Higher differential objects in additive categories

JOURNAL OF ALGEBRA

卷:549

Article

Tang, Xi¹  Huang, Zhaoyong² 

[1] Guilin Univ Technol, Coll Sci, Guilin 541004, Guangxi Provinc, Peoples R China

[2] Nanjing Univ, Dept Math, Nanjing 210093, Jiangsu, Peoples R China

关键词: n-th differential objects;    Additive categories;    Exact categories;    Homological conjectures;    Triangulated categories;    Derived categories;   

DOI  :  10.1016/j.jalgebra.2019.12.011

来源: Elsevier

【 摘 要 】

Given an additive category C and an integer n >= 2. We form a new additive category C[epsilon](n) consisting of objects X in C equipped with an endomorphism epsilon(X) satisfying epsilon(n)(X) = 0. First, using the descriptions of projective and injective objects in C[epsilon](n), we not only establish a connection between Gorenstein flat modules over a ring R and R[t]/(t(n)), but also prove that an Artinian algebra R satisfies some homological conjectures if and only if so does R[t]/(t(n)). Then we show that the corresponding homotopy category K(C[epsilon](n)) is a triangulated category when C is an idempotent complete exact category. Moreover, under some conditions for an abelian category A, the natural quotient functor Q from K(A[epsilon](n)) to the derived category D(A[epsilon](n)) produces a recollement of triangulated categories. Finally, we prove that if A is an Ab4-category with a compact projective generator, then D(A[epsilon](n)) is a compactly generated triangulated category. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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