On d-term silting objects, torsion classes, and cotorsion classes

2024/07/15 by Esha Gupta, Gupta, Esha · 2 citations
Mathematics · Physics and Astronomy · #16E35 #16G10 #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2407.10562

openalex publication_date 2024/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a finite-dimensional algebra Λ over an algebraically closed field K, it is known that the poset of 2-term silting objects in K^b(projΛ) is isomorphic to the poset of functorially finite torsion classes in modΛ, and to that of complete cotorsion classes in K^[-1,0](projΛ). In this work, we generalise this result to the case of d-term silting objects for arbitrary d≥ 2 by introducing the notion of torsion classes for extriangulated categories. In particular, we show that the poset of d-term silting objects in K^b(projΛ) is isomorphic to the poset of complete and hereditary cotorsion classes in K^[-d+1,0](projΛ), and to that of positive and functorially finite torsion classes in D^[-d+2,0](modΛ), an extension-closed subcategory of D^b(modΛ). We further show that the posets cotorsK^[-d+1,0](projΛ) and tors D^[-d+2,0](modΛ) are lattices, and that the truncation functor τ_{≥ -d+2} gives an isomorphism between the two.

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