Triangulated Categories Research Articles

On the levels of maps and topological realization of objects in a triangulated category

The level of a module over a differential graded algebra measures the number of steps required to build the module in an appropriate triangulated category. Based on this notion, we introduce a new homotopy invariant of spaces over a fixed space, called the level of a map. Moreover, we provide a method to compute the invariant for spaces over a K -formal space. This enables us to determine the level of the total space of a bundle over the 4-dimensional sphere with the aid of Auslander–Reiten theory for spaces due to Jørgensen. We also discuss the problem of realizing an indecomposable object in the derived category of the sphere by the singular cochain complex of a space. The Hopf invariant provides a criterion for the realization.

Formal deformations and their categorical general fibre

We study the general fibre of a formal deformation over the formal disk of a projective variety from the view point of abelian and derived categories. The abelian category of coherent sheaves of the general fibre is constructed directly from the formal deformation and is shown to be linear over the field of Laurent series. The various candidates for the derived category of the general fibre are compared. If the variety is a surface with trivial canonical bundle, we show that the derived category of the general fibre is again a linear triangulated category with a Serre functor given by the square of the shift functor. The paper is a companion to [9], where the results are applied to Fourier–Mukai equivalences of K3 surfaces.

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

Stable Projective Homotopy Theory of Modules, Tails, and Koszul Duality

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein–Gelfand–Gelfand correspondence.

Rouquier's Cocovering Theorem and Well-generated Triangulated Categories

AbstractWe study cocoverings of triangulated categories, in the sense of Rouquier, and prove that for any regular cardinal α the condition of α-compactness, in the sense of Neeman, is local with respect to such cocoverings. This was established for ordinary compactness by Rouquier. Our result yields a new technique for proving that a given triangulated category is well-generated. As an application we describe the α-compact objects in the unbounded derived category of a quasi-compact and semi-separated scheme.

Matrix problems, triangulated categories and stable homotopy types

We show how the matrix problems can be used in studying triangulated categories. Then we apply the general technique to the classi cation of stable homotopy of polyhedra, and out the \representation types of such problems and give a description of stable homotopy in nite and tame cases.

Determinant functors on triangulated categories

AbstractWe study determinant functors which are defined on a triangulated category and take values in a Picard category. The two main results are the existence of a universal determinant functor for every small triangulated category, and a comparison theorem for determinant functors on a triangulated category with a non-degenerate bounded t-structure and determinant functors on its heart. For a small triangulated category Τ we give a natural definition of groups K0(Τ) and K1(Τ) in terms of the universal determinant functor on Τ, and we show that Ki(Τ) ≅ Ki(ε) for i = 0 and 1 if Τ has a non-degenerate bounded t-structure with heart ε.

The Popescu–Gabriel theorem for triangulated categories

The Popescu–Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topological triangulated categories has recently been obtained by A. Heider.

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.

Extensions of covariantly finite subcategories

Gentle and Todorov proved that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is covariantly finite. We give an example to show that Gentle–Todorov’s theorem may fail in an arbitrary abelian category; however we prove a triangulated version of Gentle–Todorov’s theorem which holds for arbitrary triangulated categories; we apply Gentle–Todorov’s theorem to obtain short proofs of a classical result by Ringel and a recent result by Krause and Solberg.

Inducing stability conditions

We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety, we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application, we examine stability conditions on Kummer and Enriques surfaces and we improve the derived version of the Torelli Theorem for the latter surfaces already present in the literature. We also study the relationship between stability conditions on projective spaces and those on their canonical bundles.

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.

Dimensions of triangulated categories

AbstractWe define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.

Dimensions of triangulated categories

AbstractWe define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.

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.

Universal Coefficient Theorem in Triangulated Categories

We consider a homology theory on a triangulated category with values in an abelian category . If the functor h reflects isomorphisms, is full and is such that for any object x in there is an object X in with an isomorphism between h(X) and x, we prove that is a hereditary abelian category, all idempotents in split and the kernel of h is a square zero ideal which as a bifunctor on is isomorphic to .

Moduli of objects in dg-categories

The purpose of this work is to prove the existence of an algebraic moduli classifying objects in a given triangulated category. To any dg-category T (over some base ring k), we define a D − -stack M T in the sense of [ Toën B., Vezzosi G., Homotopical algebraic geometry II: Geometric stacks and applications, Mem. Amer. Math. Soc., in press], classifying certain T op -dg-modules. When T is saturated, M T classifies compact objects in the triangulated category [ T ] associated to T. The main result of this work states that under certain finiteness conditions on T (e.g. if it is saturated) the D − -stack M T is locally geometric (i.e. union of open and geometric sub-stacks). As a consequence we prove the algebraicity of the group of auto-equivalences of saturated dg-categories. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as well as complexes of representations of a finite quiver.

Homological and homotopical aspects of torsion theories

Introduction Torsion pairs in abelian and triangulated categories Torsion pairs in pretriangulated categories Compactly generated torsion pairs in triangulated categories Hereditary torsion pairs in triangulated categories Torsion pairs in stable categories Triangulated torsion(-free) classes in stable categories Gorenstein categories and (co)torsion pairs Torsion pairs and closed model structures (Co)torsion pairs and generalized Tate-Vogel cohomology Nakayama categories and Cohen-Macaulay cohomology Bibliography Index.

Support varieties: an ideal approach

We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can classify ideals in terms of their support. Applications arise from studying abelian or triangulated tensor categories. Specific examples from algebraic geometry and modular representation theory are discussed, illustrating the power of this approach which is inspired by recent work of Balmer.

Triangulated categories of singularities and equivalences between Landau-Ginzburg models

The existence of a certain type of equivalence between triangulated categories of singularities for varieties of different dimensions is proved. This class of equivalences generalizes the so-called Knörrer periodicity. As a consequence, equivalences between the categories of D-branes of type B on Landau-Ginzburg models of different dimensions are obtained.