Contravariant functors on the category of finitely presented modules
Contravariant functors on the category of finitely presented modules
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DOI：
10.1007/s11856-008-1052-8
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发表时间：
2008-10
影响因子：
1
通讯作者：
I. Herzog
中科院分区：
文献类型：
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作者：
I. Herzog
如果R是一个有单位元的结合环，则在反变函子G：（R-mod）op→ Ab的范畴（（R-mod）op，Ab）中，从R-表示左R-模的范畴R-mod到Abel群的范畴Ab，建立了极小平坦分解的理论.对左R-模M，证明了平坦反变函子（-，M）是余挠的当且仅当M是纯内射的.这被应用于刻画当一个对象Fin（（R-mod）op，Ab）的平坦归结是最小的，并且被用于构造F的最小平坦归结，给出一个射影表示.它表明（（R-mod）op，Ab）的内射对象正好是那些形式为Ext 1（−，M）的对象，其中M是纯内射的，并且如果m：M→ PE（M）是M的纯内射包络，那么Ext 1（−，m）：Ext 1（−，M）→ Ext 1（−，PE（M））是Ext 1（−，M）在（（R-mod）op，Ab）中的内射包络。M <$Ext1（−，M）给出了（（R-mod）op，Ab）的内射对象的子范畴和纯内射左R-模范畴之间的显式等价性。最小平坦归结的刻画也被用来描述稳定范畴上函子F在（（R-mod）op，Ab）中的最小平坦归结与它在（（R-mod）op，Ab）中的最小内射互表示之间的关系，最后一个应用是对函子范畴（（R-mod）op，Ab）的不可分解内射对象集R的反变Gabriel谱的刻画.这些点与纯内射不可分解左R-模的集合是双射对应的，这些点对应于R的协变Gabriel谱的点。证明了R的两个Gabriel谱都可以被划分为开集和闭集，使得这一典型双射限制于每一个上的同胚。
IfRis an associative ring with identity, a theory of minimal flat resolutions is developed in the category ((R-mod)op, Ab) of contravariant functorsG: (R-mod)op→ Ab from the categoryR-mod of finitely presented leftR-modules to the category Ab of abelian groups. For a leftR-moduleM, it is shown that the flat contravariant functor (−,M) is cotorsion if and only ifMis pure-injective. This is applied to characterize when a flat resolution of an objectFin ((R-mod)op, Ab) is minimal, and is used to construct a minimal flat resolution ofF, given a projective presentation.It is shown that the injective objects of ((R-mod)op, Ab) are precisely those of the form Ext1(−,M), whereMis pure-injective, and ifm:M→ PE(M) is the pure-injective envelope ofM, then Ext1(−,m): Ext1(−,M) → Ext1(−, PE(M)) is an injective envelope of Ext1(−,M) in ((R-mod)op, Ab).M↦ Ext1(−,M) yields an explicit equivalence between the subcategory of injective objects of ((R-mod)op, Ab) and the category of pure-injective leftR-modules, modulo morphisms that factor through an injective. The characterization of minimal flat resolutions is also used to describe the relationship between the minimal flat resolution in ((R-mod)op, Ab) of a functorFon the stable category and its minimal injective copresentation in ((R-mod)op, Ab).A final application is a description of the contravariant Gabriel spectrum ofR, the set of indecomposable injective objects of the functor category ((R-mod)op, Ab). The points are in bijective correspondence with the set of pure-injective indecomposable leftR-modules, which correspond to the points of the covariant Gabriel spectrum ofR. It is proved that both Gabriel spectra ofRmay be partitioned into an open and a closed set such that this canonical bijection restricts to a homeomorphism on each.