arXiv subjects

J. A. Lima

Publications and source records attributed to J. A. Lima.

4 recordsLinked to original sources

arXiv2025

Hilbert-Kunz Multiplicity of Fiber Product Rings and Nagata Idealizations

The main purpose of this paper is to provide formulas for the Hilbert-Kunz multiplicity of fiber product rings and Nagata idealizations. We give explicit formulas for the Hilbert-Kunz multiplicity of a fiber product $R \times_T S$, where $R$, $S$, and $T$ are Noetherian local rings sharing the same characteristic and residue field. We compute the Hilbert-Kunz multiplicity of a general Nagata idealization $R \ltimes M$, where $M$ is a finitely generated $R$-module. Additionally, we provide examples, structural results and establish new bounds for the Hilbert-Kunz multiplicity.

math.AC↗

arXiv2025

On Betti numbers for symmetric powers of modules

Let $M$ be a finitely generated module over a local ring $(R,\mathfrak{m})$. By $\mathcal{S}_j(M)$, we denote the $j$th symmetric power of $M$ ($j$th graded component of the symmetric algebra $\mathcal{S}_R(M)$). The purpose of this paper is to investigate the minimal free resolutions $\mathcal{S}_j(M)$ as $R$-module for each $j\geq 2$ and determine the Betti numbers of $\mathcal{S}_j(M)$ in terms of the Betti numbers of $M$. This has some applications, for example for linear type ideals $I$, we obtain formulas of the Betti numbers $I^j$ in terms of the Betti numbers of $I$. In addition, we establish upper and lower bounds of Betti numbers of $\mathcal{S}_j(M)$ in terms of Betti numbers of $M$. In particular, obtain some applications of the famous Buchsbaum-Eisenbud-Horrocks conjecture.

math.AC↗

arXiv2024

On General fiber product rings, Poincaré series and their structure

The present paper deals with the investigation of the structure of general fiber product rings $R\times_TS$, where $R$, $S$ and $T$ are local rings with common residue field. We show that the Poincaré series of any $R$-module over the fiber product ring $R\times_TS$ is bounded by a rational function. In addition, we give a description of ${\rm depth}(R\times_TS)$, which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over $R\times_TS$ obtained, we provide certain cases of the Cohen-Macaulayness of $R\times_TS$ and, in particular, we show that $R\times_TS$ is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over $R\times_TS$ are also established.

math.AC↗

arXiv2024

On the Gluing of germs of complex analytic spaces, Betti numbers and their structure

In this paper we introduce new classes of gluing of complex analytic spaces germs, called weakly large, large and strongly large. We give a description of their Poincaré series and, as applications, we give numerical criteria to determine when these classes of gluing of germs of complex analytic spaces are smooth, singular, complete intersections and Gorenstein in terms of their Betti numbers. In particular, we show that the gluing of the same germ of complex analytic space along of any subspace is always a singular germ.

math.AG↗