Rank functions on triangulated categories

We introduce the notion of a rank function on a triangulated category 𝒞 {\mathcal{C}} which generalizes the Sylvester rank function in the case when 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} is the perfect derived category of a ring A. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn’s matrix localization of rings and has independent interest.

Almost split morphisms in subcategories of triangulated categories

For a suitable triangulated category [Formula: see text] with a Serre functor [Formula: see text] and a full precovering subcategory [Formula: see text] closed under summands and extensions, an indecomposable object [Formula: see text] in [Formula: see text] is called Ext-projective if Ext[Formula: see text]. Then there is no Auslander–Reiten triangle in [Formula: see text] with end term [Formula: see text]. In this paper, we show that if, for such an object [Formula: see text], there is a minimal right almost split morphism [Formula: see text] in [Formula: see text], then [Formula: see text] appears in something very similar to an Auslander–Reiten triangle in [Formula: see text]: an essentially unique triangle in [Formula: see text] of the form [Formula: see text] where [Formula: see text] is an indecomposable not in [Formula: see text] and [Formula: see text] is a [Formula: see text]-envelope of [Formula: see text]. Moreover, under some extra assumptions, we show that removing [Formula: see text] from [Formula: see text] and replacing it with [Formula: see text] produces a new subcategory of [Formula: see text] closed under extensions. We prove that this process coincides with the classic mutation of [Formula: see text] with respect to the rigid subcategory of [Formula: see text] generated by all the indecomposable Ext-projectives in [Formula: see text] apart from [Formula: see text]. When [Formula: see text] is the cluster category of Dynkin type [Formula: see text] and [Formula: see text] has the above properties, we give a full description of the triangles in [Formula: see text] of the form [Formula: see text] and show under which circumstances replacing [Formula: see text] by [Formula: see text] gives a new extension closed subcategory.

Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories

We prove that if the Auslander–Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangulated converse to a theorem of Butler and Auslander–Reiten on the relations for Grothendieck groups. Our approach has applications in the context of Frobenius categories.

Homology and K-theory of dynamical systems I. Torsion-free ample groupoids

Given an ample groupoid, we construct a spectral sequence with groupoid homology with integer coefficients on the second sheet, converging to the K-groups of the (reduced) groupoid C $^*$ -algebra, provided the groupoid has torsion-free stabilizers and satisfies a strong form of the Baum–Connes conjecture. The construction is based on the triangulated category approach to the Baum–Connes conjecture developed by Meyer and Nest. We also present a few applications to topological dynamics and discuss the HK conjecture of Matui.

Recollements, comma categories and morphic enhancements

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$ , there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$ . Examples related to inflation categories and weighted projective lines are discussed.

APPLICATION OF THE TRIANGULATED CATEGORY TO THE EXPLANATION OF FULLY-CHARM TETRAQUARK MASS

The discovery in July 2020 of fully-charm tetraquark led to the need for its theoretical explanation. For investigation of such complex four-quark formation, the modern mathematical apparatus of the theory of derived categories is used. By representing diquarks as solitonic objects in terms of sheaves, one can explain the measured mass of the broad resonance of fully-charm tetraquarks consisting of di-charmonia.

Tropical Duality in $$(d+2)$$-Angulated Categories

Let $$\mathscr {C}$$ C be a 2-Calabi–Yau triangulated category with two cluster tilting subcategories $$\mathscr {T}$$ T and $$\mathscr {U}$$ U . A result from Jørgensen and Yakimov (Sel Math (NS) 26:71–90, 2020) and Demonet et al. (Int Math Res Not 2019:852–892, 2017) known as tropical duality says that the index with respect to $$\mathscr {T}$$ T provides an isomorphism between the split Grothendieck groups of $$\mathscr {U}$$ U and $$\mathscr {T}$$ T . We also have the notion of c-vectors, which using tropical duality have been proven to have sign coherence, and to be recoverable as dimension vectors of modules in a module category. The notion of triangulated categories extends to the notion of $$(d+2)$$ ( d + 2 ) -angulated categories. Using a higher analogue of cluster tilting objects, this paper generalises tropical duality to higher dimensions. This implies that these basic cluster tilting objects have the same number of indecomposable summands. It also proves that under conditions of mutability, c-vectors in the $$(d+2)$$ ( d + 2 ) -angulated case have sign coherence, and shows formulae for their computation. Finally, it proves that under the condition of mutability, the c-vectors are recoverable as dimension vectors of modules in a module category.

On the Voevodsky motive of the moduli space of Higgs bundles on a curve

We study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

Resolving resolution dimensions in triangulated categories

Let T {\mathcal{T}} be a triangulated category with a proper class ξ \xi of triangles and X {\mathcal{X}} be a subcategory of T {\mathcal{T}} . We first introduce the notion of X {\mathcal{X}} -resolution dimensions for a resolving subcategory of T {\mathcal{T}} and then give some descriptions of objects having finite X {\mathcal{X}} -resolution dimensions. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors and characterize objects having finite X {\mathcal{X}} -resolution dimensions in terms of a notion of ξ \xi -cellular towers. We also construct a new resolving subcategory from a given resolving subcategory and reformulate some known results.

On uniqueness of p-adic period morphisms, II

We prove equality of the various rational $p$-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the $K$-theoretical uniqueness criterion we had found earlier for proper smooth schemes extends to proper finite simplicial schemes in the good reduction case and to cohomology with compact support in the semistable reduction case. It yields the equality of the period morphisms for cohomology with compact support defined using the syntomic, almost étale, and motivic constructions. We continue with showing that the $h$-cohomology period morphism agrees with the syntomic and almost étale period morphisms whenever the latter morphisms are defined (and up to a change of Hyodo–Kato cohomology). We do it by lifting the syntomic and almost étale period morphisms to the $h$-site of varieties over a field, where their equality with the $h$-cohomology period morphism can be checked directly using the Beilinson Poincaré lemma and the case of dimension $0$. This also shows that the syntomic and almost étale period morphisms have a natural extension to the Voevodsky triangulated category of motives and enjoy many useful properties (since so does the $h$-cohomology period morphism).