arXiv subjects

Berrin Şentürk

Publications and source records attributed to Berrin Şentürk.

4 recordsLinked to original sources

arXiv2026

On the Regularity and Clean Properties of Matrices over Dual Numbers

In this paper, we study regularity and clean-type decompositions for matrices over dual numbers. We characterize the von Neumann regularity of dual matrices by establishing a necessary and sufficient compatibility condition between their real and dual components. We determine the conditions under which a dual matrix $\mathcal{M} = A + Bε$ is regular, $π$-regular, nil-clean, or strongly nil-clean. Using the solvability of Sylvester matrix equations, we show that the matrix ring over dual numbers $M_n(\mathbb{D})$ is strongly $π$-regular and strongly clean. Moreover, $M_n(\mathbb{D})$ is $π$-regular, clean, $r$-clean, strongly $r$-clean, $\mathrm{NR}$-clean, and strongly $\mathrm{NR}$-clean. On the other hand, $M_n(\mathbb{D})$ fails to be regular, strongly regular, nil-clean, strongly nil-clean and uniquely $\mathrm{NR}$-clean, although it inherits several strong decomposition properties from $M_n(\mathbb{R})$.

math.RA↗

arXiv2025

The rank 8 case of a conjecture on square-zero upper triangular matrices

Let $A$ be the polynomial algebra in $r$ variables with coefficients in an algebraically closed field $k$. When the characteristic of $k$ is $2$, Carlsson conjectured that any $\mathrm{dg}$-$A$-module that is free of rank $N$ as an $A$-module and whose homology is nontrivial and finite dimensional as a $k$-vector space satisfies $N\geq 2^r$. In this paper, we examine a stronger conjecture concerning varieties of square-zero upper triangular $N\times N$ matrices. Stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when $N = 8$ without any restriction on the characteristic of $k$. This result also verifies that if $X$ is a product of $3$ spheres of any dimensions, then the elementary abelian $2$-group of rank $4$ cannot act freely on $X$.

math.AC↗

arXiv2022

Minimal models of some differential graded modules

Minimal models of chain complexes associated with free torus actions on spaces have been extensively studied in the literature. In this paper, we discuss these constructions using the language of operads. The main goal of this paper is to define a new Koszul operad that has projections onto several of the operads used in these minimal model constructions.

math.AT↗

arXiv2018

Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

Let $k$ be an algebraically closed field and $A$ the polynomial algebra in $r$ variables with coefficients in $k$. In case the characteristic of $k$ is $2$, Carlsson conjectured that for any $DG$-$A$-module $M$ of dimension $N$ as a free $A$-module, if the homology of $M$ is nontrivial and finite dimensional as a $k$-vector space, then $2^r\leq N$. Here we state a stronger conjecture about varieties of square-zero upper-triangular $N\times N$ matrices with entries in $A$. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when $N < 8$ or $r < 3$ without any restriction on the characteristic of $k$. As a consequence, we attain a new proof for many of the known cases of Carlsson's conjecture and give new results when $N > 4$ and $r = 2$.

math.AC↗