Special Semester on Commutative Algebra

Hoàng Lê Trường (2/10/2024)

Title: On the Ulrich complexity of complete intersections

Abstract: In this talk, we establish the Buchweitz-Greuel-Schreyer Conjecture for cubic hypersurfaces with small Ulrich complexity and complete intersections of two quadrics.

Ngô Việt Trung (8/10/2024)

Title: Asymptotic behaviour of the saturation degree

Abstract: Recently, Ein-Ha-Lazarsfeld proved that if I is a homogeneous ideal whose zero locus is a non-singular complex projective scheme, then the saturation degree sdeg I^n is bounded above by a linear function of n whose slope is less or equal the maximal generating degree of I. Inspired by the asymptotic behavior of the Castelnuovo-Mumford regularity, we show that for an arbitrary graded ideal I in an arbitrary graded ring, sdeg I^n is either a constant or a linear function for n large enough whose slope is one of the generating degrees of I.

Trần Nam Trung(15/10/2024)

Title: Cohen-Macaulay of the second powers of edge ideals of weighted-edge very-well covered graphs

Abstract. This is  a joint work with Guanjun Zhu.Let G be a simple graph and w is a weighted-edge on G. We characterize w such that I(G_w)^2 is Cohen-Macaulay.

Sergey Gaifullin (16/10/2024)

Title: Makar-Limanov invariant of cylinders over trinomial varieties

Abstract. This  is based on a joint work with Mikhail Petrov. A trinomial variety is an affine irreducible variety defined by a system of trinomial equations of special form. The coordinates are divided onto groups, and for each group, a monomial in the variables in this group is fixed. Each equation consists of three such monomials. In 2017, J. Hausen and M. Wrobel proved that each variety with a torus action of complexity one, satisfying certain conditions, can be obtained from a trinomial variety via Cox's construction.

We are interested in computing the Makar-Limanov invariant of trinomial varieties. Recall that the Makar-Limanov invariant is the intersection of all kernels of locally nilpotent derivations, or equivalently, the invariant subalgebra for all algebraic actions of the additive group of the ground field. If it coincides with the ring of regular functions, the variety is called rigid. In general, computing the Makar-Limanov invariant of a trinomial variety is a challenging problem. However, we prove that a cylinder over a non-rigid trinomial variety has a trivial Makar-Limanov invariant.

Alexander Perepechko (18/10)

Title: Automorphisms of affine varieties: flexibility and unipotent group actions

Abstract: A Ga-action on an affine variety X is an algebraic action of the additive group Ga(K) of the base field K.  A variety is called flexible, if for each smooth point, its tangent space is spanned by tangent vectors to orbits of Ga-actions. We will discuss the connection of flexibility and multiple transitivity of the automorphism group action. We will also survey families of varieties known to be flexible.

 In the case of an affine space A^n, there is a natural notion of a subgroup of triangular automorphisms, which is an infinite-dimensional analogue of upper-triangular matrices U(n) in the matrix group GL(n). It is well known that any unipotent subgroup of GL(n) is conjugated to a subgroup of U(n). Unfortunately, this result does not hold for the subgroup of triangular automorphisms.

We will present a generalization of a triangular automorphism subgroup for an arbitrary affine variety X that describes all maximal unipotent subgroups of Aut(X). We will also discuss its properties, construction, and connection to additive actions. In particular, any unipotent subgroup of Aut(X) happens to be closed in the Zariski topology.

 Phạm Hùng Quý (29/10/2024)

Title: Some perturbation problems in Commutative Algebra

Abstract: Many fundamental questions in singularity theory arise from studying deformations. One particular way of deforming a singularity is by changing the defining equations by adding terms of high order. This problem often arises while working with analytic singularities. The first instance is the problem of finite determinacy which asks whether for a singularity defined analytically, e.g., as a quotient of a (convergent) power series ring, can be transformed into an equivalent algebraic singularity by truncating the defining equations. In this talk, we discuss two parts: domain, reduced, normal properties of local rings under small perturbation; and Hilbert functions under small perturbations.  

Trần Nam Trung (6/11/2024)

Title: Cohen-Macaulay of powers of edge ideals of weighted-edge trees

Abstract. This is  a joint work with Guanjun Zhu.Let G be a tree  and w is a weighted-edge on G. We characterize G_w such that I(G_w)^n is Cohen-Macaulay for all n>=1.

