Acta Mathematica Vietnamica
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Issue 1, Volume 47, March 2022
2022
Special Issue: Algebra, Geometry and Related Topics
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Skew Polynomial Rings: the Schreier Technique Phạm Ngọc Ánh PDF5-17
▸ Abstract
Schreier bases are introduced and used to show that skew polynomial rings are free ideal rings, i.e., rings whose one-sided ideals are free of unique rank, as well as to compute a rank of one-sided ideals together with a description of corresponding bases. The latter fact, a so-called Schreier-Lewin formula (Lewin Trans. Am. Math. Soc. 145, 455–465 1969), is a basic tool determining a module type of perfect localizations which reveal a close connection between classical Leavitt algebras, skew polynomial rings, and free associative algebras.
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Maximal Generating Degrees of Powers of Homogeneous Ideals Le Tuan Hoa PDF19-37
▸ Abstract
The degree excess function $\epsilon(I,n)$ is the difference between the maximal generating degree $d(I^n)$ of the n-th power of a homogeneous ideal $I$ of a polynomial ring and $p(I)n$, where $p(I)$ is the leading coefficient of the asymptotically linear function $d(I^n)$. It is shown that any non-increasing numerical function can be realized as a degree excess function, and there is a monomial ideal I whose $\epsilon(I,n)$ has exactly a given number of local maxima. In the case of monomial ideals, an upper bound on $\epsilon(I,n)$ is provided. As an application, it is shown that in the worst case, the so-called stability index of the Castelnuovo-Mumford regularity of a monomial ideal $$ must be at least an exponential function of the number of variables.
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Problems About Torsors over Regular Rings Kęstutis Česnavičius PDF39-107
▸ Abstract
We overview a web of conjectures about torsors under reductive groups over regular rings and survey some techniques that have been used for making progress on such problems.
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On Shifted Principles for Attached Primes of the Top Local Cohomology Modules Tran Do Minh Chau , Nguyen Thi Kieu Nga , Le Thanh Nhan PDF109-121
▸ Abstract
Let $(R, \mathfrak{m})$ be a Noetherian local ring, let $I$ be an ideal of $R$, and let $M$ be a finitely generated $R$-module with $d=\dim(M)$. In this paper, we establish shifted principles under localization and completion for attached primes of the top local cohomology module $H^d_I(M)$. We characterize the catenarity, the weak going-up property, and the strong Lichstenbaum-Hartshorne vanishing property of the base ring $R$ in terms of these shifted principles of the top local cohomology modules.
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Comparisons Between Annihilators of Tor and Ext Souvik Dey , Ryo Takahashi PDF123-139
▸ Abstract
In this paper, we compare annihilators of Tor and Ext modules of finitely generated modules over a commutative noetherian ring. For local Cohen–Macaulay rings, one of our results refines a theorem of Dao and Takahashi.
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F-Stable Secondary Representations and Deformation of F-Injectivity Alessandro De Stefani , Linquan Ma PDF141-150
▸ Abstract
We prove that deformation of F-injectivity holds for local rings $(R,\mathfrak{m})$ that admit secondary representations of $H^i_{\mathfrak{m}}(R)$ which are stable under the natural Frobenius action. As a consequence, F-injectivity deforms when $(R,\mathfrak{m})$ is sequentially Cohen–Macaulay (or more generally when all the local cohomology modules $H^i_{\mathfrak{m}}(R)$ have no embedded attached primes). We obtain some additional cases if $R/\mathfrak{m}$ is perfect or if $R$ is $\mathbb N$-graded.
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Frobenius Test Exponent for Ideals Generated by Filter Regular Sequences Duong Thi Huong , Pham Hung Quy PDF151-159
▸ Abstract
Let $(R,\mathfrak m)$ be a Noetherian local ring of prime characteristic $p>0$, and $t$ an integer such that $H_{\mathfrak m}^{j}(R)/0^{F}_{H^{j}_{\mathfrak m}(R)}$ has finite length for all $j
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Canonical Stretched Rings Nguyen Thi Anh Hang , Do Van Kien , Hoang Le Truong PDF161-179
▸ Abstract
In this paper, we introduce the concept of canonical stretched rings, sparse stretched rings, and maximum sparse ideals. Then, we give characterizations of canonical stretched rings and sparse stretched rings, and a characterization of Gorenstein rings in terms of their maximum sparse ideals. Several explicit examples are provided along the paper to illustrate such rings.
