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Register a new user$n$-APR tilting and $tau$-mutations
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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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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.
Let $A$ be a finite-dimensional gentle algebra over an algebraically closed field. We investigate the combinatorial properties of support $tau$-tilting graph of $A$. In particular, it is proved that the support $tau$-tilting graph of $A$ is connected and has the so-called reachable-in-face property. The property was conjectured by Fomin and Zelevinsky for exchange graphs of cluster algebras which was recently confirmed by Cao and Li.
In this paper, we introduce the notions of Gorenstein projective $tau$-rigid modules, Gorenstein projective support $tau$-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reitens bijection theorem on support $tau$-tilting modules. More precisely, for an algebra $Lambda$, we show that there is a bijection between the set of Gorenstein projective $tau$-rigid pairs in $mod Lambda$ and the set of Gorenstein injective $tau^{-1}$-rigid pairs in $mod Lambda^{rm op}$. We prove that there is a bijection between the set of Gorenstein projective support $tau$-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-$tau$-tilting finite algebras and show that $Lambda$ is CM-$tau$-tilting finite if and only if $Lambda^{rm {op}}$ is CM-$tau$-tilting finite.
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 $Lambda$.
We study which algebras have tilting modules that are both generated and cogenerated by projective-injective modules. Crawley-Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension $2$, Auslander algebras are classified by the existence of such tilting modules. In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least $2$, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting-cotilting module generated-cogenerated by projective-injective modules are precisely $1$-Auslander-Gorenstein algebras. When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.
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