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
Φ
\Phi
is a finite dimensional algebra, then each functorially finite wide subcategory of
mod
(
Φ
)
\operatorname {mod}( \Phi )
is of the form
ϕ
∗
(
mod
(
Γ
)
)
\phi _{ {\textstyle *}}\big ( \operatorname {mod}( \Gamma ) \big )
in an essentially unique way, where
Γ
\Gamma
is a finite dimensional algebra and
Φ
⟶
ϕ
Γ
\Phi \stackrel { \phi }{ \longrightarrow } \Gamma
is an algebra epimorphism satisfying
Tor
1
Φ
(
Γ
,
Γ
)
=
0
\operatorname {Tor}^{ \Phi }_1( \Gamma ,\Gamma ) = 0
.
Let
F
⊆
mod
(
Φ
)
\mathscr {F} \subseteq \operatorname {mod}( \Phi )
be a
d
d
-cluster tilting subcategory as defined by Iyama. Then
F
\mathscr {F}
is a
d
d
-abelian category as defined by Jasso, and we call a subcategory of
F
\mathscr {F}
wide if it is closed under sums, summands,
d
d
-kernels,
d
d
-cokernels, and
d
d
-extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of
F
\mathscr {F}
is of the form
ϕ
∗
(
G
)
\phi _{ {\textstyle *}}( \mathscr {G} )
in an essentially unique way, where
Φ
⟶
ϕ
Γ
\Phi \stackrel { \phi }{ \longrightarrow } \Gamma
is an algebra epimorphism satisfying
Tor
d
Φ
(
Γ
,
Γ
)
=
0
\operatorname {Tor}^{ \Phi }_d( \Gamma ,\Gamma ) = 0
, and
G
⊆
mod
(
Γ
)
\mathscr {G} \subseteq \operatorname {mod}( \Gamma )
is a
d
d
-cluster tilting subcategory.
We illustrate the theory by computing the wide subcategories of some
d
d
-cluster tilting subcategories
F
⊆
mod
(
Φ
)
\mathscr {F} \subseteq \operatorname {mod}( \Phi )
over algebras of the form
Φ
=
k
A
m
/
(
rad
k
A
m
)
ℓ
\Phi = kA_m / (\operatorname {rad}\,kA_m )^{ \ell }
.