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A Brief Survey on Pure Cohen–Macaulayness in a Fixed Codimension M. R. Pournaki , M. Poursoltani , N. Terai , S. Yassemi PDF181-196
▸ Abstract
A concept of Cohen–Macaulay in codimension $t$ is defined and characterized for arbitrary finitely generated modules and coherent sheaves by Miller, Novik, and Swartz in 2011. Soon after, Haghighi, Yassemi, and Zaare-Nahandi defined and studied $\mathrm{CM}_t$ simplicial complexes, which is the pure version of the abovementioned concept and naturally generalizes both Cohen–Macaulay and Buchsbaum properties. The purpose of this paper is to survey briefly recent results of $\mathrm{CM}_t$ simplicial complexes.
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Maximal Subgroups of Almost Subnormal Subgroups in Division Rings Bui Xuan Hai PDF197-209
▸ Abstract
In this paper, we investigate maximal subgroups of an almost subnormal subgroup $G$ in a division ring $D$ whose center is infinite. Among results, we prove that if $M$ is such a maximal subgroup, then $M$ is abelian provided $M$ is either locally finite or $FC$-group, and $D$ is weakly locally finite. Also, we prove a theorem on the existence of non-cyclic free subgroups of a maximal subgroup $M$ in such a $G$. -
Zeros of Differential Polynomials of Meromorphic Functions Ta Thi Hoai An , Nguyen Viet Phuong PDF211-221
▸ Abstract
Let $f$ be a transcendental meromorphic function on $\mathbb C, k$ be a positive integer, and $Q_0,Q_1,\dots,Q_k$ be polynomials in $\mathbb C[z]$. In this paper, we will prove that the frequency of distinct poles of $f$ is governed by the frequency of zeros of the differential polynomial form $Q_0(f)Q_1(f^\prime)\dots Q_k(f^{(k})$ in $f$. We will also prove that the Nevanlinna defect of the differential polynomial form $Q_0(f)Q_1(f^\prime)\dots Q_k(f^{(k})$ in $f$ satisfies $$\sum\limits_{a\in\mathbb{C}}\delta\left( a,Q_{0}(f)Q_{1}(f^{\prime}){\dots} Q_{k}\left( f^{(k)}\right)\right)\leq 1$$ with suitable conditions on $k$ and the degree of the polynomials. Thus, our work is a generalization of Mues’s conjecture and Goldberg’s conjecture for the more general differential polynomials.
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Lelong Number and the Log Canonical Thresholds of Plurisubharmonic Functions on Analytic Subsets Le Mau Hai , Pham Hoang Hiep , Trinh Tung PDF223-241
▸ Abstract
The aim of this paper is to introduce the notion of Lelong number and the log canonical thresholds of plurisubharmonic functions on analytic subsets $A$ in an open subset $\Omega$ of $\mathbb C^n$. Next, we establish some results about the relationship between these quantities in the relation with the analyticity of $A$.
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“Infinite” Properties of Certain Local Cohomology Modules of Determinantal Rings Peter Schenzel PDF243-250
▸ Abstract
For given integers $m,n\geq 2$ there are examples of ideals $I$ of complete determinantal local ring $(R, \mathfrak{m}),$ $ \dim R=m+n-1, $ $\mathrm{grade\,}I=n-1$, with the canonical module $\omega_R$ and the property that the socle dimensions of $H_I^{m+n-2}(\omega_R)$ and $H^m_{\mathfrak{m}}(H^{n-1}_I(\omega_R))$ are not finite. In the case of $m=n$, i.e., a Gorenstein ring, the socle dimensions provide further information about the $\tau$-numbers as studied in Mahmood and Schenzel (J. Algebra 372, 56–67, 10). Moreover, the endomorphism ring of $H^{n-1}_I(\omega_R)$ is studied and shown to be an $R$-algebra of finite type but not finitely generated as $R$-module generalizing an example of Schenzel (J. Algebra 344, 229–245, 15).
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The First Hilbert Coefficient of Stretched Ideals Kazuho Ozeki PDF251-267
▸ Abstract
In this paper, we explore the almost Cohen-Macaulayness of the associated graded ring of stretched $\mathfrak{m}$-primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local ring $(A, \mathfrak{m})$. In particular, we explore the structure of stretched $\mathfrak{m}$-primary ideals satisfying the equality $\mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+4$, where $\mathrm{e}_0(I)$ and $\mathrm{e}_1(I)$ denote the multiplicity and the first Hilbert coefficient, respectively.
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Formal Partial Derivatives and the Commutativity of Squaring Operations Nguyễn Đ. Ngà , Ngô A. Tuấn PDF269-277
▸ Abstract
We introduce the notion of (formal) partial derivative and develop an application of it to get a new proof for the commutativity of the classical squaring and the Kameko squaring.
