Recollements from Ding Injective Modules
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Recollements from Ding Injective Modules Miao Wang1 · Zhanping Wang1
· Pengfei Yang1
Received: 16 June 2020 / Revised: 25 August 2020 / Accepted: 1 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
A 0 Let A and B be rings, U a (B, A)-bimodule and T = a triangular matrix U B ring. In this paper, we firstly construct a right recollement of stable categories of Ding injective B-modules, Ding injective T -modules and Ding injective A-modules and then establish a recollement of stable categories of these modules. Abstract
Keywords Recollement · Ding injective module · Triangular matrix ring Mathematics Subject Classification 18G65 · 18G25 · 16D50 · 16E05
1 Introduction The theory of Gorenstein homological algebra for modules has been well developed, where the study of Gorenstein injective and Gorenstein projective modules has been drawn wide attention, refer to [1,4,5] and so on. As a special case of Gorenstein injective module, Mao and Ding introduced and studied in [13] Gorenstein FP-injective
Communicated by Shiping Liu.
B
Zhanping Wang [email protected] Miao Wang [email protected] Pengfei Yang [email protected]
1
Department of Mathematics, Northwest Normal University, Lanzhou, China
123
M. Wang et al.
module, and several well-known classes of rings are characterized in terms of these modules, which shares nice analogous properties with Gorenstein injective modules. Gillespie [7] renamed these modules as Ding injective modules. An right R-module M is called Ding injective if there exists an exact complex of injective right R-modules I• : · · · → I −1 → I 0 → I 1 → · · · with M = Ker(I 0 → I 1 ), which remains exact after applying Hom R (E, −) for any FP-injective right R-module E. By definitions, Ding injective modules are Gorenstein injective. Conversely, if R is a left Noetherian ring, then each FP-injective module is injective by [15, Theorem1.6], and so, each Gorenstein injective module is Ding injective. Ding injective modules have been studied by many authors, refer to [7,8,14,17–19] and so on. For a ring R, we write Mod-R (resp., R-Mod) for the category of right (left) R-modules. Denote by I(R), GI(R) and DI(R) the full subcategories of Mod-R consisting of injective right R-modules, Gorenstein injective right R-modules and Ding injective right R-modules, respectively. Recollements of triangulated categories were introduced by Beilinson, Bernstein and Deligne in [2] and play an important role in algebraic geometry and representation theory. It is well known that GI(R) is a Frobenius category with injective right R-modules as relative projective–injective objects, andthen, the stable category A 0 be the triangular GI(R) := GI(R)/I(R) is a triangulated category. Let T = U B matrix ring with A, B rings and U a (B, A)-bimodule. Zhang introduced compatible bimodules in [20] and then showed that there is a left recollement of the stable categories of Gorenstein projective left A-modules, Gorenstein projective left B-modules and Gorenstein pr
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