Related papers: The countable Telescope Conjecture for module cate…

For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exact sequence is constructed. The pairs of finite type are…

The Countable Telescope Conjecture arose in the framework of stable homotopy theory, as a tool conceived to study the chromatic filtration. It turned out, however, to trigger extremely fertile research within the framework of Module…

For a commutative noetherian ring $R$, we classify all the hereditary cotorsion pairs cogenerated by pure-injective modules of finite injective dimension. The classification is done in terms of integer-valued functions on the spectrum of…

Let $R$ be a ring and $\mathsf S$ be a class of strongly finitely presented (FP${}_\infty$) $R$-modules closed under extensions, direct summands, and syzygies. Let $(\mathsf A,\mathsf B)$ be the (hereditary complete) cotorsion pair…

For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules…

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context.…

Silting modules are abundant. Indeed, they parametrise the definable torsion classes over a noetherian ring, and the hereditary torsion pairs of finite type over a commutative ring. Also the universal localisations of a hereditary ring, or…

Let $R$ be a ring and denote by $\mathcal{FM}$ the class of all flat and Mittag-Leffler left $R$-modules. In \cite{BazzoniStovicek2} it is proved that, if $R$ is countable, the orthogonal class of $\mathcal{FM}$ consists of all cotorsion…

We revisit a construction of wide subcategories going back to work of Ingalls and Thomas. To a torsion pair in the category $ R\operatorname{-}\operatorname{mod}$ of finitely presented modules over a left artinian ring $R$, we assign two…

We show that bounded type implies finite type for a constructible subcategory of the module category of a finitely generated algebra over a field, which is a variant of the first Brauer-Thrall conjecture. A full subcategory is constructible…

We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an…

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 consider the question of whether the injective modules generate the unbounded derived category of a ring as a triangulated category with arbitrary coproducts. We give an example of a non-Noetherian commutative ring where they don't, but…

Given a hereditary complete cotorsion pair $(\mathsf A,\mathsf B)$ generated by a set of objects in a Grothendieck category $\mathsf K$, we construct a natural equivalence between the Becker coderived category of the left-hand class…

For any ring $R$, we investigate balanced pairs of classes of modules and their relations to cotorsion triples. We characterize the case when a balanced pair generates a tilting cotorsion pair, and dually, when it cogenerates a cotilting…

A torsion class $\mathcal{T}$ of the module category $\operatorname{\mathsf{mod}} A$ of a finite dimensional algebra $A$ over a field $K$ is said to be compact if there exists a module $M \in \operatorname{\mathsf{mod}} A$ such that…

We prove that the category of countable Tate modules over an arbitrary discrete ring embeds fully faithfully into that of condensed modules. If the base ring is of finite type, we characterize the essential image as generated by the free…

We show that any homotopically smashing t-structure in the derived category of a commutative noetherian ring is compactly generated. This generalizes the validity of the telescope conjecture for commutative noetherian rings due to Neeman.…

Let $\mathcal{C}$ be a small category. In this paper, we mainly study the category of modules $\mathfrak{M}\mbox{od-}\mathfrak{R}$ on ringed sites $(\mathbf{C},\mathfrak{R})$. We firstly reprove the Theorem A of the paper (M. Wu and F. Xu.…

We study the notions of $n$-hereditary rings and its connection to the classes of finitely $n$-presented modules, FP$_n$-injective modules, FP$_n$-flat modules and $n$-coherent rings. We give characterizations of $n$-hereditary rings in…