On tau-tilting subcategories
The main theme of this paper is to study $\tau $ -tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $ -cotorsion torsion triples and investigate a bijection between the collection of $\tau $ -cotorsion torsion triples in $\ma...
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Veröffentlicht in: | Canadian journal of mathematics 2024-03, p.1-38 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | The main theme of this paper is to study
$\tau $
-tilting subcategories in an abelian category
$\mathscr {A}$
with enough projective objects. We introduce the notion of
$\tau $
-cotorsion torsion triples and investigate a bijection between the collection of
$\tau $
-cotorsion torsion triples in
$\mathscr {A}$
and the collection of support
$\tau $
-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 modules. If
$\mathscr {A}=\mathrm{Mod}\mbox {-}R$
, where
R
is a unitary associative ring, we characterize all support
$\tau $
-tilting (resp. all support
$\tau ^-$
-tilting) subcategories of
$\mathrm{Mod}\mbox {-}R$
in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support
$\tau $
-tilting (resp. support
$\tau ^{-}$
-tilting) subcategory of
$\mathrm{Mod}\mbox {-}R$
. We also study the theory in
$\mathrm {Rep}(Q, \mathscr {A})$
, where
Q
is a finite and acyclic quiver. In particular, we give an algorithm to construct support
$\tau $
-tilting subcategories in
$\mathrm {Rep}(Q, \mathscr {A})$
from certain support
$\tau $
-tilting subcategories of
$\mathscr {A}$
. |
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ISSN: | 0008-414X 1496-4279 |
DOI: | 10.4153/S0008414X24000221 |