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We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of $T$. Given a chain of…

In the last few years, Lopez-Permouth and several collaborators have introduced a new approach in the study of the classical projectivity, injectivity and flatness of modules. This way, they introduced subprojectivity domains of modules as…

We study abelian envelopes for pseudo-tensor categories with the property that every object in the envelope is a quotient of an object in the pseudo-tensor category. We establish an intrinsic criterion on pseudo-tensor categories for the…

Let $\mathcal{M}$ be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in $\mathcal{M}$. This is based on…

Let A and B be abelian categories with enough projective and injective objects, and T : A-B a left exact additive functor. Then one has a comma category (B*T). It is shown that If T : A-B is X-exact, then (*X, X) is a (hereditary) cotorsion…

We define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories. We develop the theory of thin exact categories, an exact-category…

Let B be an extriangulated category with enough projectives and enough injectives. Let C be a fully rigid subcategory of B which admits a twin cotorsion pair ((C,K),(K,D)). The quotient category B/K is abelian, we assume that it is…

Let $\mathscr{A}$ be an abelian category and $\mathscr{P}(\mathscr{A})$ the subcategory of $\mathscr{A}$ consisting of projective objects. Let $\mathscr{C}$ be a full, additive and self-orthogonal subcategory of $\mathscr{A}$ with…

Given a $1$-tilting cotorsion pair over a commutative ring, we characterise the rings over which the $1$-tilting class is an enveloping class. To do so, we consider the faithful finitely generated Gabriel topology $\mathcal{G}$ associated…

We introduce the notion of noncompact (partial) silting and (partial) tilting sets and objects in any triangulated category D with arbitrary (set-indexed) coproducts. We show that equivalence classes of partial silting sets are in bijection…

We construct Abelian model structures on the category of chain complexes over a ring $R$, from the notion homological dimensions of modules. Given an integer $n > 0$, we prove that the left modules over a ringoid $\mathfrak{R}$ with…

In this paper, we classify certain subcategories of modules over a ring R. A wide subcategory of R-modules is an Abelian subcategory of R-Mod that is closed under extensions. We give a complete classification of wide subcategories of…

We present a comonadic approach to pretorsion theories on semiexact categories, i.e. categories equipped with a closed ideal of null morphisms that admits all kernels and all cokernels. We first prove that bihereditary pretorsion theories…

The notion of a pseudo cluster tilting subcategory $\mathcal X$ in an extriangulated category $\mathcal C$ is defined in this article. We prove that the quotient category $\mathcal C/\mathcal X$, obtained by factoring an extriangulated…

This article introduces pre-Hilbert $*$-categories: an abstraction of categories exhibiting "algebraic" aspects of Hilbert-space theory. Notably, finite biproducts in pre-Hilbert $*$-categories can be orthogonalised using the Gram-Schmidt…

Extriangulated category was introduced by Nakaoka and Palu to give a unification of properties in exact categories and triangulated categories. A notion of tilting (or cotilting) subcategories in an extriangulated category is defined in…

We establish connections between silting and tilting objects in an abelian category $\mathcal{B}$ and those in a cleft extension $\mathcal{A}$ of $\mathcal{B}$, which provides a method for constructing more silting and tilting objects. Then…

Let $k$ be an algebraic extension of $\mathbb F_p$ and $K/k$ a regular extension of fields (e.g. $\mathbb F_p(T)/\mathbb F_p$). Let $A$ be a $K$-abelian variety such that all the isogeny factors are neither isotrivial nor of $p$-rank zero.…

We study certain toric Gorenstein varieties with isolated singularities which are the quotient spaces of generic unimodular representations by the one-dimensional torus, or by the product of the one-dimensional torus with a finite abelian…

We show that various derived categories of torsion modules and contramodules over the adic completion of a commutative ring by a weakly proregular ideal are full subcategories of the related derived categories of modules. By the work of…