Related papers: Phantom stable category of $n$-Frobenius categorie…

Let $n$ be a non-negative integer. {Motivated by the universal property of the stable category of Frobenius categories, the authors in \cite{bfss} extended the stabilization of Frobenius categories to $n$-Frobenius categories, and called it…

We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius…

In a recent article Facchini and Finocchiaro considered a natural pretorsion theory in the category of preordered sets inducing a corresponding stable category. In the present work we propose an alternative construction of the stable…

For a Frobenius abelian category $\mathcal{A}$, we show that the category ${\rm Mon}(\mathcal{A})$ of monomorphisms in $\mathcal{A}$ is a Frobenius exact category; the associated stable category $\underline{\rm Mon}(\mathcal{A})$ modulo…

A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity…

We prove that the stable category associated with the category $\mathsf{PreOrd}(\mathbb C)$ of internal preorders in a pretopos $\mathbb C$ satisfies a universal property. The canonical functor from $\mathsf{PreOrd}(\mathbb C)$ to the…

We introduce continuous Frobenius categories. These are topological categories which are constructed using representations of the circle over a discrete valuation ring. We show that they are Krull-Schmidt with one indecomposable object for…

Given a complete, cocomplete category $\mathcal C$, we investigate the problem of describing those small categories $I$ such that the diagonal functor $\Delta:\mathcal C\to {\rm Functors}(I,\mathcal C)$ is a Frobenius functor. This…

From certain triangle functors, called non-negative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the…

We give a new characterization of silting subcategories in the stable category of a Frobenius extriangulated category, generalizing the result of Di et al. (J. Algebra 525 (2019) 42-63) about the Auslander-Reiten type correspondence for…

We call a tensor functor $F:\mathcal{C}\to\mathcal{D}$ between finite tensor categories $\otimes$-Frobenius if its left and right adjoints are isomorphic as $\mathcal{C}$-bimodule functors. We give several characterizations of this notion…

We develop a theory of Frobenius functors for symmetric tensor categories (STC) $\mathcal{C}$ over a field $\bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear…

The main goal of this paper is to prove the following: for a triangulated category $ \underline{C}$ and $E\subset \operatorname{Obj} \underline{C}$ there exists a cohomological functor $F$ (with values in some abelian category) such that…

Let $\mathcal{C}$ be an additive category. The nilpotent category $\mathrm{Nil} (\mathcal{C})$ of $\mathcal{C}$, consists of objects pairs $(X, x)$ with $X\in\mathcal{C}, x\in\mathrm{End}_{\mathcal{C}}(X)$ such that $x^n=0$ for some…

This paper consists of three results on Frobenius categories: (1) we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an…

In this paper, we first introduce stable functors with respect to a preenveloping/precovering subcategory and investigate some of their properties. Using that we then introduce and study a relative complete cohomology theory in abelian…

We consider several types of non-existence theorems for functors. For example, there are no nontrivial functors from the category of groups (or the category of pointed sets, or vector spaces) to any small category. Another type of questions…

Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be a commutative ring and let $\mathcal{F}$ be an algebraically closed field of characteristic $0$. We introduce the category $\overline{\mathcal{F}_{Rpp_k}}$ of…

We introduce a notion of total acyclicity associated to a subcategory of an abelian category and consider the Gorenstein objects they define. These Gorenstein objects form a Frobenius category, whose induced stable category is equivalent to…

We establish relations between Gorenstein projective precovers linked by Frobenius functors. This is motivated by an open problem that how to find general classes of rings for which modules have Gorenstein projective precovers. It is shown…