Relative homology and representation theory III
Relative homology and representation theory III
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M. Auslander;.. Solberg-Solberg-2103230017
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作者：
M. Auslander;.. Solberg-Solberg-2103230017
本文的最后一个系列的三个文件研究使用相对同调代数的表示理论的阿廷代数，是致力于给一个相当明确的连接之间的相对cotilting理论介绍了在前一份文件和标准cotilting理论。读者可以参考本系列的前几篇文章[2，3]，以了解本文中所用的相对同调代数和相对余倾斜模理论的基本定义和结果以及符号。我们现在描述本文的主要结果。设mod Λ是Artin代数Λ上的n-生成左模范畴.设F是加法双函子ExtΛ（，）：（mod Λ）×modΛ → Ab的加法子函子。设F有足够多的投射子和内射子，且mod Λ中由F -投射子构成的子范畴P（F）对mod Λ中的某个G是addG.则G是mod Λ的生成元，因为P（F）包含Λ。设G = EndΛ（G）.设T在mod Λ中是F -余倾斜模.我们证明了Γ G-模HomΛ（G，T）=（G，T）是一个标准的共倾斜模，并且代数EndΓG（（G，T））自然同构于Γ = EndΛ（T）。此外，我们还证明了相对共倾函子HomΛ（，T）：modΛ → mod τ与函子Hom Λ（G，）：mod Λ → mod τ G和标准共倾函子Hom τ G（，（G，T））：mod τ G的合成Hom τ G（，（G，T）o Hom Λ（G，）正则同构。这表明如何相对cotilting函子可以描述的标准cotilting函子。这个结果的证明使用了文[1]中引入的Wedderburn对应的概念，并且给出了Wedderburn对应与标准倾斜理论、相对倾斜理论和共倾斜理论之间的以下联系。假设G是mod Λ的生成元。用F表示Fadd G。则F有足够的投射和内射，且P（F）= addG。再次设G = EndΛ（G），我们得到G是F倾斜模.因此HomΛ（G，）：mod Λ → mod ΓG是相对倾斜函子的一个特例。然后我们得到HomΛ（G，）：mod Λ → mod ΓG规范同构于合成HomΓ（，（G，T））o HomΛ（，T）其中Hom Λ（，T）：modΛ → mod Γ是相对共倾函子并且HomΓ（，（G，T））：mod Γ → mod ΓG是idΓ（G，T）= 2的标准共倾函子。进一步证明了Grothendieck群F-K 0（mod Λ）与K 0（mod ΓG）是同构的.
This paper, the last of a series of three papers studying the uses of relative homological algebra in the representation theory of artin algebras, is devoted to giving a rather explicit connection between the relative cotilting theory introduced in the previous paper and standard cotilting theory. The reader is referred to the previous papers in this series [2, 3] for basic definitions and results, as well as notations, concerning the relative homological algebra and theory of relative cotilting modules used in this paper. We now describe the main result of this paper. Let mod Λ be the category of finitely generated left modules over an artin algebra Λ. Suppose F is an additive subfunctor of the additive bifunctor ExtΛ( , ): (mod Λ) ×modΛ → Ab. Assume that F has enough projectives and injectives and that P(F ), the subcategory of mod Λ consisting of the F -projectives, is addG for some G in mod Λ. Then G is a generator for mod Λ since P(F ) contains Λ. Let ΓG = EndΛ(G) . Let T in mod Λ be an F -cotilting module. We show that the ΓG-module HomΛ(G, T ) = (G, T ) is a standard cotilting module and that the algebra EndΓG((G, T )) is naturally isomorphic to Γ = EndΛ(T ). In addition we show that the relative cotilting functor HomΛ( , T ): modΛ → mod Γ is canonically isomorphic to the composition HomΓG( , (G, T ))◦HomΛ(G, ) of the functors HomΛ(G, ): mod Λ → mod ΓG and the standard cotilting functor HomΓG( , (G, T )): mod ΓG → mod Γ. This shows how relative cotilting functors can be described in terms of standard cotilting functors. The proof of this result uses the notion of the Wedderburn correspondence introduced in [1] and also gives the following connection between the Wedderburn correspondence, and both standard and relative tilting and cotilting theory. Suppose G is a generator for mod Λ. Denote Fadd G by F . Then F has enough projectives and injectives and P(F ) = addG. Again letting ΓG = EndΛ(G) , we get that G is an F tilting module. Thus HomΛ(G, ): mod Λ → mod ΓG is a special case of a relative tilting functor. We then get that HomΛ(G, ): mod Λ → mod ΓG is canonically isomorphic to the composition HomΓ( , (G, T ))◦HomΛ( , T ) where HomΛ( , T ): modΛ → mod Γ is a relative cotilting functor and HomΓ( , (G, T )): mod Γ → mod ΓG is a standard cotilting functor with idΓ(G, T ) = 2. Further, we have that the Grothendieck groups F -K0(mod Λ) and K0(mod ΓG) are isomorphic.