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 $\phi _{ {\textstyle *}}\big ( \operatorname {mod}( \Gamma ) \big )$ 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 $\mathscr {F} \subseteq \operatorname {mod}( \Phi )$ be a $d$-cluster tilting subcategory as defined by Iyama. Then $\mathscr {F}$ is a $d$-abelian category as defined by Jasso, and we call a subcategory of $\mathscr {F}$ 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 $\mathscr {F}$ is of the form $\phi _{ {\textstyle *}}( \mathscr {G} )$ in an essentially unique way, where $\Phi \stackrel { \phi }{ \longrightarrow } \Gamma$ is an algebra epimorphism satisfying $\operatorname {Tor}^{ \Phi }_d( \Gamma ,\Gamma ) = 0$, and $\mathscr {G} \subseteq \operatorname {mod}( \Gamma )$ is a $d$-cluster tilting subcategory.

We illustrate the theory by computing the wide subcategories of some $d$-cluster tilting subcategories $\mathscr {F} \subseteq \operatorname {mod}( \Phi )$ over algebras of the form $\Phi = kA_m / (\operatorname {rad} kA_m )^{ \ell }$.