arXiv subjects

Lorenzo Martini

Publications and source records attributed to Lorenzo Martini.

5 recordsLinked to original sources

arXiv2025

Locally finitely presented Grothendieck categories with a flat generator

A problem raised by Cuadra and Simson in 2007 asks whether any locally finitely presented Grothendieck category with enough flat objects also has enough projectives. In this paper, we start from a key observation: a locally finitely presented Grothendieck category has enough flat objects if, and only if, it has exact products. This enables several equivalent reformulations of the problem, allowing us to identify a counterexample (thus providing a negative solution to the problem), while also connecting it to a classical ring-theoretical question posed by Miller in 1975, and even to the Telescope Conjecture for compactly generated triangulated categories. Moreover, we describe several classes of Grothendieck categories where the problem can be answered affirmatively. For example, we show that a locally finitely presented Grothendieck category whose category of finitely presented objects is Krull--Schmidt has enough flats if, and only if, it is generated by a family of finitely generated projectives.

math.CT↗

arXiv2025

inMOTIFin: a lightweight end-to-end simulation software for regulatory sequences

The accurate development, assessment, interpretation, and benchmarking of bioinformatics frameworks for analyzing transcriptional regulatory grammars rely on controlled simulations to validate the underlying methods. However, existing simulators often lack end-to-end flexibility or ease of integration, which limits their practical use. We present inMOTIFin, a lightweight, modular, and user-friendly Python-based software that addresses these gaps by providing versatile and efficient simulation and modification of DNA regulatory sequences. inMOTIFin enables users to simulate or modify regulatory sequences efficiently for the customizable generation of motifs and insertion of motif instances with precise control over their positions, co-occurrences, and spacing, as well as direct modification of real sequences, facilitating a comprehensive evaluation of motif-based methods and interpretation tools. We demonstrate inMOTIFin applications for the assessment of de novo motif discovery prediction, the analysis of transcription factor cooperativity, and the support of explainability analyses for deep learning models. inMOTIFin ensures robust and reproducible analyses for studying transcriptional regulatory grammars. inMOTIFin is available at PyPI https://pypi.org/project/inMOTIFin/ and Docker Hub https://hub.docker.com/r/cbgr/inmotifin. Detailed documentation is available at https://inmotifin.readthedocs.io/en/latest/. The code for use case analyses is available at https://bitbucket.org/CBGR/inmotifin_evaluation/src/main/.

q-bio.GN↗

arXiv2025

Cotilting duality for Artinian rings

A classical result due to Morita and Azumaya establishes that given two arbitrary rings, any duality between their finitely generated modules is representable by a faithfully balanced bimodule which is a finitely generated injective cogenerator of both rings and, equivalently, these latter are one-sided artinian. We extend this well-known result to the case of a cotilting bimodule, by analysing the duality it represents in the bounded derived categories of the given rings.

math.RA↗

arXiv2024

Product-complete tilting complexes and Cohen-Macaulay hearts

We show that the cotilting heart associated to a tilting complex $T$ is a locally coherent and locally coperfect Grothendieck category (i.e. an Ind-completion of a small artinian abelian category) if and only if $T$ is product-complete. We then apply this to the specific setting of the derived category of a commutative noetherian ring $R$. If $\dim(R)<\infty$, we show that there is a derived duality $\mathcal{D}^b_{fg}(R) \cong \mathcal{D}^b(\mathcal{B})^{op}$ between $\mathrm{mod} R$ and a noetherian abelian category $\mathcal{B}$ if and only if $R$ is a homomorphic image of a Cohen--Macaulay ring. Along the way, we obtain new insights about t-structures in $\mathcal{D}^b_{fg}(R)$. In the final part, we apply our results to obtain a new characterization of the class of those finite-dimensional Noetherian rings that admit a Gorenstein complex.

math.RT↗

arXiv2022

Local Coherence of Hearts Associated with Thomason Filtrations

Any Thomason filtration of a commutative ring yields (at least) two t-structures in the derived category of the ring, one of which is compactly generated [Hrb20,HHZ21]. We study the hearts of these two t-structures and prove that they coincide in case of a weakly bounded below filtration. Prompted by [SS20], in which it is proved that the heart of a compactly generated t-structure in a triangulated category with coproduct is a locally finitely presented Grothendieck category, we study the local coherence of the hearts associated with a weakly bounded below Thomason filtration, achieving a useful recursive characterisation in case of a finite length filtration. Low length cases involve hereditary torsion classes of finite type of the ring, and even their Happel-Reiten-Smalo hearts; in these cases, the relevant characterisations are given by few module-theoretic conditions.

math.RT↗