Triangulated Categories Research Articles

On n-hereditary algebras and n-slice algebras

In this paper we show that acyclic n-slice algebras are exactly acyclic n-hereditary algebras whose (n+1)-preprojective algebras are (q+1,n+1)-Koszul. We also list the equivalent triangulated categories arising from the algebra constructions related to an n-slice algebra. We show that higher slice algebras of finite type appear in pairs and they share the Auslander-Reiten quiver for their higher-preprojective modules.

The Structure of Aisles and Co-aisles of t-Structures and Co-t-structures

Right triangulated categories can be thought of as triangulated categories whose shift functor is not an equivalence. We give intrinsic characterisations of when such categories are appearing as the (co-)aisle of a (co-)t-structure in an associated triangulated category. Along the way, we also give an interpretation of these structures in the language of extriangulated categories.

A partial compactification of the Bridgeland stability manifold

AbstractBridgeland stability manifolds of Calabi–Yau categories are of noticeable interest both in mathematics and physics. By looking at some of the known examples, a pattern clearly emerges and gives a fairly precise description of how they look like. In particular, they all seem to have missing loci, which tend to correspond to degenerate stability conditions vanishing on spherical objects. Describing such missing strata is also interesting from a mirror-symmetric perspective, as they conjecturally parametrize interesting types of degenerations of complex structures. All the naive attempts at constructing modular partial compactifications show how elusive and subtle the problem in fact is: ideally, the missing strata would correspond to stability manifolds of quotient triangulated categories, but establishing such a correspondence on the geometric level and viewing stability conditions on quotients of the original triangulated category as suitable degenerations of stability conditions is not straightforward. In this paper, we will present a method to construct such partial compactifications if some additional hypotheses are satisfied, by realizing our space of interest as a suitable metric completion of the stability manifold.

Tilting theory for periodic triangulated categories

Tilting theory for periodic triangulated categories

T-Structures on stable derivators and Grothendieck hearts

We prove that, given any strong and stable derivator and a t-structure in its base triangulated category D, the t-structure canonically lifts to all the (coherent) diagram categories and each incoherent diagram in the heart uniquely lifts to a coherent one. We use this to show that the t-structure being compactly generated implies that the coaisle is closed under directed homotopy colimits which, in turn, implies that the heart is an (Ab.5) Abelian category. If, moreover, D is a well-generated algebraic or topological triangulated category, then the heart of any accessibly embedded (in particular, compactly generated) t-structure has a generator. As a consequence, it follows that the heart of any compactly generated t-structure of a well-generated algebraic or topological triangulated category is a Grothendieck Abelian category.

Silting Objects over the Stable Monomorphism Category of Higher Differential Objects

Higher differential objects are investigated and used for addressing three general problems. Torsionless differential modules over path algebras are characterized. The adjoint triples between triangulated categories, involving derived categories and singularity categories, are allowed to be constructed from those between the abelian categories [Formula: see text] and [Formula: see text]. The partial silting properties between an abelian category [Formula: see text] and [Formula: see text] are transferred, and if moreover,[Formula: see text] is Frobenius, the partial silting objects of the stable monomorphism categories of [Formula: see text] are constructed from those of [Formula: see text].

Approximating triangulated categories by spaces

We initiate a systematic study of lattices of thick subcategories for arbitrary essentially small triangulated categories. To this end we give several examples illustrating the various properties these lattices may, or may not, have and show that as soon as a lattice of thick subcategories is distributive it is automatically a spatial frame. We then construct two non-commutative spectra, one functorial and one with restricted functoriality, that give universal approximations of these lattices by spaces.

On Spectral Sequences for the Postnikov Towers

A natural spectral sequence corresponding to a pair of a Postnikov tower (system) and a (co)homological functor from a triangulated category is constructed. The results are applied to some classical filtrations in the homotopy category K(A). In triangulated categories endowed with weight structures, the weight spectral sequences (defined by M. V. Bondarko) with some additional properties are studied.

Reductions of triangulated categories and simple‐minded collections

AbstractSilting and Calabi–Yau reductions are important processes in representation theory to construct new triangulated categories from given ones, which are similar to Verdier quotient. In this paper, first we introduce a new reduction process of triangulated category, which is analogous to the silting (Calabi–Yau) reduction. For a triangulated category with a pre‐simple‐minded collection (pre‐SMC) , we construct a new triangulated category such that the SMCs in bijectively correspond to those in containing . Second, we give an analogue of Buchweitz's theorem for the singularity category of a SMC quadruple : the category can be realized as the stable category of an extriangulated subcategory of . Finally, we show the simple‐minded system (SMS) reduction due to Coelho Simões and Pauksztello is the shadow of our SMC reduction. This is parallel to the result that Calabi–Yau reduction is the shadow of silting reduction due to Iyama and Yang.

