# Wide subcategories of $d$-cluster tilting subcategories

Research paper by **Martin Herschend, Peter Jorgensen, Laertis Vaso**

Indexed on: **05 May '17**Published on: **05 May '17**Published in: **arXiv - Mathematics - Representation Theory**

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