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Let $Lambda$ be a finite-dimensional algebra. A wide subcategory of $mathsf{mod}Lambda$ is called left finite if the smallest torsion class containing it is functorially finite. In this paper, we prove that the wide subcategories of $mathsf{mod}Lambda$ arising from $tau$-tilting reduction are precisely the Serre subcategories of left finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $tau$-tilting reduction. This leads to a natural way to extend the definition of the $tau$-cluster morphism category of $Lambda$ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan-Marsh in the $tau$-tilting finite case and by Igusa-Todorov in the hereditary case.
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Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(mathcal A ,mathcal B,mathcal C)$ of extriangulated categories, we show that $n$-tilting (resp. $n$-cotilting) subcategories in $mathcal A$ and $mathcal C$ can be glued to get $n$-tilting (resp. $n$-cotilting) subcategories in $mathcal B$ under certain conditions.
We define evaluation forms associated to objects in a module subcategory of Ext-symmetry generated by finitely many simple modules over a path algebra with relations and prove a multiplication formula for the product of two evaluation forms. It is analogous to a multiplication formula for the product of two evaluation forms associated to modules over a preprojective algebra given by Geiss, Leclerc and Schroer in cite{GLS2006}.
Let $Lambda$ be an artin algebra and $mathcal{M}$ be an n-cluster tilting subcategory of mod$Lambda$. We show that $mathcal{M}$ has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of $mathcal{M}$ if and only if every effaceable functor $mathcal{M}rightarrow Ab$ has finite length. As a consequence we show that if mod$Lambda$ has n-cluster tilting subcategory of finite type then the n-almost split sequences form a basis for the relations for the Grothendieck group of $Lambda$.
Let $mathscr C$ be a Krull-Schmidt $(n+2)$-angulated category and $mathscr A$ be an $n$-extension closed subcategory of $mathscr C$. Then $mathscr A$ has the structure of an $n$-exangulated category in the sense of Herschend-Liu-Nakaoka. This construction gives $n$-exangulated categories which are not $n$-exact categories in the sense of Jasso nor $(n+2)$-angulated categories in the sense of Geiss-Keller-Oppermann in general. As an application, our result can lead to a recent main result of Klapproth.
APR tilts for path algebra $kQ$ can be realized as the mutation of the quiver $Q$ in $mathbb Z Q$ with respect to the translation. In this paper, we show that we have similar results for the quadratic dual of truncations of $n$-translation algebras, that is, under certain condition, the $n$-APR tilts of such algebras are realized as $tau$-mutations.For the dual $tau$-slice algebras with bound quiver $Q^{perp}$, we show that their iterated $n$-APR tilts are realized by the iterated $tau$-mutations in $mathbb Z|{n-1}Q^{perp}$.
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