2022/07/01 by Asadollahi, Javad, Sadeghi, Somayeh, Treffinger, Hipolito
#Category Theory (math.CT) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2207.00457
The main theme of this paper is to study τ-tilting subcategories in an abelian category \mathscrA 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 \mathscrA and the collection of τ-tilting subcategories of \mathscrA, generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of \mathscrA. General definitions and results are exemplified using persistent modules. If \mathscrA=\rmModR, where R is an unitary associative ring, we characterize all support τ-tilting, resp. all support τ--tilting, subcategories of \rmModR in term of finendo quasitilting, resp. quasicotilting, modules. As a result, it will be shown that every silting module, respectively every cosilting module, induces a support τ-tilting, respectively support τ--tilting, subcategory of \rmModR. We also study the theory in \rm Rep(Q, \mathscrA), where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support τ-tilting subcategories in \rm Rep(Q, \mathscrA) from certain support τ-tilting subcategories of \mathscrA and present a systematic way to construct (n+1)-tilting subcategories in \rm Rep(Q, \mathscrA) from n-tilting subcategories in \mathscrA.