Nguyễn Đăng Hợp (20/11/2024)

 Title: Depth and regularity of Inc-invariant chains of monomial ideals

 Abstract: Chains of ideals in polynomial rings of varying Krull dimensions that are invariant under the action of the infinite symmetric group Sym or the semigroup Inc of increasing functions N --> N, have attracted attention of various researchers in recent years. The behaviour of minimal generating sets, Hilbert series, codimensions of such an invariant chains of homogeneous ideals is well-understood. Howerver, the behaviour of finer homological invariants like projective dimension, Castelnuovo--Mumford regularity, and depth, remains quite mysterious, even for invariant chains of monomial ideals. Combinatorially, an Inc-invariant chain of edge ideals with a fixed stablity index and fixed number of Inc-orbits can be thought of as a chain of unions of a fixed number of isoceles right triangles in R^2 with fixed lowest vertices and varying legs. We use this combinatorial description to describe recent results on the depth and regularity of Inc-invariant chains of edge ideals. This is from joint work with T.Q. Hoa, D.T. Hoang, D.V. Le and T.T. Nguyen.

Đỗ Trọng Hoàng (13/11/2024)

Title: On the skew tableau ideals

Abstract: A Young diagram \(\lambda = (\lambda_1, \ldots, \lambda_n)\) is a collection of boxes arranged in a left-justified shape with \(n\) rows, where the length of the \(i\)-th row is \(\lambda_i\), and \(\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n\). A skew Young diagram \(\lambda/\mu\) consists of the boxes in the Young diagram of \(\lambda\) that do not belong to the Young diagram of \(\mu\), where \(\mu_i \leq \lambda_i\) for all \(i\).
    
    A \textit{filling} of a skew Young diagram is an assignment of positive integers \( \omega(i,j) \) to each box \( (i,j) \) in the skew diagram. The {\it skew tableau ideal} associated with a skew Young diagram \(\lambda/\mu\) with a given filling is an ideal in a standard graded polynomial ring \(S = k[x_1, \ldots, x_n, y_1, \ldots, y_m]\) over a field \(k\), where \(m = \lambda_1\), and is defined as follows:  
$$I_{\lambda/\mu}(\mathbf{w}):= ((x_iy_j)^{\omega(i,j)}\mid 1\le i\le n, \mu_i+1\le j \le \lambda_i).$$
In this talk, we will provide  a characterization for Cohen--Macaulayness of   $I_{\lambda/\mu}(\mathbf{w})$ and explore  related problems. This work is based on a collaboration with Thanh Vu from the Vietnam Academy of Science and Technology.  

A. V. Jayanthan (27/11/2024)

Title: On the unmixedness and Cohen-Macaulayness of parity binomial edge ideals of chordal graphs.

Abstract: While the Cohen-Macaulayness of binomial edge ideal has been extensively studied in terms of combinatorial properties of the graphs, nothing is not known about the unmixedness and Cohen-Macaulayness of the parity binomial edge ideals. In this talk I will classify all unmixed and Cohen-Macaulay parity binomial edge ideals of chordal graphs. If I have time, I will discuss the proof which is very combinatorial in nature

Jugal Verma (11 & 18/12/2024)

 Title: Hilbert-Kunz multiplicity of powers of an ideal I - III

 Abstract: Let $(R, \mathfrak m)$ be a $d$-dimensional local ring, $I$ be an $\mathfrak m$-primary ideal and let  $R$ have prime characteristic $p.$ 

K.-I. Watanabe and K.-I. Yoshida investigated the  Hilbert-Kunz multiplicity  of powers of $I$ in terms of Hilbert  coefficients of I and its Frobenius powers $I^{[q]}$  where $q=p^n.$  It was proved by V. Trivedi that if $I$ is zero-dimensional graded ideal of a standard graded ring $R$ of dimension $d$ over a field, then $L_1(I)=\lim_{q\to \infty} e_1(I^{[q]})/q^d$ exists. Illya Smirnov proved Trivedi's result for $\mathfrak m$-primary ideals of all local rings. Smirnov asked if  $L_k(I)=\lim_{q\to \infty} e_k(I^{[q]})/q^d$ exists for $k=2,3,\ldots, d$ and whether the HK multiplicity of $I^n$ for all large $n$ is given by the formula

\[e_{HK}(I^n)=\sum_{k=0}^d (-1)^k L_k(I)\binom{n+d-k-1}{d-k}.\]

Smirnov also conjectured that ideals of reduction number one can be characterised in terms of the HK multiplicity.

I will report on joint works with {\bf Kriti Goel, Arindam Banerjee and  Shreedevi Masuti, Marilina Rossi and Alessandro De Stefani, } which provide partial answers to  Smirnov's questions.