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Faithfullness of Geometric Action of Skein Algebras Thang T. Q. Lê PDF279-291
▸ Abstract
We show that the action of the Kauffman bracket skein algebra of a surface Σ on the skein module of the handlebody bounded by Σ is faithful if and only if the quantum parameter is not a root of 1.
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Odd-degree Rational Irreducible Characters Pham Huu Tiep , Hung P. Tong-Viet PDF293-304
▸ Abstract
We study finite groups whose rational-valued irreducible characters are all of odd degrees. We conjecture that in such groups, all rational elements must be 2-elements.
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Degrees $d \geqslant \left (\sqrt {n} \log n ight )^{n}$ and $d \geqslant \left (n \log n ight )^{n}$ in the Conjectures of Green-Griffiths and of Kobayashi Joël Merker , The-Anh Ta PDF305-358
▸ Abstract
Once first answers in any dimension to the Green-Griffiths and Kobayashi conjectures for generic algebraic hypersurfaces $\mathbb {X}^{n-1} \subset \mathbb {P}^{n}(\mathbb {C})$ have been reached, the principal goal is to decrease (to improve) the degree bounds, knowing that the ‘celestial’ horizon lies near $d\geqslant 2n$.
For Green-Griffiths algebraic degeneracy of entire holomorphic curves, we obtain $$d\geqslant (\sqrt{n}\log n)^n,$$ and for Kobayashi-hyperbolicity (constancy of entire curves), we obtain $$ d\geqslant \left (n\log n\right )^{n}. $$ The latter improves $d \geqslant n^{2n}$ obtained by Merker in arxiv.org/1807/11309/.
Admitting a certain technical conjecture $I_{0} \geqslant \widetilde {I}_{0} $, the method employed (Diverio-Merker-Rousseau, Bérczi, Darondeau) conducts to constant power $n$, namely to $$ d\geqslant 2^{5n}\quad \text {and, respectively, to}\quad d\geqslant 4^{5n}. $$
In Spring 2021, a forthcoming prepublication based on intensive computer explorations will present several subconjectures supporting the belief that $I_{0} \geqslant \widetilde {I}_{0}$ a conjecture which will be established up to dimension $n=50$.
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On Monomial Golod Ideals Hailong Dao , Alessandro De Stefani PDF359-367
▸ Abstract
We study ideal-theoretic conditions for a monomial ideal to be Golod. For ideals in a polynomial ring in three variables, our criteria give a complete characterization. Over such rings, we show that the product of two monomial ideals is Golod.
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Regularity and h-polynomials of Binomial Edge Ideals Takayuki Hibi , Kazunori Matsuda PDF369-374
▸ Abstract
Let $G$ be a finite simple graph on the vertex set $[n]=\{1,\dots,n\}$ and $K[\mathbf{x},\mathbf{y}]=K[x_1,\dots,x_n,y_1,\dots,y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $\deg x_i=\deg y_j=1$. The binomial edge ideal of $G$ is the binomial ideal $J_G \subset K[\mathbf{x},\mathbf{y}]$ which is generated by those binomials $x_iy_j-x_jy_i$ for which $\{i, j\}$ is an edge of $G$. The Hilbert series $H_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )$ of $K[\mathbf{x},\mathbf{y}]/J_G$ is of the form $H_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )/(1 - \lambda )^{d} $ where $d = \dim K[\mathbf {x}, \mathbf { y}]/J_{G} $ and where $h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = h_{0} + h_{1}\lambda + h_{2}\lambda ^{2} + {\cdots } + h_{s}\lambda ^{s} $ with each $h_{i} \in \mathbb Z$ and with $h_s\not=0$ is the $h$-polynomial of $K[\mathbf{x},\mathbf{y}]/J_G$. It is known that, when $K[\mathbf{x},\mathbf{y}]/J_G$ is Cohen–Macaulay, one has $\operatorname {reg}(K[\mathbf {x}, \mathbf {y}]/J_{G}) = \deg h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda )$, where $\mathrm{reg}(K[\mathbf{x},\mathbf{y}]/J_G)$ is the (Castelnuovo–Mumford) regularity of $K[\mathbf{x},\mathbf{y}]/J_G$. In the present paper, given arbitrary integers $r$ and $s$ with $2\leq r\leq s$, a finite simple graph $G$ for which $\mathrm{reg}(K[\mathbf{x},\mathbf{y}]/J_G)=r$ and $\deg h_{K[\mathbf {x}, \mathbf {y}]/J_{G}}(\lambda ) = s$ will be constructed.