Wide subcategories of $d$-cluster tilting subcategories

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 $\mathrm{\Phi }$ is a finite dimensional algebra, then each functorially finite wide subcategory of $\mathrm{mod}(\mathrm{\Phi })$ is of the form $\phi_{ * }\big( \operatorname{mod}( \Gamma ) \big)$inanessentiallyuniqueway,where$\Gamma$isafinitedimensionalalgebraand$\Phi \stackrel{ \phi }{ \longrightarrow } \Gamma$isanalgebraepimorphismsatisfying$\operatorname{Tor}^{ \Phi }_1( \Gamma,\Gamma ) = 0$. Let $\mathcal{F}\subseteq \mathrm{mod}(\mathrm{\Phi })$ 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 ${\varphi }_{\ast }(\mathcal{G})$ in an essentially unique way, where $\Phi \stackrel{ \phi }{ \longrightarrow } \Gamma$ is an algebra epimorphism satisfying ${\mathrm{Tor}}_d^{\mathrm{\Phi }}(\mathrm{\Gamma },\mathrm{\Gamma })=0$, and ${\mathcal G} \subseteq \operatorname{mod}( \Gamma )$isa$d$-cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some $d$-cluster tilting subcategories ${\mathcal F} \subseteq \operatorname{mod}( \Phi )$overalgebrasoftheform$\Phi = kA_m / (\operatorname{rad},kA_m )^{ \ell }$.