Localization in Coalgebras. Stable Localizations and Path Coalgebras
Localization in Coalgebras. Stable Localizations and Path Coalgebras
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DOI：
10.1080/00927870600637066
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发表时间：
2006-08
影响因子：
0.7
通讯作者：
P. Jara;L. Merino;G. Navarro;J. F. Ruiz
中科院分区：
文献类型：
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作者：
P. Jara;L. Merino;G. Navarro;J. F. Ruiz
利用对偶代数C* 中局部化子范畴与幂等元等价类之间的对应关系，研究了域上余代数C的余范畴的局部化与余局部化子范畴.在这个框架下，我们给出了一个有用的描述的局部化函子的局部化子范畴的拟有限内射余生成元所定义的Morita-Takeuchi上下文的装置。应用这一描述，首先，我们刻画了一个局部化子范畴，与相关的幂等元e ∈ C*，是共局部化的当且仅当eC是一个拟有限的eCe-余模，此外，当eC是内射时，ε是完美的。其次，我们证明了局部化子范畴e是稳定的当且仅当e是C* 的半中心幂等元。我们应用该理论的道路余代数，并获得，特别是，“本地化”余代数的道路余代数是再次道路余代数。
We study localizing and colocalizing subcategories of a comodule category of a coalgebra C over a field, using the correspondence between localizing subcategories and equivalence classes of idempotent elements in the dual algebra C*. In this framework, we give a useful description of the localization functor by means of the Morita–Takeuchi context defined by the quasi-finite injective cogenerator of the localizing subcategory. Applying this description; first we characterize that a localizing subcategory 𝒯, with associated idempotent element e ∈ C*, is colocalizing if and only if eC is a quasi-finite eCe-comodule and, in addition, 𝒯 is perfect whenever eC is injective. And second, we prove that a localizing subcategory 𝒯 is stable if and only if e is a semicentral idempotent element of C*. We apply the theory to path coalgebras and obtain, in particular, that the “localized” coalgebra of a path coalgebra is again a path coalgebra.