Gluing approximable triangulated categories

AbstractGiven a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful.Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category isapproximableif this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that$\mathsf {D}_{\mathsf {qc}}( X )$is approximable whenXis a quasi compact, separated scheme led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier.In this article, we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable. Furthermore, the lemmas and techniques developed in this article form a powerful toolbox which, in conjunction with the groundwork laid in [16], has interesting applications in existing and forthcoming work by the authors.

Noncommutative tensor triangular geometry

We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M$\Delta$C, ${\bf K}$, and then to associate to ${\bf K}$ a topological space--the Balmer spectrum ${\rm Spc}{\bf K}$. We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that ${\rm Spc}{\bf K}$ is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M$\Delta$C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of ${\bf K}$, which in turn can be applied to classify the thick two-sided ideals and ${\rm Spc}{\bf K}$.

On the Composition of Two Spherical Twists

Abstract Segal [17] proved that any autoequivalence of an enhanced triangulated category can be realised as a spherical twist. However, when exhibiting an autoequivalence as a spherical twist one has various choices for the source category of the spherical functor. We describe a construction that realises the composition of two spherical twists as the twist around a single spherical functor whose source category semi-orthogonally decomposes into the source categories for the spherical functors we started with. We give a description of the cotwist for this spherical functor and prove, in the special case when our starting twists are around spherical objects, that the cotwist is the Serre functor (up to a shift). We finish with an explicit treatment for the case of $\mathbb {P}$-objects.

Groups generated by two twists along spherical sequences

AbstractWe 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.

Yoneda algebras and their singularity categories

For a finite dimensional algebra Λ $\Lambda$ of finite representation type and an additive generator M $M$ for mod Λ $\operatorname{mod}\Lambda$ , we investigate the properties of the Yoneda algebra Γ = ⨁ i ⩾ 0 Ext Λ i ( M , M ) $\Gamma =\bigoplus _{i \geqslant 0}\operatorname{Ext}_\Lambda ^i(M,M)$ . We show that Γ $\Gamma$ is graded coherent and Gorenstein of self-injective dimension at most 1, and the graded singularity category D sg Z ( Γ ) $\mathrm{D_{sg}^\mathbb {Z}}(\Gamma )$ of Γ $\Gamma$ is triangle equivalent to the derived category of the stable Auslander algebra of Λ $\Lambda$ . These results remain valid for representation-infinite algebras. For this we introduce the Yoneda category Y $\mathcal {Y}$ of Λ $\Lambda$ as the additive closure of the shifts of the Λ $\Lambda$ -modules in the derived category D b ( mod Λ ) $\mathrm{D^b}(\operatorname{mod}\Lambda )$ . We show that Y $\mathcal {Y}$ is coherent and Gorenstein of self-injective dimension at most 1, and the singularity category of Y $\mathcal {Y}$ is triangle equivalent to the derived category D b ( mod ( mod ̲ Λ ) ) $\mathrm{D^b}(\operatorname{mod}(\operatorname{\underline{\operatorname{mod}}}\Lambda ))$ of the stable category mod ̲ Λ $\operatorname{\underline{\operatorname{mod}}}\Lambda$ . To give a triangle equivalence, we apply the theory of realization functors. We show that any algebraic triangulated category has an f-category over itself by formulating the filtered derived category of a DG category, which assures the existence of a realization functor.

Idempotent completion of right suspended categories

<abstract><p>We show that the idempotent completion of a right suspended category has a natural structure of right suspended category and dually this is true for a left suspended category. This unifies and extends results of Balmer-Schlichting, Bühler and Liu-Sun for triangulated, exact and right triangulated categories, respectively.</p></abstract>

New model structures and projective (injective) cotorsion pairs

Let [Formula: see text] be either the category of R-modules or the category of chain complexes of R-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer n, and hence, one can get new algebraic triangulated categories. Second, it is shown that any n-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let M be an R-module with finite flat dimension and n a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then M is injective.

Rank functions on triangulated categories

Abstract 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.

Lifting of recollements and gluing of partial silting sets

This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.

Relative stable categories and birationality

We propose a general method to construct new triangulated categories, relative stable categories, as additive quotients of a given one. This construction enhances results of Beligiannis, particularly in the tensor-triangular setting. We prove a birationality result showing that the original category and its relative stable quotient are equivalent on some open piece of their spectrum.

On the E-infinity Algebras

In this paper we study an elementary use of E-infinity modules and E-infinity algebras as together they have a use in terms of describing triangulated categories. Also, we show an interpretation of E-infinity algebras where the modules are fibrant objects within the categories of differential graded co-algebras and co-modules.