Abstract:
in our work [9], we complete jans' classification of ttf-triples [8] by giving a precise description of those two-sided ideals of a ring associated to one-sided split ttf-triples in the corresponding module category.

Abstract:
We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: Whenever a direct product $\prod_{n \in \Bbb N} M_n$ of finitely generated indecomposable modules $M_n$ is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the $M_n$. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type.

Abstract:
We study the behavior of direct limits in the heart of a t-structure. We prove that, for any compactly generated t-structure in a triangulated category with exact coproducts, countable direct limits are exact in its heart. Then, for a given Grothendieck category G and a torsion pair t = (T ;F) in G, we show that if the heart of the associated t-structure in the derived category D(G) is AB5, then F is closed under taking direct limits. The reverse implication is true, even implying that the heart is a Grothendieck category, for a wide class of torsion pairs which include the hereditary ones, those for which T is a cogenerating class and those for which F is a generating class. The results allow to extend well-known results by Buan-Krause, Bazzoni and Colpi-Gregorio to the general context of Grothendieck categories and to improve some results of (co)tlting theory of modules.

Abstract:
This paper is devoted to the study of the endo-structure of infinite direct sums $\bigoplus_{i \in I} M_i$ of indecomposable modules $M_i$ over a ring $R$. It is centered on the following question: If $S = \text{End}_R \bigl( \bigoplus_{i \in I} M_i \bigr)$, how much pressure, in terms of the $S$-structure of $\bigoplus_{i \in I} M_i$, is required to force the $M_i$ into finitely many isomorphism classes? In case the $M_i$ are endofinite (i.e., of finite length over their endomorphism rings), the number of isomorphism classes among the $M_i$ is finite if and only if $\bigoplus_{i \in I} M_i$ is endo-noetherian and the $M_i$ form a right $T$-nilpotent class. This is a corollary of a more general theorem in the paper which features the weaker conditions of (right or left) semi-$T$-nilpotence as well as the endosocle of a module. This result is sharpened in the case of Artin algebras, by showing that then, if the $M_i$ are finitely generated, the direct sum $\bigoplus_{i \in I} M_i$ is endo-Artinian if and only if it is $\Sigma$-algebraically compact.

Abstract:
Let $\mathcal{G}$ be a Grothendieck category, let $\mathbf{t}=(\mathcal{T},\mathcal{F})$ be a torsion pair in $\mathcal{G}$ and let $(\mathcal{U}_\mathbf{t},\mathcal{W}_\mathbf{t})$ be the associated Happel-Reiten-Smal$\o$ t-structure in the derived category $\mathcal{D}(\mathcal{G})$. We prove that the heart of this t-structure is a Grothendieck category if, and only if, the torsionfree class $\mathcal{F}$ is closed under taking direct limits in $\mathcal{G}$.

Abstract:
Suppose that $\mathcal{A}$ is an abelian category whose derived category $\mathcal{D}(\mathcal{A})$ has $Hom$ sets and arbitrary (small) coproducts, let $T$ be a (not necessarily classical) ($n$-)tilting object of $\mathcal{A}$ and let $\mathcal{H}$ be the heart of the associated t-structure on $\mathcal{D}(\mathcal{A})$. We show that the inclusion functor $\mathcal{H}\hookrightarrow\mathcal{D}(\mathcal{A})$ extends to a triangulated equivalence of unbounded derived categories $\mathcal{D}(\mathcal{H})\stackrel{\cong}{\longrightarrow}\mathcal{D}(\mathcal{A})$. The result admits a straightforward dualization to cotilting objects in abelian categories whose derived category has $Hom$ sets and arbitrary products.

Abstract:
Given a torsion pair $\mathbf{t} = (\mathcal{T} ;\mathcal{F})$ in a module category $R$-Mod we give necessary and sufficient conditions for the associated Happel-Reiten-Smal\o $\text{ }$ t-structure in $\mathcal{D}(R)$ to have a heart $\mathcal{H}_{\mathbf{t}}$ which is a module category. We also study when such a pair is given by a 2-term complex of projective modules in the way described by Hoshino-Kato-Miyachi ([HKM]). Among other consequences, we completely identify the hereditary torsion pairs $\mathbf{t}$ for which $\mathcal{H}_{\mathbf{t}}$ is a module category in the following cases: i) when $\mathbf{t}$ is the left constituent of a TTF triple, showing that $\mathbf{t}$ need not be HKM; ii) when $\mathbf{t}$ is faithful; iii) when $\mathbf{t}$ is arbitrary and the ring $R$ is either commutative, semi-hereditary, local, perfect or Artinian. We also give a systematic way of constructing non-tilting torsion pairs for which the heart is a module category generated by a stalk complex at zero

Abstract:
Let $R$ be a commutative Noetherian ring and let $\mathcal D(R)$ be its (unbounded) derived category. We show that all compactly generated t-structures in $\mathcal D(R)$ associated to a left bounded filtration by supports of Spec$(R)$ have a heart which is a Grothendieck category. Moreover, we identify all compactly generated t-structures in $\mathcal D(R)$ whose heart is a module category. As geometric consequences for a compactly generated t-structure $(\mathcal{U},\mathcal{U}^\perp [1])$ in the derived category $\mathcal{D}(\mathbb{X})$ of a Noetherian scheme $\mathbb{X}$, we get the following: 1) If the sequence $(\mathcal{U}[-n]\cap\mathcal{D}^{\leq 0}(\mathbb{X}))_{n\in\mathbb{N}}$ is stationnary, then the heart $\mathcal{H}$ is a Grothendieck category; 2) If $\mathcal{H}$ is a module category, then $\mathcal{H}$ is always equivalent to $\text{Qcoh}(\mathbb{Y})$, for some affine subscheme $\mathbb{Y}\subseteq\mathbb{X}$; 3) If $\mathbb{X}$ is connected, then: a) when $\bigcap_{k\in\mathbb{Z}}\mathcal{U}[k]=0$, the heart $\mathcal{H}$ is a module category if, and only if, the given t-structure is a translation of the canonical t-estructure in $\mathcal{D}(\mathbb{X})$; b) when $\mathbb{X}$ is irreducible, the heart $\mathcal{H}$ is a module category if, and only if, there are an affine subscheme $\mathbb{Y}\subseteq\mathbb{X}$ and an integer $m$ such that $\mathcal{U}$ consists of the complexes $U\in\mathcal{D}(\mathbb{X})$ such that the support of $H^j(U)$ is in $\mathbb{X}\setminus\mathbb{Y}$, for all $j>m$.