Wide subcategories of $d$-cluster tilting subcategories
| Autor: | Herschend, Martin, Jorgensen, Peter, Vaso, Laertis |
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| Rok vydání: | 2017 |
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| Druh dokumentu: | Working Paper |
| Popis: | A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If $\Phi$ is a finite dimensional algebra, then each functorially finite wide subcategory of $\operatorname{mod}( \Phi )$ is of the form $\phi_{ * }\big( \operatorname{mod}( \Gamma ) \big)$ in an essentially unique way, where $\Gamma$ is a finite dimensional algebra and $\Phi \stackrel{ \phi }{ \longrightarrow } \Gamma$ is an algebra epimorphism satisfying $\operatorname{Tor}^{ \Phi }_1( \Gamma,\Gamma ) = 0$. Let ${\mathcal F} \subseteq \operatorname{mod}( \Phi )$ be a $d$-cluster tilting subcategory as defined by Iyama. Then ${\mathcal F}$ is a $d$-abelian category as defined by Jasso, and we call a subcategory of ${\mathcal F}$ wide if it is closed under sums, summands, $d$-kernels, $d$-cokernels, and $d$-extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of ${\mathcal F}$ is of the form $\phi_{ * }( {\mathcal G} )$ in an essentially unique way, where $\Phi \stackrel{ \phi }{ \longrightarrow } \Gamma$ is an algebra epimorphism satisfying $\operatorname{Tor}^{ \Phi }_d( \Gamma,\Gamma ) = 0$, and ${\mathcal G} \subseteq \operatorname{mod}( \Gamma )$ is a $d$-cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some $d$-cluster tilting subcategories ${\mathcal F} \subseteq \operatorname{mod}( \Phi )$ over algebras of the form $\Phi = kA_m / (\operatorname{rad}\,kA_m )^{ \ell }$. Comment: Dedicated to Idun Reiten on the occasion of her 75th birthday. This is the final version which has been accepted for publication in the Transactions of the American Mathematical Society. 27 pages |
| Databáze: | arXiv |
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