Results will be given in Part I in the workshop, while the proofs will be given in Parts II and III.

Lê Văn Đính (25/12/2024)

 Title: Invariant chains in algebra and discrete geometry

 Abstract:In various areas of mathematics, including algebra and discrete geometry, one frequently encounters chains of objects (e.g. ideals, cones, monoids, lattices) of increasing dimensions that are invariant under the action of symmetric groups. A central problem is to study finiteness and asymptotic properties up to symmetry of such chains. In this talk, I will discuss some recent developments in the study of this problem.

Nguyen Thu Hang

Title: Regularity of powers and symbolic powers of edge ideals of cubic circulant graphs] {Regularity of powers and symbolic powers of edge ideals of cubic circulant graphs

Abstract. We compute the regularity of powers and symbolic powers of edge ideals of all cubic circulant graphs. In particular, we establish Minh's conjecture for cubic circulant graphs,  as follows:

\begin{thm}\label{non-bipartite-theorem} Let $G = C_{2n}(1,n)$ or $G= C_{2n}(2,n)$ where $n > 1$ is an odd number. Then $$\reg(I(G)^t) = \reg (I(G)^{(t)}) = 2t- 1 + \lfloor \frac{n}{2} \rfloor,$$
for all $t \ge 2$.
\end{thm}

Joint work with My Hanh Pham,  and Thanh Vu.


Suprajo Das

Saturation density functions and some applications

Abstract.  Let $R=\oplus_{m\geq 0}R_m$ be a $d$-dimensional standard graded finitely generated domain over an algebraically closed field $R_0=k$ and $\m$ be the unique homogeneous maximal ideal of $R$. Given a homogeneous ideal $I$ in $R$, we are interested in studying the growth of the graded components $\left(I^n\colon_R \m^{\infty}\right)_m$ as $m$ and $n$ vary over integers. In general, this is a difficult problem because the saturated Rees algebra $\oplus_{n\geq 0}\left(I^n\colon_R \m^{\infty}\right)t^n$ can be non-Noetherian.

In this talk, we shall analyze this by introducing a function, $$f^{\mathrm{sat}}_{I}(x) = \limsup_{n\to\infty}\dfrac{\dim_k\left(I^n\colon_R \m^{\infty}\right)_{\lfloor xn\rfloor}}{n^{d-1}/d!},$$ which we call the saturation density function of $I$. We shall show that $f^{\mathrm{sat}}_{I}(x)$ exists as a limit for all real numbers $x$ and it is continuous. We shall also prove a Rees-type statement for ideal sheaves by using the equality of saturation density functions. Our proofs will use the theory of volume functions developed by Lazarsfeld and others. If time permits, we shall give some examples of such functions in low dimensions.

This talk will be based on two ongoing joint projects with Sudeshna Roy and Vijaylaxmi Trivedi.


Suprajo Das

Title: Density functions for filtrations of ideals in a polynomial ring

Abstract. Let $R=k[x_1,\ldots,x_d]$ be $d$-dimensional standard graded polynomial ring over a field $k$. Let $\mathcal{I} = \{I_n\}_{n\geq 0}$ and $\mathcal{J} = \{J_n\}_{n\geq 0}$ be two (not necessarily Noetherian) filtrations of homogeneous ideals in $R$ with $I_n \subseteq J_n$ for each $n\geq 0$. Suppose that there exists an integer $c>0$ such that ${\left(I_n\right)}_m = {\left(J_n\right)}_m$ for all $n\geq 0$ and $m\geq cn$. The limit $\lim_{n\to\infty}\frac{\lambda_R\left(J_n/I_n\right)}{n^d}$ exists due to a result of Cutkosky.

In this talk, we shall consider the function $$f_{\mathcal{J}/\mathcal{I}}(x) = \limsup_{n\to\infty}\dfrac{\dim_k\big({(J_n)}_{\lfloor xn\rfloor}/{(I_n)}_{\lfloor xn\rfloor}\big)}{n^d},$$ which we call the density function for $\mathcal{J}/\mathcal{I}$. We shall show that $f_{\mathcal{J}/\mathcal{I}}(x)$ exists as a limit for all real numbers $x$. We shall also prove that $f_{\mathcal{J}/\mathcal{I}}$ is a compactly supported continuous function and $$\int_{0}^{\infty} f_{\mathcal{J}/\mathcal{I}}(x)dx = \lim_{n\to\infty}\frac{\lambda_R\left(J_n/I_n\right)}{n^d}.$$ Moreover, we shall describe this function as a difference of volumes of slices of appropriate Newton-Okounkov bodies.

