arXiv subjects

Manuel Saorín

Publications and source records attributed to Manuel Saorín.

24 records · Page 2Linked to original sources

arXiv2015

Addendum to 'Direct limits in the heart of a t-structure: The case of a torsion pair'

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$ø$ 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 $\mathcal{G}$.

math.CT↗

arXiv2015

On hearts which are module categories

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-Smalø$\text{ }$ 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

math.RT↗

arXiv2014

Pseudo-Frobenius graded algebras with enough idempotents

We introduce and study the notion of pseudo-Frobenius graded algebra with enough idempotents, showing that it follows the pattern of the classical concept of pseudo-Frobenius (PF) and Quasi-Frobenius (QF) rings, in particular finite dimensional self-injective algebras, as studied by Nakayama, Morita, Faith, Tachikawa, etc. We show that such an algebra is characterized by the existence of a graded Nakayama form. Moreover, we prove that the pseudo-Frobenius property is preserved and reflected by covering functors, a fact that makes the concept useful in Representation Theory.

math.RT↗

arXiv2014

Direct limits in the heart of a t-structure: the case of a torsion pair

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.

math.RT↗

arXiv2014

Direct sums of representations as modules over their endomorphism rings

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 $Σ$-algebraically compact.

math.RT↗

arXiv2014

Direct products of modules and the pure semisimplicity conjecture. Part II

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.

math.RT↗