A quiver representation assigns a vector space to each vertex, and a linear map to each arrow of a quiver. When one considers the category $\mathrm {Vect}(\mathbb {F}_1)$ of vector spaces “over $\mathbb {F}_1$” (the field with one element), one obtains $\mathbb {F}_1$-representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficient quivers. To be precise, we prove that the category $\mathrm {Rep}(Q,\mathbb {F}_1)$ is equivalent to the (suitably defined) category of coefficient quivers over Q. This provides a conceptual way to see Euler characteristics of a class of quiver Grassmannians as the number of “$\mathbb {F}_1$-rational points” of quiver Grassmannians. We generalize techniques originally developed for string and band modules to compute the Euler characteristics of quiver Grassmannians associated with $\mathbb {F}_1$-representations. These techniques apply to a large class of $\mathbb {F}_1$-representations, which we call the $\mathbb {F}_1$-representations with finite nice length: we prove sufficient conditions for an $\mathbb {F}_1$-representation to have finite nice length, and classify such representations for certain families of quivers. Finally, we explore the Hall algebras associated with $\mathbb {F}_1$-representations of quivers. We answer the question of how a change in orientation affects the Hall algebra of nilpotent $\mathbb {F}_1$-representations of a quiver with bounded representation type. We also discuss Hall algebras associated with representations with finite nice length, and compute them for certain families of quivers.

We establish some properties of $\tau$-exceptional sequences for finite-dimensional algebras. In an earlier paper, we established a bijection between the set of ordered support $\tau$-tilting modules and the set of complete signed $\tau$-exceptional sequences. We describe the action of the symmetric group on the latter induced by its natural action on the former. Similarly, we describe the effect on a $\tau$-exceptional sequence obtained by mutating the corresponding ordered support $\tau$-tilting module via a construction of Adachi-Iyama-Reiten.

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 article, 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.

We investigate symmetry of the silting quiver of a given algebra which is induced by an anti-automorphism of the algebra. In particular, one shows that if there is a primitive idempotent fixed by the anti-automorphism, then the 2-silting quiver ($=$ the support $\tau$-tilting quiver) has a bisection. Consequently, in that case, we obtain that the cardinality of the 2-silting quiver is an even number (if it is finite).

In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi–Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi–Yau algebras for cluster algebras with polarized principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables) and obtain Frobenius categorifications in the acyclic case. Via partial stabilization, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh’s result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$-Calabi–Yau.

Let R be a commutative Noetherian ring. We prove that if R is either an equidimensional finitely generated algebra over a perfect field, or an equidimensional equicharacteristic complete local ring with a perfect residue field, then the annihilator of the singularity category of R coincides with the Jacobian ideal of R up to radical. We establish a relationship between the annihilator of the singularity category of R and the cohomological annihilator of R under some mild assumptions. Finally, we give an upper bound for the dimension of the singularity category of an equicharacteristic excellent local ring with isolated singularity. This extends a result of Dao and Takahashi to non-Cohen–Macaulay rings.

We prove the flow tree formula conjectured by Alexandrov and Pioline, which computes Donaldson–Thomas invariants of quivers with potentials in terms of a smaller set of attractor invariants. This result is obtained as a particular case of a more general flow tree formula reconstructing a consistent scattering diagram from its initial walls.

Motivated by a new conjecture on the behavior of bricks, we start a systematic study of minimal $\tau $-tilting infinite (min-$\tau $-infinite, for short) algebras. In particular, we treat min-$\tau $-infinite algebras as a modern counterpart of minimal representation-infinite algebras and show some of the fundamental similarities and differences between these families. We then relate our studies to the classical tilting theory and observe that this modern approach can provide fresh impetus to the study of some old problems. We further show that in order to verify the conjecture, it is sufficient to treat those min-$\tau $-infinite algebras where almost all bricks are faithful. Finally, we also prove that minimal extending bricks have open orbits, and consequently obtain a simple proof of the brick analogue of the first Brauer–Thrall conjecture, recently shown by Schroll and Treffinger using some different techniques.

