We calculate the Grothendieck group $K_0(\cal A)$, where $\cal A$ is an additive category, locally finite over a Dedekind ring and satisfying some additional conditions. The main examples are categories of modules over finite algebras and the stable homotopy category $\mathsf{SW}$ of finite CW-complexes. We show that this group is a direct sum of a free group arising from localizations of the category $\cal A$ and a group analogous to the groups of ideal classes of maximal orders. As a corollary, we obtain a new simple proof of the Freyd's theorem describing the group $K_0(\mathsf{SW})$.
The basic properties of locally finite triangulated categories are discussed. The focus is on Auslander--Reiten theory and the lattice of thick subcategories.
We define and give axioms for Garside and locally Garside categories. We give an application to Coxeter and Artin groups and Deligne-Lusztig varieties.
We show that the homotopy category of injective $A$-modules is generically trivial if and only if the derived category of all modules is generically trivial for an algebra $A$. Moreover we show some connections between the generic objects, locally finiteness and Krull-Gabriel dimension.
We define locally wide finitary 2-categories by relaxing the definition of finitary 2-categories to allow infinitely many objects and isomorphism classes of 1-morphisms and infinite dimensional hom-spaces of 2-morphisms. After defining related concepts including transitive 2-representations in this setting, we provide a new method of constructing coalgebra 1-morphisms associated to transitive 2-representations of locally wide weakly fiat 2-categories, and demonstrate that any such transitive 2-representation is equivalent to a certain subcategory of the category of comodule 1-morphisms over th…
We show that almost split sequences in the category of comodules over a coalgebra with finite-dimensional right-hand term are direct limits of almost split sequences over finite dimensional subcoalgebras. In previous work we showed that such almost split sequences exist if the right hand term has a quasifinitely copresented linear dual. Conversely, taking limits of almost split sequences over finte-dimensional comodule categories, we then show that, for countable-dimensional coalgebras, certain exact sequences exist which satisfy a condition weaker than being almost split, which we call ``fini…
If $X$ is a quasi-compact and quasi-separated scheme, the category $Qcoh(X)$ of quasi-coherent sheaves on $X$ is locally finitely presented. Therefore categorical flat quasi-coherent sheaves naturally arise. But there is also the standard definition of flatness in $Qcoh(X)$ from the stalks. So it makes sense to wonder the relationship (if any) between these two notions. In this paper we show that there are plenty of locally finitely presented categories having no other categorical flats than the zero object. As particular instance, we show that $Qcoh(\mathbf{P}^n(R)))$ has no other categorical…
For a locally presentable abelian category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived category $\mathsf D(\mathsf B)$ is generated, as a triangulated category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck abelian category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's prin…
A new class of locally unital and locally finite dimensional algebras $A$ over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of an upper finite triangular decomposition. Any locally unital algebra which admits an upper finite Cartan decomposition is Morita equivalent to some special locally unital algebra $A$ which admits an upper finite weakly triangular decomposition. It is established that the category $A$-lfdmod of locally finite dimensional left $A$-modules is an upper finite fully stratified ca…
We are concerned with relating derived categories of all modules of two dual Koszul algebras defined by a locally bounded quiver. We first generalize the well known Acyclic Assembly Lemma and formalize an old method of extending a functor from an additive category into a complex category to its complex category. Applying this to the Koszul functor associated with a Koszul algebra defined by a gradable quiver, we obtain a Koszul complex functor, that descends to an equivalence of a continuous family of pairs of triangulated subcategories of doubly unbounded complexes of the respective derived c…
In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.
We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent abelian categories and coherent morphisms. These categories link algebra, model theory and "geometry".
For a symmetrizable GCM $C$ and its symmetrizer $D$, Geiss-Leclerc-Schröer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra $Π$ associated to $C$ and $D$, that contains a class of modules, called locally free modules. We show that any basic support $τ$-tilting $Π$-module is locally free and gives a classification theorem of torsion-free classes in $\operatorname{\mathbf{rep}}Π$ as the generalization of the work of Mizuno [Math. Z. 277 (2014)].
We study generic objects in triangulated categories and characterize the finite dimensional algebras $A$ such that the derived categories $D(\Mod A)$ are generically trivial. This is an analogue of a result of Crawley-Boevey for module categories. As a consequence, we show that $D(\Mod A)$ is generically trivial if and only if the category of perfect complexes $K^b(\proj A)$ is locally finite.
Let $A$ be the locally unital algebra associated to a cyclotomic oriented Brauer category over an arbitrary algebraically closed field $\Bbbk$ of characteristic $p\ge 0$. The category of locally finite dimensional representations of $A $ is used to give the tensor product categorification (in the general sense of Losev and Webster) for an integrable lowest weight with an integrable highest weight representation of the same level for the Lie algebra $\mathfrak g$, where $\mathfrak g$ is a direct sum of copies of $\mathfrak {sl}_\infty$ (resp., $ \hat{\mathfrak {sl}}_p$ ) if $p=0$ (resp., $p>0$)…
We show that every k-linear abelian Ext-finite hereditary category with Serre duality which is generated by preprojective objects is derived equivalent to the category of representations of a strongly locally finite thread quiver.
It is shown that the idempotent completion of the additive hull of the tensor product of the residue category of the category of paths of a locally finite quiver modulo an admissible ideal and a dualizing category is dualizing. Furthermore, the category of finitely presented functors over such tensor product category is dualizing and has almost split sequences. As applications, the categories of all kinds of complexes are proved to have almost split sequences.