This talk will be based on an ongoing joint project with Sudeshna Roy.


Do Trong Hoang

Title: On the skew tableau ideals

Abstract. A Young diagram \(\lambda = (\lambda_1, \ldots, \lambda_n)\) is a collection of boxes arranged in a left-justified shape with \(n\) rows, where the length of the \(i\)-th row is \(\lambda_i\), and \(\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n\). A skew Young diagram \(\lambda/\mu\) consists of the boxes in the Young diagram of \(\lambda\) that do not belong to the Young diagram of \(\mu\), where \(\mu_i \leq \lambda_i\) for all \(i\).
   
    A \textit{filling} of a skew Young diagram is an assignment of positive integers \( \omega(i,j) \) to each box \( (i,j) \) in the skew diagram. The {\it skew tableau ideal} associated with a skew Young diagram \(\lambda/\mu\) with a given filling is an ideal in a standard graded polynomial ring \(S = k[x_1, \ldots, x_n, y_1, \ldots, y_m]\) over a field \(k\), where \(m = \lambda_1\), and is defined as follows:  
$$I_{\lambda/\mu}(\mathbf{w}):= ((x_iy_j)^{\omega(i,j)}\mid 1\le i\le n, \mu_i+1\le j \le \lambda_i).$$
In this talk, we will provide  a characterization for Cohen--Macaulayness of   $I_{\lambda/\mu}(\mathbf{w})$ and explore  related problems. This work is based on a collaboration with Thanh Vu from the Vietnam Academy of Science and Technology.  
    

Jugal Verma

Title: Hilbert-Kunz multiplicity of powers of an ideal

Abstract. Let $(R, \mathfrak m)$ be a $d$-dimensional local ring, $I$ be an $\mathfrak m$-primary ideal and let  $R$ have prime characteristic $p.$  
K.-I. Watanabe and K.-I. Yoshida investigated the  Hilbert-Kunz multiplicity  of powers of $I$ in terms of Hilbert  coefficients of I and its Frobenius powers $I^{[q]}$  where $q=p^n.$  It was proved by V. Trivedi that if $I$ is zero-dimensional graded ideal of a standard graded ring $R$ of dimension $d$ over a field, then $L_1(I)=\lim_{q\to \infty} e_1(I^{[q]})/q^d$ exists. Illya Smirnov proved Trivedi's result for $\mathfrak m$-primary ideals of all local rings. Smirnov asked if  $L_k(I)=\lim_{q\to \infty} e_k(I^{[q]})/q^d$ exists for $k=2,3,\ldots, d$ and whether the HK multiplicity of $I^n$ for all large $n$ is given by the formula
\[e_{HK}(I^n)=\sum_{k=0}^d (-1)^k L_k(I)\binom{n+d-k-1}{d-k}.\]
Smirnov also conjectured that ideals of reduction number one can be characterised in terms of the HK multiplicity.
I will report on joint works with {\bf Kriti Goel, Arindam Banerjee and  Shreedevi Masuti, Marilina Rossi and Alessandro De Stefani, } which provide partial answers to  Smirnov's questions.


Sudeshna Roy

Title: Density function associated to a module and integral dependence

Abstract. A density function for an algebraic invariant is a measurable function on the set of real numbers which measures the invariant on a real scale. In this talk, we will discuss density functions for Noetherian filtrations of homogeneous ideals in a standard graded Noetherian domain over a field $k$. This was inspired by the Hilbert-Kunz density functions developed by V. Trivedi. We will also demonstrate a density function for a finitely generated bigraded module over a bigraded Noetherian $k$-algebra, which is generated in bidegrees $(1,0), (d_1, 1), \ldots, (d_r,1)$ for some $d_i \geq 0$. Our main ingredients are the method of Gr\"obner bases and Sturmfels' structure theorem for vector partition functions. As an application, we will provide a new numerical criterion for integral dependence of arbitrary homogeneous ideals in terms of computable and well-studied invariants, such as, mixed multiplicities of ideals and Hilbert-Samuel multiplicities of certain standard graded algebras. A novelty of our approach is that it does not involve localizations. This talk is based on joint works with Suprajo Das and Vijaylaxmi Trivedi.