It was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild character varieties described as moduli spaces of points on $\mathbb {P}^1$ by Sibuya. Furthermore, Boalch noticed that these varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which are attached to the quiver $\Gamma _n$ on two vertices and n equioriented arrows. In this article, we go a step further by unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modelled on the quiver $\Gamma _n$. We prove that the Poisson structure carried by these varieties is induced, via the Kontsevich–Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver $\Gamma _n$ such that its noncommutative multiplicative moment map is given in terms of Euler continuants. This result generalises the Hamiltonian double quasi-Poisson algebra associated with the quiver $\Gamma _1$ by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to $\Gamma _n$ admits a factorisation in terms of n copies of the algebra attached to $\Gamma _1$.

Let Q be a quiver of type $\tilde {A}_n$. Let $\alpha =\alpha _1+\alpha _2+\cdots +\alpha _s$ be the canonical decomposition. For the polynomials $M_Q(\alpha ,q)$ that count the number of isoclasses of representations of Q over ${\mathbb F}_q$ with dimension vector $\alpha $, we obtain a precise relation between the degree of $M_Q(\alpha ,q)$ and that of $\prod _{i=1}^{s} M_Q(\alpha _i,q)$ for an arbitrary dimension vector $\alpha $.

In this note, we correct an oversight regarding the modules from Definition 4.2 and proof of Lemma 5.12 in Baur et al. (Nayoga Math. J., 2020, 240, 322–354). In particular, we give a correct construction of an indecomposable rank $2$ module $\operatorname {\mathbb {L}}\nolimits (I,J)$, with the rank 1 layers I and J tightly $3$-interlacing, and we give a correct proof of Lemma 5.12.

We prove that a finite-dimensional algebra $ \Lambda $ is $ \tau $-tilting finite if and only if all the bricks over $ \Lambda $ are finitely generated. This is obtained as a consequence of the existence of proper locally maximal torsion classes for $ \tau $-tilting infinite algebras.

We show that the dimer algebra of a connected Postnikov diagram in the disc is bimodule internally $3$-Calabi–Yau in the sense of the author’s earlier work [43]. As a consequence, we obtain an additive categorification of the cluster algebra associated to the diagram, which (after inverting frozen variables) is isomorphic to the homogeneous coordinate ring of a positroid variety in the Grassmannian by a recent result of Galashin and Lam [18]. We show that our categorification can be realised as a full extension closed subcategory of Jensen–King–Su’s Grassmannian cluster category [28], in a way compatible with their bijection between rank $1$ modules and Plücker coordinates.

We consider a quiver with potential (QP) $(Q(D),W(D))$ and an iced quiver with potential (IQP) $(\overline {Q}(D), F(D), \overline {W}(D))$ associated with a Postnikov Diagram D and prove that their mutations are compatible with the geometric exchanges of D. This ensures that we may define a QP $(Q,W)$ and an IQP $(\overline {Q},F,\overline {W})$ for a Grassmannian cluster algebra up to mutation equivalence. It shows that $(Q,W)$ is always rigid (thus nondegenerate) and Jacobi-finite. Moreover, in fact, we show that it is the unique nondegenerate (thus rigid) QP by using a general result of Geiß, Labardini-Fragoso, and Schröer (2016, Advances in Mathematics 290, 364–452).

Then we show that, within the mutation class of the QP for a Grassmannian cluster algebra, the quivers determine the potentials up to right equivalence. As an application, we verify that the auto-equivalence group of the generalized cluster category ${\mathcal {C}}_{(Q, W)}$ is isomorphic to the cluster automorphism group of the associated Grassmannian cluster algebra ${{\mathcal {A}}_Q}$ with trivial coefficients.

