Algebraic Triangulated Category Research Articles

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.

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.

Definability and approximations in triangulated categories

We give criteria for subcategories of a compactly generated algebraic triangulated category to be precovering or preenveloping. These criteria are formulated in terms of closure conditions involving products, coproducts, directed homotopy colimits and further conditions involving the notion of purity. In particular, we provide sufficient closure conditions for a subcategory of a compactly generated algebraic triangulated category to be a torsion class. Finally we explore applications of the previous results to the theory of recollements.

Classifying Subcategories in Quotients of Exact Categories

We classify certain subcategories in quotients of exact categories. In particular, we classify the triangulated and thick subcategories of an algebraic triangulated category, i.e. the stable category of a Frobenius category.

Silting theory in triangulated categories with coproducts

We introduce the notion of noncompact (partial) silting and (partial) tilting sets and objects in any triangulated category D with arbitrary (set-indexed) coproducts. We show that equivalence classes of partial silting sets are in bijection with t-structures generated by their co-heart whose heart has a generator, and in case D is compactly generated, this restricts to: i) a bijection between equivalence classes of self-small partial silting objects and left nondegenerate t-structures in D whose heart is a module category and whose associated cohomological functor preserves products; ii) a bijection between equivalence classes of classical silting objects and nondegenerate smashing and co-smashing t-structures whose heart is a module category.We describe the objects in the aisle of the t-structure associated to a partial silting set T as Milnor (or homotopy) colimits of sequences of morphisms with successive cones in Sum(T)[n]. We use this fact to develop a theory of tilting objects in very general AB3 abelian categories, a setting and its dual in which we show the validity of several well-known results of tilting and cotilting theory of modules. Finally, we show that if T is a bounded tilting set in a compactly generated algebraic triangulated category D and H is the heart of the associated t-structure, then the inclusion H↪D extends to a triangulated equivalence D(H)⟶∼D which restricts to bounded levels.

On two Hall algebra approaches to odd periodic triangulated categories

Given an odd-periodic algebraic triangulated category, we compare Bridgelands Hall algebra in thesense of Bridgeland (2013) and Gorsky (2014), and the derived Hall algebra in the sense of Toen (2006), Xiao and Xu (2008) and Xu and Chen (2013), andshow that the former one is the twisted form of the tensor product of the latter one and a suitable group algebra.

A construction of derived equivalent pairs of symmetric algebras

Recently, Hu and Xi have exhibited derived equivalent endomorphism rings arising from (relative) almost split sequences as well as AR-triangles in triangulated categories. We present a broader class of triangles (in algebraic triangulated categories) for which the endomorphism rings of different terms are derived equivalent. We then study applications involving 0-Calabi-Yau triangulated categories. In particular, applying our results in the category of perfect complexes over a symmetric algebra gives a nice way of producing pairs of derived equivalent symmetric algebras. Included in the examples we work out are some of the algebras of dihedral type with two or three simple modules. We also apply our results to stable categories of Cohen-Macaulay modules over odd-dimensional Gorenstein hypersurfaces having an isolated singularity.

A Note on Derived Equivalences for Φ-Green Algebras

In this note, we construct derived equivalences for Φ-Green algebras from a class of triangles in algebraic triangulated categories, where Φ is a finite admissible set of a set $\mathbb {N}$ of the natural numbers.

The p -order of topological triangulated categories

In this paper we explain certain systematic differences between algebraic and topological triangulated categories. A triangulated category is algebraic if it admits a differential graded model, and topological if it admits a model in the form of a stable cofibration category. The precise statements use the 'n-order' of a triangulated category, for a natural number n. The n-order is a non-negative integer (or infinity) and measures `how strongly' n annihilates objects of the form Y/n. We show the following results: the n-order of an algebraic triangulated category is infinite; for every prime p, the p-order of a topological triangulated category is at least p-1; the p-order of the p-local stable homotopy category is exactly p-1. In particular, the p-local stable homotopy category is not algebraic for any prime p. As a tool we develop certain foundations about enrichments of cofibration categories by Delta-sets; in particular we generalize the theory of `framings' (or `cosimplicial resolutions') from model categories to cofibration categories.

The n -order of algebraic triangulated categories

We quantify certain features of algebraic triangulated categories using the ‘n-order’, an invariant that measures how strongly n annihilates objects of the form Y/n. We show that the n-order of an algebraic triangulated category is infinite, and that the p-order of the p-local stable homotopy category is exactly p−1 for any prime p. In particular, the p-local stable homotopy category is not algebraic.

Cluster tilting vs. weak cluster tilting in Dynkin type A infinity

Abstract This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category 𝖢 with the following properties. On the one hand, the d-cluster tilting subcategories of 𝖢 have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of 𝖢 which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 ≤ ℓ ≤ d - 1, we show a weakly d-cluster tilting subcategory 𝖳 ℓ ${\mathsf {T}_{\ell }}$ which has an indecomposable object with precisely ℓ mutations. The category 𝖢 is the algebraic triangulated category generated by a (d + 1)-spherical object and can be thought of as a higher cluster category of Dynkin type A ∞.

Sparseness of t-structures and negative Calabi-Yau dimension in triangulated categories generated by a spherical object

Let k be an algebraically closed field and let T be the k-linear algebraic triangulated category generated by a w-spherical object for an integer w. For certain values of w this category is classical. For instance, if w = 0 then it is the compact derived category of the dual numbers over k. As main results of the paper we show that for w \leq 0, the category T has no non-trivial t-structures, but does have one family of non-trivial co-t-structures, whereas for w \geq 1 the opposite statement holds. Moreover, without any claim to originality, we observe that for w \leq -1, the category T is a candidate to have negative Calabi-Yau dimension since \Sigma^w is the unique power of the suspension functor which is a Serre functor.

The Ringel-Hall Lie algebra of a spherical object

For an integer w, let 𝒮w be the algebraic triangulated category generated by a w-spherical object. We determine the Picard group of 𝒮w and show that each orbit category of 𝒮w is triangulated and is triangle equivalent to a certain orbit category of the bounded derived category of a standard tube. Moreover, we describe the Ringel–Hall Lie algebra associated to 𝒮w in the sense of Peng and Xiao.

On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon

Let k be a field and let D be a k-linear algebraic triangulated category with split idempotents. Let be the suspension functor of D and let s be a 2-spherical object of D, that is, the morphism space D(s, i s) is k for i = 0 and i = 2 and vanishes otherwise. Assume that s classically generates D, that is, each object of D can be built from s using (de)suspensions, direct sums, direct summands, and distinguished triangles. It was proved in [15, thm. 2.1] that D is uniquely determined by these properties. As we will explain, D is a good candidate for a cluster category of Dynkin type A∞. For instance, we show that there is a bijection between the cluster tilting subcategories of D and certain triangulations of the∞-gon. We use this to give an example of a subcategory A which is weak cluster tilting, that is, satisfies A = ( −1A )⊥ = ⊥( A ), but fails to be functorially finite. Perpendicular

The Hall algebra of a spherical object

We determine the Hall algebra, in the sense of Toen, of the algebraic triangulated category generated by a spherical object.

Homotopy theory of well-generated algebraic triangulated categories

AbstractFor every regular cardinal α, we construct a cofibrantly generated Quillen model structure on a category whose objects are essentially dg categories which are stable under suspensions, cosuspensions, cones and α-small sums.Using results of Porta, we show that the category of well-generated (algebraic) triangulated categories in the sense of Neeman is naturally enhanced by our Quillen model category.

Realising Smashing Localisations as Morphisms of DG Algebras

For a smashing localisation L of the derived category of a differential graded (dg) algebra A we construct a dg algebra A L and a morphism of dg algebras A→A L that induces the canonical map in cohomology. As a first application we obtain a localisations $A_{\mathfrak{p}}$ of a dg algebra A with graded commutative homology at a prime ideal $\mathfrak{p}$ in the homology H * A, namely a morphism $A\to A_{\mathfrak{p}}$ of dg algebras. As a second application we can use results of Keller to “model” every smashing localisation of compactly generated algebraic triangulated categories by a morphism of dg algebras.