JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2016, Vol. 51 ›› Issue (2): 79-84.doi: 10.6040/j.issn.1671-9352.0.2015.076

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Gorenstein injective objects in Abelian categories

CHENG Hai-xia, YIN Xiao-bin   

  1. School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, Anhui, China
  • Received:2015-02-27 Online:2016-02-16 Published:2016-03-11

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Abstract: Let A be an abelian category with enough injective objects and X a full subcategory of A. The definitions of X -Gorenstein injective object and YX -Gorenstein injective object are given, where YX={Y∈Ch(A)|Y is acyclic and KerdnY∈X. Under certain conditions, these two Gorenstein injective objects are related in a nice way. In particular, if I(A)⊆X, X∈Ch(A)is YX -Gorenstein injective if and only if Xi is X -Gorenstein injective for each i, when X is a self-orthogonal class. Subsequently, the relationships between X -Gorenstein injective dimension and YX -Gorenstein injective dimension are considered.

Key words: precover, X -Gorenstein injective object, X -Gorenstein injective dimension

CLC Number: 

  • O154.2
[1] ENOCHS E E, JENDA O M G. Gorenstein injective and projective modules[J]. Math Z, 1995, 220(4):611-633.
[2] ENOCHS E E, GARCÍA ROZAS J R. Gorenstein injective and projective complexes[J]. Comm Algebra, 1998, 26(5):1657-1674.
[3] HOLM H. Gorenstein homological dimensions[J]. J Pure Appl Algebra, 2004, 189:167-193.
[4] LIU Zhongkui, ZHANG Chunxia. Gorenstein projective dimensions of complexes[J]. Acta Math Sinica, 2011, 27(7):1395-1404.
[5] YANG Xiaoyan, LIU Zhongkui. Gorenstein projective, injective and flat complexes[J]. Comm Algebra, 2011, 39(5):1705-1721.
[6] ENOCHS E E, JENDA O M G. Xu J Z. Orthogonality in the category of complexes[J]. Math J Okayama Univ, 1996, 38:25-46.
[7] ENOCHS E E, JENDA O M G. Relative homological algebra[M]. Berlin: Walter de Gruyter, 2000: 13-36.
[8] ENOCHS E E, JENDA O M G. Relative homological algebra 2[M]. Berlin: Walter de Gruyter, 2011: 105-162.
[9] AVRAMOV L L, Foxby H B. Homological dimensions of unbounded complexes[J]. J Pure Appl Algebra,1991, 71:129-155.
[10] BENNIS D, MAHDOU N, OUARGHI K. Rings over which all modules are strongly Gorenstein projective[J]. J Rocky Mountain Math, 2010, 40(3):749-759.
[11] ZHU Xiaosheng. Resolving resolution dimensions[J]. Algebra Represent Theor, 2013, 16(4):1165-1191.
[12] GILLESPIE J. The flat model structure on Ch(R)[J]. Trans Amer Math Soc, 2004, 356(8):3369-3390.
[13] EKLOF P, TRLIFAJ J. How to make Ext vanish[J]. Bull London Math Soc, 2001, 33(4):41-51.
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