For a triangulated category A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S we introduce a dg-category F(S,A) parametrizing systems of exact triangles in A labelled by triangles of S. Our main result is that F(S,A) is independent on the choice of a triangulation of S up to essentially unique Morita equivalence. In particular, it admits a canonical action of the mapping class group. The proof is based on general properties of cyclic 2-Segal spaces.
In the simplest case, where A is the category of 2-periodic complexes of vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya category of the surface S. Therefore, our construction can be seen as implementing a 2-dimensional instance of Kontsevich's program on localizing the Fukaya category along a singular Lagrangian spine.
We study the problem of when triangulated categories admit unique infinity-categorical enhancements. Our results use Lurie's theory of prestable infinity-categories to give conceptual proofs of, and in many cases strengthen, previous work on the subject by Lunts--Orlov and Canonaco--Stellari. We also give a wide range of examples involving quasi-coherent sheaves, categories of almost modules, and local cohomology to illustrate the theory of prestable infinity-categories. Finally, we propose a theory of stable $n$-categories which would interpolate between triangulated categories and stable inf…
We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to Fourier-Mukai transforms are discussed. In particular, we obtain a large number of such transforms for K3 surfaces.
We prove an equivalence of triangulated categories between Orlov's triangulated category of singularities for a Gorenstein cyclic quotient singularity and the derived category of representations of a quiver with relations which is obtained from the McKay quiver by removing one vertex and half of the arrows.
We introduce the notion of log motivic triangulated categories, which is the theoretical framework for understanding the motivic aspect of cohomology theories for fs log schemes. Then we study the Grothendieck six operations formalism for log motivic triangulated categories.
We prove a stronger version of the octahedral axiom in a pre-triangulated category. The proof uses a new lemma about exact sequences in pointed additive categories which is based on a weak converse of the snake lemma.
Let $R$ be a commutative ring. We introduce the notion of support of objects in an $R$-linear triangulated category. As an application, we study the non-existence of Bridgeland stability conditions on $R$-linear triangulated categories.
An alternative approach to the classical Morel-Voevodsky stable motivic homotopy theory $SH(k)$ is suggested. The triangulated category of framed bispectra $SH_{nis}^{fr}(k)$ and effective framed bispectra $SH_{nis}^{fr,eff}(k)$ are introduced in the paper. Both triangulated categories only use Nisnevich local equivalences and have nothing to do with any kind of motivic equivalences. It is shown that $SH_{nis}^{fr}(k)$ and $SH_{nis}^{fr,eff}(k)$ recover the classical Morel-Voevodsky triangulated categories of bispectra $SH(k)$ and effective bispectra $SH^{eff}(k)$ respectively. We also recover…
We define and study the functorial spectrum for every triangulated tensor category. A reconstruction result for topologically noetherian schemes similar to (and based on) a theorem by Balmer is proved. An alternative proof of the reconstruction theorem by Bondal-Orlov for smooth projective varieties with ample (anti-)canonical bundles is given.
We develop Tannaka duality theory for dg categories. To any dg functor from a dg category $\mathcal{A}$ to finite-dimensional complexes, we associate a dg coalgebra $C$ via a Hochschild homology construction. When the dg functor is faithful, this gives a quasi-equivalence between the derived dg categories of $\mathcal{A}$-modules and of $C$-comodules. When $\mathcal{A}$ is Morita fibrant (i.e. an idempotent-complete pre-triangulated category), it is thus quasi-equivalent to the derived dg category of compact $C$-comodules. We give several applications for motivic Galois groups.
In this note we discuss three notions of dimension for triangulated categories: Rouquier dimension, diagonal dimension and Serre dimension. We prove some basic properties of these dimensions, compare them and discuss open problems.
In this paper, we compute triangular spectrum (as defined by P. Balmer) of two classes of tensor triangulated categories which are quite common in algebraic geometry. One of them is the derived category of $G$-equivariant sheaves on a smooth scheme $X$ for a finite group $G$ which acts on $X$ with some smoothness condition on the quotient. The other class is the derived category of split superschemes.
The paper contains general results on the uniqueness of a DG enhancement for triangulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded derived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded derived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derive…
We consider a closure operator $c$ of finite type on the space $SMod(\mathcal M)$ of thick $\mathcal K$-submodules of a triangulated category $\mathcal M$ that is a module over a tensor triangulated category $(\mathcal K,\otimes,1)$. Our purpose is to show that the space $SMod^c(\mathcal M)$ of fixed points of the operator $c$ is a spectral space that also carries the structure of a topological monoid.
These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.
In this survey we explain the results of the recent article arXiv:1806.06471. Following a 1973 article by Lawvere one can define metrics on categories, and following Kelly's 1982 book one can complete a category with respect to its metric. We specialize these general constructions to triangulated categories, and restrict our attention to "good metrics". And the remarkable new theorem is that, when we start with a triangulated category $\mathcal S$ with a good metric, its completion $\mathfrak{L}(\mathcal{S})$ contains an interesting subcategory $\mathfrak{S}(\mathcal{S})$ which is always trian…
Small, finite entities are easier and simpler to manipulate than gigantic, infinite ones. Consequently huge chunks of mathematics are devoted to methods reducing the study of big, cumbersome objects to an analysis of their finite building blocks. The manifestation of this general pattern, in the study of derived and triangulated categories, dates back almost to the beginnings of the subject -- more precisely to articles by Illusie in SGA6, way back in the early 1970s. What's new, at least new in the world of derived and triangulated categories, is that one gets extra mileage from analysing mor…
In this article, we investigate the category $\mathcal{A}^G$ of equivariant objects of an additive category $\mathcal{A}$ with respect to an action of a finite group $G$. We show that if $G$ is solvable then we can reconstruct $\mathcal{A}$ from $\mathcal{A}^G$ via a finite sequence of equivariantization. We also consider the possibility of a triangulated structure on $\mathcal{A}^G$ canonical in certain sense when $\mathcal{A}$ is triangulated and give several instances in which $\mathcal{A}^G$ is indeed canonically triangulated.
