Issue 1, Volume 49, March 2024

2024

1. Preface Preface PDF

   1
2. On the Infinitely Generated Locus of Frobenius Algebras of Rings of Prime Characteristic Alberto F. Boix , Danny A. J. Gómez–Ramírez , Santiago Zarzuela PDF

   ▸ Abstract

   Let $R$ be a commutative Noetherian ring of prime characteristic $p$. The main goal of this paper is to study in some detail when $$\{\mathfrak {p}\in {\text {Spec}} (R)\,:\, \mathcal {F}^{E_{\mathfrak {p}}}\text { is finitely generated as a ring over its degree zero piece}\}$$ is an open set in the Zariski topology, where $\mathcal {F}^{E_{\mathfrak {p}}}$ denotes the Frobenius algebra attached to the injective hull of the residue field of $R_{\mathfrak {p}}.$ We show that this is true when $R$ is a Stanley–Reisner ring; moreover, in this case, we explicitly compute its closed complement, providing an algorithmic method for doing so.

   3-18
3. Taylor Polynomials of Rational Functions Aldo Conca , Simone Naldi , Giorgio Ottaviani , Bernd Sturmfels PDF

   ▸ Abstract

   A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on Hankel matrices. Inversion of the natural parametrization is known as Padé approximation. We study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters. We explain this with Fröberg’s Conjecture in commutative algebra.

   19-37
4. Bounds for the Hilbert-Kunz Multiplicity of Singular Rings Nicholas O. Cox-Steib , Ian M. Aberbach PDF

   ▸ Abstract

   In this paper, we prove that the Watanabe-Yoshida conjecture holds up to dimension 7. Our primary new tool is a function, $\varphi_J(R;z^t),$ that interpolates between the Hilbert-Kunz multiplicities of a base ring, $R$, and various radical extensions, $R_n$. We prove that this function is concave and show that its rate of growth is related to the size of $e_{\textrm{HK}}(R)$. We combine techniques from Celikbas et al. (Nagoya Math. J. 205, 149–165, 2012) and Aberbach and Enescu (Nagoya Math. J. 212, 59–85, 2013) to get effective lower bounds for $\varphi ,$ which translate to improved bounds on the size of Hilbert-Kunz multiplicities of singular rings. The improved inequalities are powerful enough to show that the conjecture of Watanabe and Yoshida holds in dimension 7.

   39-60
5. The Asymptotic Samuel Function of a Filtration Steven Dale Cutkosky , Smita Praharaj PDF

   ▸ Abstract

   We extend the asymptotic Samuel function of an ideal to a filtration and show that many of the good properties of this function for an ideal are true for filtrations. There are, however, interesting differences, which we explore. We study the notion of projective equivalence of filtrations and the relation between the asymptotic Samuel function and the multiplicity of a filtration. We further consider the case of discrete valued filtrations and show that they have particularly nice properties.

   61-81
6. Cone of Maximal Subextensions of the Plurisubharmonic Functions Le Mau Hai , Pham Hoang Hiep , Trinh Tung PDF

   ▸ Abstract

   In this note, we give some results on maximal subextensions of plurisubharmonic functions on hyperconvex domains in $\mathbb C^n$ and introduce the notion about cone of maximal subextensions of plurisubharmonic functions. Furthermore, we establish the invariant of this cone through proper holomorphic surjections.

   83-97
7. Uniform Symbolic Topologies and Hypersurfaces Craig Huneke , Daniel Katz PDF

   ▸ Abstract

   We study the question of which rings, and which families of ideals, have uniform symbolic topologies. In particular, we show that the uniform symbolic topology property holds for all dimension one primes in any normal complete local domain, provided dimension one primes in hypersurfaces have the uniform symbolic topology property. We also discuss bootstrapping techniques and provide a strong bootstrapping statement in positive characteristic. We apply these techniques to give families of primes in hypersurfaces of positive characteristic which have uniform symbolic topologies.

   99-113
8. On Nilpotent-invariant One-sided Ideals Truong Cong Quynh , Truong Thi Thuy Van PDF

   ▸ Abstract

   The notion of a nilpotent-invariant module was introduced and thoroughly investigated in Koşan and Quynh (Comm. Algebra 45, 2775–2782 2017) as a proper extension of an automorphism-invariant module. In this paper a ring is called a right $\mathfrak {n}$-ring if every right ideal is nilpotent-invariant. We show that a right $\mathfrak {n}$-ring is the direct sum of a square full semisimple artinian ring and a right square-free ring. Moreover, right $\mathfrak {n}$-rings are shown to be stably finite, and if the ring is also an exchange ring then it satisfies the substitution property, has stable range 1. These results are non-trivial extensions of similar ones on rings every right ideal is automorphism-invariant.

   115-128
9. On Segre Products, $F$-regularity, and Finite Frobenius Representation Type Anurag K. Singh , Kei-ichi Watanabe PDF

   ▸ Abstract

   We study the behavior of various properties of commutative Noetherian rings under Segre products, with a special focus on properties in positive prime characteristic defined using the Frobenius endomorphism. Specifically, we construct normal graded rings of finite Frobenius representation type that are not Cohen-Macaulay.

   129-138
10. On the Set of Chern Numbers in Local Rings Le Truong Hoang , Ngoc Yen Hoang PDF

    ▸ Abstract

    The purpose of this paper is to characterize Noetherian local rings $(R, \mathfrak {m})$ such that the Chern numbers of certain $\mathfrak {m}$-primary ideals in $R$ are bounded above or range among only finitely many values. Consequently, we characterize the Gorensteinness, Cohen-Macaulayness, and generalized Cohen-Macaulayness of local rings in terms of the behavior of their Chern numbers.

    139-157