Given a negatively graded Calabi-Yau algebra, we regard it as a DG algebra with vanishing differentials and study its cluster category. We show that this DG algebra is sign-twisted Calabi-Yau and realise its cluster category as a triangulated hull of an orbit category of a derived category and as the singularity category of a finite-dimensional Iwanaga-Gorenstein algebra. Along the way, we give two results that stand on their own. First, we show that the derived category of coherent sheaves over a Calabi-Yau algebra has a natural cluster tilting subcategory whose dimension is determined by the Calabi-Yau dimension and the a-invariant of the algebra. Second, we prove that two DG orbit categories obtained from a DG endofunctor and its homotopy inverse are quasi-equivalent. As an application, we show that the higher cluster category of a higher representation infinite algebra is triangle equivalent to the singularity category of an Iwanaga-Gorenstein algebra, which is explicitly described. Also, we demonstrate that our results generalise the context of Keller–Murfet–Van den Bergh on the derived orbit category involving a square root of the AR translation.

Postnikov constructed a cellular decomposition of the totally nonnegative Grassmannians. The poset of cells can be described (in particular) via Grassmann necklaces. We study certain quiver Grassmannians for the cyclic quiver admitting a cellular decomposition, whose cells are naturally labeled by Grassmann necklaces. We show that the posets of cells coincide with the reversed cell posets of the cellular decomposition of the totally nonnegative Grassmannians. We investigate algebro-geometric and combinatorial properties of these quiver Grassmannians. In particular, we describe the irreducible components, study the action of the automorphism groups of the underlying representations, and describe the moment graphs. We also construct a resolution of singularities for each irreducible component; the resolutions are defined as quiver Grassmannians for an extended cyclic quiver.

We describe all groups that can be generated by two twists along spherical sequences in an enhanced triangulated category. It will be shown that with one exception such a group is isomorphic to an abelian group generated by not more than two elements, the free group on two generators or the braid group of one of the types $A_2$, $B_2$ and $G_2$ factorised by a central subgroup. The last mentioned subgroup can be nontrivial only if some specific linear relation between length and sphericity holds. The mentioned exception can occur when one has two spherical sequences of length 3 and sphericity 2. In this case the group generated by the corresponding two spherical twists can be isomorphic to the nontrivial central extension of the symmetric group on three elements by the infinite cyclic group. Also we will apply this result to give a presentation of the derived Picard group of selfinjective algebras of the type $D_4$ with torsion 3 by generators and relations.

Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $. Let $\Lambda $ be an R-order such that $Q\Lambda $ is a separable Q-algebra. Maranda showed that there exists $k\in \mathbb {N}$ such that for all $\Lambda $-lattices L and M, if $L/L\pi ^k\simeq M/M\pi ^k$, then $L\simeq M$. Moreover, if R is complete and L is an indecomposable $\Lambda $-lattice, then $L/L\pi ^k$ is also indecomposable. We extend Maranda’s theorem to the class of R-reduced R-torsion-free pure-injective $\Lambda $-modules.

As an application of this extension, we show that if $\Lambda $ is an order over a Dedekind domain R with field of fractions Q such that $Q\Lambda $ is separable, then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of $\Lambda $ is isomorphic to the lattice of open subsets of the R-torsion-free part of the left Ziegler spectrum of $\Lambda $.

Furthermore, with k as in Maranda’s theorem, we show that if M is R-torsion-free and $H(M)$ is the pure-injective hull of M, then $H(M)/H(M)\pi ^k$ is the pure-injective hull of $M/M\pi ^k$. We use this result to give a characterization of R-torsion-free pure-injective $\Lambda $-modules and describe the pure-injective hulls of certain R-torsion-free $\Lambda $-modules.

We continue the work started in parts (I) and (II) of this series. In this paper, we classify which continuous quivers of type A are derived equivalent. Next, we define the new ${\mathcal {C}(A_{{\mathbb {R}},S})}$, which we call weak continuous cluster category. It is a triangulated category, it does not have cluster structure but it has a new weaker notion of “cluster theory.” We show that the original continuous cluster category of [15] is a localization of this new weak continuous cluster category. We define cluster theories to be appropriate groupoids and we show that cluster structures satisfy the conditions for cluster theories. We describe the relationship between different cluster theories: some new and some obtained from cluster structures. The notion of continuous mutation which appears in cluster theories (but not in cluster structures) appears in the next paper [20].

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of surfaces with orbifold points, establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices.