arXiv subjects

Alejandro Argudin Monroy

Publications and source records attributed to Alejandro Argudin Monroy.

2 recordsLinked to original sources

arXiv2026

Tilting objects in the extended heart of a $t$-structure

Building on the recent work of Adachi, Enomoto and Tsukamoto on a generalization of the Happel-Reiten-Smalø tilting process, we study extended tilting objects in extriangulated categories with negative first extension. These objects coincide with the 1-tilting objects in abelian categories as in the work of Parra, Saor{í}n and Virili. We will be particularly interested in the case where the extriangulated category in question is the heart $\mathcal{H}_{[\mathbf{t}_{1},\mathbf{t}_{2}]}$ of an interval of $t$-structures $[\mathbf{t}_{1},\mathbf{t}_{2}]$. Our main results consist of a characterization of the extended tilting objects of a heart $\mathcal{H}_{[\mathbf{t}_{1},\mathbf{t}_{2}]}$ for the case when $\text{\ensuremath{\mathbf{t}}}_{2}\leqΣ^{-1}\mathbf{t}_{1}$, and another one for the case when $Σ^{-2}\mathbf{t}_{1}<\mathbf{t}_{2}$. In the first one, we give conditions for these tilting objects to coincide with the quasi-tilting objects of the abelian category $\mathcal{H}_{[\mathbf{t}_{1},Σ^{-1}\mathbf{t}_{1}]}$. In the second one, it is given conditions for these to coincide with projective generators in the extriangulated category $\mathcal{H}_{[\mathbf{t}_{1},Σ\mathbf{t}_{2}]}$

math.RT↗

arXiv2023

Relative tilting theory in abelian categories II: $n$-$\mathcal{X}$-tilting theory

We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander-Solberg relative tilting modules on Artin algebras to infinitely generated tilting modules on arbitrary rings. Furthermore, we see that it presents a tool for developing new tilting theories in categories that can be embedded nicely in an abelian category. In particular, we will show how the tilting theory in exact categories built this way, coincides with tilting objects in extriangulated categories introduced recently. We will review Bazzoni\textquoteright s tilting characterization, the relative homological dimensions on the induced tilting classes and parametrise certain cotorsion-like pairs by using $n$-$\mathcal{X}$-tilting classes. As an application, we show how to construct relative tilting classes and cotorsion pairs in $\operatorname{Rep}(Q,\mathcal{C})$ (the category of representations of a quiver $Q$ in an abelian category $\mathcal{C}$) from tilting classes in $\mathcal{C},$ where $Q$ is finite-cone-shape.

math.RT↗