Starting with a Grothendieck category $\mathcal{G}$ and a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal G$, we study the local finite presentability and local coherence of the heart $\mathcal{H}_{\mathbf{t}}$ of the associated Happel-Reiten-Smalø $t$-structure in the derived category $\mathrm{Der} (\mathcal{G})$. We start by showing that, in this general setting, the torsion pair $\mathbf t$ is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those $\mathbf t$ for which $\mathcal{H}_{\mathbf{t}}$ is locally finite…
Let $\cal C$ be a $(d+2)$-angulated category. In this note, we show that if $\cal C$ is a locally finite, then $\cal C$ has Auslander-Reiten $(d+2)$-angles. This extends a result of Xiao-Zhu for triangulated categories.
The aim of the paper is to discuss the relation subgroups of the Grothendieck groups of extriangulated categories and certain other subgroups. It is shown that a locally finite extriangulated category $\C$ has Auslander-Reiten $\E-$triangles and the relations of the Grothendieck group $K_{0}(\C)$ are generated by the Auslander-Rieten $\E-$triangles. A partial converse result is given when restricting to the triangulated categories with a cluster tilting subcategory: in the triangulated category $\C$ with a cluster tilting subcategory, the relations of the Grothendieck group $K_0(\C)$ are gener…
Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category $\mathcal C$ as modules over the smash extension $\mathcal C\# H$. We construct Grothendieck spectral sequences for the cohomologies as well as the $H$-locally finite cohomologies of these objects. We also introduce relative $(\mathcal D,H)$-Hopf modules over a Hopf comodule category $\mathcal D$. These generalize relative $(A,H)$-Hopf modules over an $H$-comodule algebra $A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational $Hom$ objects and higher derived func…
Let $\mathcal{A}$ be a locally bounded $k$-category and $G$ a torsion-free group of $k$-linear automorphisms of $\mathcal{A}$ acting freely on the objects of $\mathcal{A},$ and $F:\mathcal{A}\rightarrow \mathcal{B}$ is a Galois functor. We extend naturally the push-down functor $F_λ$ to the functor $\rm{H}\rm{F}_λ:\rm{H}(\rm{mod}\mbox{-} \mathcal{A})\rightarrow \rm{H}(\rm{mod}\mbox{-} \mathcal{B})$, resp. $\mathcal{S} \rm{F}_λ:\mathcal{S}(\rm{mod}\mbox{-} \mathcal{A})\rightarrow \mathcal{S}(\rm{mod}\mbox{-} \mathcal{B})$, between the corresponding morphism categories, resp. monomorphism catego…
Let $G$ be a $p$-adic Lie group with reductive Lie algebra $\mathfrak{g}$. Denote by $D(G)$ the locally analytic distribution algebra of $G$. Orlik-Strauch and Agrawal-Strauch have studied certain exact functors defined on various categories of $\mathfrak{g}$-representations with image in the category of locally analytic $G$-representations or $D(G)$-modules. In this paper we prove that for suitably defined categories of $D(G)$-modules, this functor gives rise to injective homomorphisms at the level of Grothendieck groups. We also explain how this functor interacts with translation functors at…
For a finite dimensional algebra $A$, we prove that the bounded homotopy category of projective $A$-modules and the bounded derived category of $A$-modules are dual to each other via certain categories of locally-finite cohomological functors. The duality gives rise to a $2$-categorical duality between certain strict $2$-categories involving the bounded homotopy categories and bounded derived categories, respectively. We apply the $2$-categorical duality to the study of triangle autoequivalence groups. These results are analogous to the ones in [M.R. Ballard, {\em Derived categories of sheaves…
Let $\mathscr{C}$ be an $n$-exangulated category. In this note, we show that if $\mathscr{C}$ is locally finite, then $\mathscr{C}$ has Auslander-Reiten $n$-exangles. This unifies and extends results of Xiao-Zhu, Zhu-Zhuang, Zhou and Xie-Lu-Wang for triangulated, extriangulated, $(n+2)$-angulated and $n$-abelian categories, respectively.
We introduce a very general extension of the monomorphism category as studied by Ringel and Schmidmeier which in particular covers generalised species over locally bounded quivers. We prove that analogues of the kernel and cokernel functor send almost split sequences over the path algebra and the preprojective algebra to split or almost split sequences in the monomorphism category. We derive this from a general result on preservation of almost split morphisms under adjoint functors whose counit is a monomorphism. Despite of its generality, our monomorphism categories still allow for explicit c…
We introduce a new method to construct a Grothendieck category from a given colored quiver. This is a variant of the construction used to prove that every partially ordered set arises as the atom spectrum of a Grothendieck category. Using the new method, we prove that for every finite partially ordered set, there exists a locally noetherian Grothendieck category such that every nonzero object contains a compressible subobject and its atom spectrum is isomorphic to the given partially ordered set.
Xiao and Zhu has shown that if $\cal C$ is a locally finite triangulated category, then the Auslander-Reiten triangles generate the relations for the Grothendieck group of $\cal C$. The notion of $(d +2)$-angulated categories is a "higher dimensional" analogue of triangulated categories. In this article, we show that if A $(d+2)$-angulated category $\cal C$ is locally finite if and only if the Auslander-Reiten $(d+2)$-angles generate the relations for the Grothendieck group of $\cal C$. This extends the result of Xiao and Zhu, and gives the converse of Xiao and Zhu's result is also true.
Given a locally coherent Grothendieck category G, we prove that the homotopy category of complexes of injective objects (also known as the coderived category of G) is compactly generated triangulated. Moreover, the full subcategory of compact objects is none other than D^b(fp G). If G admits a generating set of finitely presentable objects of finite projective dimension, then also the derived category of G is compactly generated and Krause's recollement exists. Our main tools are (a) model theoretic techniques and (b) a systematic study of the pure derived category of an additive finitely acce…
