Auslander–Reiten theory in extriangulated categories
Auslander–Reiten theory in extriangulated categories
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DOI：
10.1090/btran/159
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发表时间：
2018-05
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影响因子：
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通讯作者：
O. Iyama;H. Nakaoka;Yann Palu
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文献类型：
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作者：
O. Iyama;H. Nakaoka;Yann Palu
外三角范畴的概念统一了正合范畴或阿贝尔范畴和三角范畴中的现有理论。在本文中，我们发展了用于外三角范畴的Auslander-Reiten理论。这统一了Auslander-Reiten理论发展的确切范畴和三角范畴独立。我们给出了两组不同的充分条件的extriangulated范畴，使几乎分裂扩张的存在成为等价的Auslander-Reiten-Serre对偶。我们还证明了几乎分裂扩张的存在性在取相对外三角范畴、理想同分范畴和扩张闭子范畴的情况下是保持的。此外，我们证明了外三角范畴C \mathscr {C}的稳定范畴C\underline {C}是τ \tau -范畴（参见O. Iyama [Algebr.代表。《理论》8（2005），pp. 297-321]）如果C \mathscr {C}有足够的投射、几乎分裂扩张和源态射。这给出了C _ \underline {\mathscr {C}}的各种结果，包括Igusa-Todorov的Radical Layers Theorem（参见K. Igusa和G. Todorov [J. Algebra 89（1984），pp. 105-147]），关于Hom-空间维数的Auslander-Reiten组合学，以及通过C _\underline {\mathscr {C}}的Auslander-Reiten类的完全网格范畴重构C_\underline {\mathscr {C}}的相关完全分次范畴的定理。最后，我们证明了任何局部有限可对称化的τ \tau -τ（=值平移τ）是具有汇态射和源态射的外三角范畴的Auslander-Reiten τ。
The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander–Reiten theory for extriangulated categories. This unifies Auslander–Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander–Reiten–Serre duality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category C _ \underline {\mathscr {C}} of an extriangulated category C \mathscr {C} is a τ \tau -category (see O. Iyama [Algebr. Represent. Theory 8 (2005), pp. 297–321]) if C \mathscr {C} has enough projectives, almost split extensions and source morphisms. This gives various consequences on C _ \underline {\mathscr {C}} , including Igusa–Todorov’s Radical Layers Theorem (see K. Igusa and G. Todorov [J. Algebra 89 (1984), pp. 105–147]), Auslander–Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of C _ \underline {\mathscr {C}} via the complete mesh category of the Auslander–Reiten species of C _ \underline {\mathscr {C}} . Finally we prove that any locally finite symmetrizable τ \tau -quiver (=valued translation quiver) is an Auslander–Reiten quiver of some extriangulated category with sink morphisms and source morphisms.