Separated monic representations I: Gorenstein-projective modules
| Autor: | Pu Zhang, Xiu-Hua Luo |
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| Rok vydání: | 2017 |
| Předmět: |
Subcategory
Pure mathematics Monomial Algebra and Number Theory 010102 general mathematics Quiver 0102 computer and information sciences 01 natural sciences Path algebra 010201 computation theory & mathematics If and only if Mathematics::Category Theory 0101 mathematics Projective test Monic polynomial Mathematics |
| Zdroj: | Journal of Algebra. 479:1-34 |
| ISSN: | 0021-8693 |
| DOI: | 10.1016/j.jalgebra.2017.01.038 |
| Popis: | For a finite acyclic quiver Q, an ideal I of a path algebra kQ generated by monomial relations, and a finite-dimensional k-algebra A, we introduce the separated monic representations of a bound quiver ( Q , I ) over A. They differ from the (usual) monic representations. The category smon ( Q , I , A ) of the separated monic representations of ( Q , I ) over A coincides with the category mon ( Q , I , A ) of the (usual) monic representations if and only if I = 0 and each vertex of Q is the ending vertex of at most one arrow. We give properties of the structural maps of separated monic representations, and prove that smon ( Q , I , A ) is a resolving subcategory of rep ( Q , I , A ) . We introduce the condition (G). Let Λ : = A ⊗ k Q / I . By the equivalence rep ( Q , I , A ) ≅ Λ -mod of categories, the main result claims that a Λ-module is Gorenstein-projective if and only if it is in smon ( Q , I , A ) and has a local A-Gorenstein-projective property (G). As consequences, the separated monic Λ-modules are exactly the projective Λ-modules if and only if A is semi-simple; and they are exactly the Gorenstein-projective Λ-modules if and only if A is self-injective, and if and only if smon ( Q , I , A ) is a Frobenius category. |
| Databáze: | OpenAIRE |
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