On cotilting cotorsion and torsion pairs

Cotorsion Theories and Splitters

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is Ext_R(G,G)=0 holds and follow Schultz to call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples for splitters. Are there others? Answering an open problem by Schultz we will show that there are more splitters, in fact we are able to prescribe their endomorphism R-algebras with a free R-module structure. As a byproduct we are able to answer a problem of Salce showing that all rational cotorsion theories have enough injectives and enough projectives.

On $τ$-tilting subcategories

The main theme of this paper is to study $τ$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $τ$-cotorsion torsion triples and show a bijection between the collection of $τ$-cotorsion torsion triples in $\mathscr{A}$ and the collection of $τ$-tilting subcategories of $\mathscr{A}$, generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr{A}$. General definitions and results are exemplified using persistent mod…

与扩展心相关的沉默子范畴和(余)挠对

Silting subcategories and (co)torsion pairs associated to extended hearts

建立了(d+1)-项沉默子范畴、d-扩展心中的函子有限s-挠对以及适当子范畴中的遗传完全余挠对之间的偏序集同构，并给出了dg代数版本。

Covers, preenvelopes, and purity

We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and Saor\'ın. We also show that if a class of modules contains the ground ring and is closed under extensions, direct sums, pure submodules, and pure quotients, then it forms the first half of a so-called perfect cotorsion pair as introduced by Salce; this is stronger than being covering. Some applications are given to concrete classes of modules such as kernels of homol…

相对语境下完备余挠对的显式自对偶构造

An explicit self-dual construction of complete cotorsion pairs in the relative context

本文在环同态诱导的模范畴中，利用余诱导函子和有限分解维数条件，显式构造了完备余挠对，并推广到可数乘积、序数指标滤过及倾斜余挠对情形。

Endomorphism rings of modules whose cardinality is cofinal to omega

The main result is Theorem: Let A be an R-algebra, mu, lambda be cardinals such that |A|<=mu=mu^{aleph_0}<lambda<=2^mu. If A is aleph_0-cotorsion-free or A is countably free, respectively, then there exists an aleph_0-cotorsion-free or a separable (reduced, torsion-free) R-module G respectively of cardinality |G|=lambda with End_RG=A oplus Fin G.

Locally finitely presented and coherent hearts

Starting with a Grothendieck category $\mathcal{G}$ and a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal G$, we study the local finite presentability and local coherence of the heart $\mathcal{H}_{\mathbf{t}}$ of the associated Happel-Reiten-Smalø $t$-structure in the derived category $\mathrm{Der} (\mathcal{G})$. We start by showing that, in this general setting, the torsion pair $\mathbf t$ is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those $\mathbf t$ for which $\mathcal{H}_{\mathbf{t}}$ is locally finite…

相对内射模、超稳定性与诺特范畴

Relative injective modules, superstability and noetherian categories

本文研究三类封闭模类的$\boldsymbol{\text{mathcal{M}}}$-内射模的类贝尔准则，用模型论稳定独立关系论证，证明超稳定性等价于诺特范畴，并以此刻画多种环类。

由投射模态射范畴诱导的$r$ICE-闭子范畴

$r$ICE-closed subcategories induced by the morphism category of projective modules

本文通过研究投射模范畴的态射范畴中的ICE-闭子范畴，建立了与$\tau$-刚性模和rICE-闭子范畴之间的双射，推广了Enomoto对遗传代数的结果，并改进了Buan和Zhou的双射。