Wide subcategories of $d$-cluster tilting subcategories

Abstract

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 $\Phi = kA_m / (\operatorname{rad}\,kA_m )^{ \ell }$.