Sudeshna Roy

Title: Computing epsilon multiplicities in two dimensional graded domains

Abstract. The notion of epsilon multiplicity, a generalization of the Hilbert-Samuel multiplicity, was introduced by B. Ulrich and J. Validashti to detect integral dependence of arbitrary ideals. This invariant is difficult to handle, as it can be irrational and so the associated length function is very far from being polynomial-like. The objective of this presentation is to discuss certain computational aspects of this invariant. Let $I$ be a homogeneous ideal in a two dimensional standard graded Cohen-Macaulay domain over a field. We will show that the $\varepsilon$-multiplicity, $\varepsilon(I)$, of $I$ is a rational number. The proof given in arXiv:2402.11935 uses some classical theory in algebraic geometry for projective curves. However, in this talk, we will sketch an alternative proof using purely algebraic methods. We will further describe $\varepsilon(I)$ in terms of various mixed multiplicities associated to $I$, which enables us to explicitly compute it using Macaulay2. This talk is based on a joint work with Suprajo Das, Saipriya Dubey, and Jugal K. Verma.

Special Semester on Commutative Algebra

 http://math.ac.vn/en/conferences/icalrepeat.detail/2024/10/02/5439/-/-.html

 Main seminars

1. Das, Suprajo (IIT Bombay, India)
2. Dubey, Saipriya (Chennai Math. Inst., India)
3. D'Cruz, Clare Chennai Math. Inst., India)
4. Ghorpade, Sudhir Ramakant (IIT Bombay, India)
5. Gaifullin, Sergey (HSE University, Russia)
6. Jayanthan, A. V.  (IIT Madras, India)
7. Mukundan, Vivek (IIT Delhi, India)
8. Perepechko, Alexander (HSE University, Russia)
9. Puthenpurakal, Tony Joseph (IIT Bombay, India)
10. Roy, Sudeshna (TIFR, India)
11. Verma, Jugal (IIT Gandhinagar, India)

From Vietnam:

1. Trần Đỗ Minh Châu, Univ.  Education – Thai Nguyen Univ.
2. Đoàn Trung Cường, Institute of Mathematics, VAST
3. Lê Xuân Dũng, Univ. Hong Duc, Thanh Hoa
4. Lê Văn Định, FPT Univ.
5. Nguyễn Thị Ngọc Giao, Univ. of Science and Technology - The Univ. of Da Nang
6. Phạm Mỹ Hạnh, An Giang Univ.
7. Nguyễn Thị Ánh Hằng, Univ.  Education – Thai Nguyen Univ.
8. Nguyễn Thị Hằng, Univ.  Sciences – Thai Nguyen Univ.
9. Hà Thu Hiền, Hanoi National Univ. Education
10. Trương Thị Hiền, Univ. Hong Duc, Thanh Hoa
11. Lê Tuấn Hoa, Institute of Mathematics, VAST
12. Đỗ Trọng Hoàng, Hanoi Univ. Sciences and Techology
13. Nguyễn Đăng Hợp,  Institute of Mathematics, VAST
14. Dương Thu Hương, Thang Long Univ.
15. Cao Phạm Tân Khải, Hanoi,  Univ. Sciences and Techology (Student)
16. Đỗ Văn Kiên, Hanoi Univ. Education No. 2
17. Hà Minh Lam, Institute of Mathematics, VAST
18. Phan Văn Lộc, Hanoi Univ. Education No. 2
19. Đồng Hữu Mậu, Hanoi Metropolitan Univ.
20. Nguyễn Đình Nam, Ha Tinh Univ.
21. Phan Hồng Nam, Univ. Sciences – Thai Nguyen Univ.
22. Nguyễn Văn Ninh, Univ.  Education – Thai Nguyen Univ.
23. Lê Thanh Nhàn, MOET
24. Phạm Hùng Quý, FPT Univ.
25. Nguyễn Thanh Tâm, Hung Vuong Univ.
26. Phan Thị Thuỷ, Hanoi National Univ. Education
27. Doãn Quang Tiến,  Institute of Mathematics, VAST (Student, Pre-PhD Program)
28. Nguyễn Thị Trà, Hanoi Univ. Education No. 2
29. Ngô Việt Trung, Institute of Mathematics, VAST
30. Trần Nam Trung, Institute of Mathematics, VAST
31. Hoàng Lê Trường,  Institute of Mathematics, VAST
32. Vũ Quang Thanh,  Institute of Mathematics, VAST (Associate member)
33. Trần Đại Tân, Institute of Mathematics, VAST
34. Nguyễn Bích Vân, Institute of Mathematics, VAST
35. Đỗ Hoàng Việt, Institute of Mathematics, VAST (Master Student)
36. Hoàng Ngọc Yến, Univ.  Education – Thai Nguyen Univ.