:
[1] N.T. Cuong, p-standard systems of parameters and p-standard ideals in local rings, Acta Mathematica Vietnamica Vol. 20 (1995), 145-161.
[2] N.T. Cuong and D.T. Cuong, Local cohomology annihilators and Macaulayfication,, Acta Mathematica Vietnamica Vol. 42 (2017), 37-60.
[3] N.T. Cuong and P.H. Quy, On the index of reducibility of parameter ideals: the stable and limit values, To appear in Acta Mathematica Vietnamica Vol. 45 (2020).
[4] N.T. Cuong and P.H. Quy, On the structure of finitely generated modules over quotients of Cohen-Macaulay local rings. Preprint arXiv:1612.07638.
Abstract: The purpose of these talks is to give a fast introduction to some problems of homological and geometrical nature related to finite dimensional representations of finitely generated, and especially, finite dimensional associative algebras over a field. Some of these results can also be extended to the situation where the field is not algebraically closed, and some of the results can even be extended to the situation where one is considering algebras over a commutative artin ring. For the results which hold true in the most general situation the proofs become most elegant since they depend on using length arguments only, and thereby forgetting about the nature of a field altogether.

# Useful Information

### IPMInstitute for Research in Fundamental Sciences

Niavaran

P.O. Box 19395-5746, Tehran - Iran

• Tel: +98 21 222 90 928, Fax: +98 21 222 90 648
• ipmcommalg@ipm.ir
Top