Ideal approximation theory
Ideal approximation theory
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DOI：
10.1016/j.aim.2013.05.020
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发表时间：
2013-09
影响因子：
1.7
通讯作者：
X. Fu;P. A. G. Asensio;I. Herzog;B. Torrecillas
中科院分区：
文献类型：
--
作者：
X. Fu;P. A. G. Asensio;I. Herzog;B. Torrecillas
设(A；E)是正则范畴，F⊆Ext是次函子。A中的态射φ是F-模当且仅当A中有足够的内射对象和投射态射时，A的理想I是特殊预包当且仅当存在具有足够的内射态射的子函子F-φExt使得I是F-模态射的理想.证明中的关键一步是对Salce关于理想余扭对的引理的推广：如果I是一个特殊的预覆盖理想，则由I在(A；E)中生成的理想余扭对(I，I⊥)是完备的。这一定理被用来证明：(1)R-Mod的纯内射模余生成的理想余扭对是完备的；(2)复CH(R-Mod)范畴中可压缩复形余生成的理想余扭对是完备的；(3)利用Auslander和Reiten的几乎分裂序列理论，(3)有限生成的Artin代数表示范畴Λ-mod的Λ根Jac(Jac-mod)余生成的理想余扭对是完备的。
Let (A; E) be an exact category and F⊆ Ext a subfunctor. A morphism φ in A is an F-phantom if the pullback of an E-conflation along φ is a conflation in F. If the exact category (A; E) has enough injective objects and projective morphisms, it is proved that an ideal I of A is special precovering if and only if there is a subfunctor F⊆ Ext with enough injective morphisms such that I is the ideal of F-phantom morphisms. A crucial step in the proof is a generalization of Salce’s Lemma for ideal cotorsion pairs: if I is a special precovering ideal, then the ideal cotorsion pair (I, I⊥) generated by I in (A; E) is complete. This theorem is used to verify:(1) that the ideal cotorsion pair cogenerated by the pure-injective modules of R-Mod is complete;(2) that the ideal cotorsion pair cogenerated by the contractible complexes in the category of complexes Ch (R-Mod) is complete; and, using Auslander and Reiten’s theory of almost split sequences,(3) that the ideal cotorsion pair cogenerated by the Jacobson radical Jac (Λ-mod) of the category Λ-mod of finitely generated representations of an Artin algebra is complete.