Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.01483

Relative quasi-Gorenstein homological dimensions in extriangulated categories

Zhenggang He·Jifen Liu·Jiaqun Wei
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $(\mathcal{C},\mathbb{E},\mathfrak s)$ be an extriangulated category equipped with a proper class $ξ$ of $\mathbb E$-triangles.We establish stability under iteration for the quasi-$ξ$-Gorenstein projective and injective subcategories and extend the relative Ext-vanishing criteria for their finite homological dimensions to larger coefficient subcategories. An example in a bounded homotopy category shows that $\mathcal{P}(ξ)\subsetneq\mathcal{QGP}(ξ)\subsetneq\mathcal{GP}(ξ)$. For module categories, every quasi-Gorenstein projective module over a left perfect ring or a commutative Noetherian ring of finite Krull dimension is projective. When a commutative ring admits a finite chain of trace and nilpotent quotients ending in an Artinian ring and the endomorphism rings at the trace steps have finite right global dimension, we construct finite families of quasi-projective test modules that detect prescribed upper bounds on projective dimension for modules of bounded cardinality through Ext vanishing in finitely many consecutive degrees.

Explore related subjects

math.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhenggang He, Jifen Liu, Jiaqun Wei. 2026-10-01. Relative quasi-Gorenstein homological dimensions in extriangulated categories. https://arxiv.org/abs/2610.01483

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Categorical Framework for Typed Extensional and Intensional Models in Formal Semantics

A morphism of semantic models is a map under which denotation is invariant. We characterize such morphisms for typed extensional and intensional semantics with finitely many index sorts. Maps on entities and frames induce logical relations on all types, and the fundamental lemma yields invariance when lexical constants are related and frame maps satisfy the appropriate boundedness conditions. The lexicon determines the resulting class of morphisms. Among entity maps, universal and existential determiners force surjectivity, proportional determiners additionally constrain fiber sizes, counting determiners force bijectivity, and identity forces injectivity. Modal operators impose parallel conditions on frame maps: necessity requires bounded morphisms, counting modalities require local bijectivity, and proportional modalities require preservation and reflection of majorities over accessible sets. For lexicons containing the relevant counting determiners, morphisms are bijective on entities. The extensional models then form a groupoid. Intensional morphisms may still act nontrivially on frames. Induced logical relations can fail to compose. We, therefore, identify a composability condition on lexical types under which the models form a category, and polarized types and the standard lexicons considered here satisfy it. We identify ordinary extensional models with the zero-index case for extensional lexicons, determine when index sorts collapse, analyze the effect of enriching the lexicon with index quantification and accessibility, and show that product frames arise from a faithful functor on the corresponding frame categories.

math.CT↗
arXiv2026

The algebras of the Giry monad

The algebras of the Giry monad on the category of measurable spaces are derived. These algebras, which we denote as $\mathbf{Cvx}_{Meas}$, are measurable convex spaces satisfying three properties: (1) all the affine sum operations are measurable functions, (2) $X$ is coseparated by affine measurable maps to $\mathbb{R}_{\infty}$, and (3) $X$ satisfies a fullness property. The category of algebras has as morphisms affine measurable functions.

math.CT↗
arXiv2026

Generalised ultracategories and conceptual completeness of geometric logic

We introduce the theory of generalised ultracategories, these are relational extensions to ultracategories as defined by Lurie. An essential example of generalised ultracategories are topological spaces, and these play a fundamental role in the theory of generalised ultracategories. Another example of these generalised ultracategories is points of toposes. In this paper, we show a conceptual completeness theorem for toposes with enough points, stating that any such topos can be reconstructed from its generalised ultracategory of points. This is done by considering left ultrafunctors from topological spaces to the category of points and paralleling this construction with another known fundamental result in topos theory, namely that any topos with enough points is a colimit of a topological groupoid.

math.CT↗