This short note surveys the constructions of 3-Calabi--Yau triangulated categories with simple-minded collections due to Ginzburg and Kontsevich--Soibelman and the constructions of 2-Calabi--Yau triangulated categories with cluster-tilting objects due to Buan--Marsh--Reineke--Reiten--Todorov and Amiot, and includes a discussion on the normal form of 2-Calabi--Yau triangulated categories with cluster-tilting objects.
In this paper we prove an existence of some type of equivalences between triangulated categories of singularities for varieties of different dimensions. This class of equivalences generalizes so called Knörrer periodicity. As consequence we get equivalences between categories of D-branes of type B on Landau-Ginzburg models of different dimensions.
Using the localization property, we construct a triangulated category of motives over quasi-projective T-schemes for any coefficient where T is a noetherian separated scheme, and we prove the Grothendieck six operations formalism. We also construct integral étale realization of motives.
This is a companion paper of arXiv:1901.09461, where different notions of dimension for triangulated categories are discussed. Here we compute dimensions for some examples of triangulated categories and thus illustrate and motivate material from loc. cit. Our examples include path algebras of finite ordered quivers, orbifold projective lines, some tensor powers of path algebras in Dynkin quivers of type A and categories, generated by an exceptional pair.
In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models.
We 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 semi-separated scheme.
In this paper we prove that the dimension of the bounded derived category of coherent sheaves on a smooth quasi-projective curve is equal to one. We also discuss dimension spectrums of these categories.
Consider a finite group $G$ acting on a triangulated category $\mathcal T$. In this paper we investigate triangulated structure on the category $\mathcal T^G$ of $G$-equivariant objects in $\mathcal T$. We prove (under some technical conditions) that such structure exists. Supposed that an action on $\mathcal T$ is induced by a DG-action on some DG-enhancement of $\mathcal T$, we construct a DG-enhancement of $\mathcal T^G$. Also, we show that the relation "to be an equivariant category with respect to a finite abelian group action" is symmetric on idempotent complete additive categories.
In this survey we present the relatively new concept of \emph{approximable triangulated categories.} We will show that the definition is natural, that it leads to powerful new results, and that it throws new light on old, familiar objects. In particular: a recent theorem says that the category $D_{\text{qc}}(X)$ is approximable whenever $X$ is a quasicompact separated scheme. As corollaries of this (seemingly technical) statement one can prove striking improvements on old theorems by Bondal, Rickard, Rouquier and Van den Bergh, about the (much smaller) categories $D^{\text{perf}}(X)$ and $D^b_…
Given a triangulated category over a field $K$ and a field extension $L/K$, we investigate how one can construct a triangulated category over $L$. Our approach produces the derived category of the base change scheme $X_L$ if the category one starts with is the bounded derived category of a smooth projective variety $X$ over $K$ and the field extension is finite and Galois. We also investigate how the dimension of a triangulated category behaves under scalar extensions.
The main goal of this paper is to prove that the idempotent completions of the triangulated categories of singularities of two schemes are equivalent if the formal completions of these schemes along singularities are isomorphic. We also discuss Thomason theorem on dense subcategories and a relation to the negative K-theory.
We show that Quillen's small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, Jørgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.
The present paper is devoted to study the homotopy category associated with a simplicial descent category (D,s,E) (arXiv:0808.3684v2). We prove that the class E of equivalences has a calculus of left fractions over a quotient category of D modulo homotopy. We study the fiber/cofiber sequences induced by a (co)simplicial descent structure. Examples of such fiber/cofiber sequences are deduced for (commutative) differential graded algebras, simplicial sets or topological spaces. We prove that the homotopy category of a stable simplicial descent category is triangulated. In addition, these triangu…
These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer's construction of the spectrum, generalised Rickard idempotents and support for compactly generated triangulated categories, and actions of tensor triangulated categories. Several examples are also given along the way. These notes are based on a series of lectures given during t…
The goal of the article is to better understand cosupport in triangulated categories since it is still quite mysterious. We study boundedness of local cohomology and local homology functors using Koszul objects, give some characterizations of cosupport and get some results that, in special cases, recover and generalize the known results about the usual cosupport. Also we include some computations of cosupport, settle the comparison of support and cosupport of cohomologically finite objects. Finally, we assign to any object of the category a subset of $\mathrm{Spec}R$, called the big cosupport.
In this note, I define a notion of a compactly supported object in a triangulated category. I prove a number of propositions relating this to traditional notions of support and give an application to the theory of derived Morita equivalence. I also discuss a connection to supersymmetric gauge theories arising from D-branes at a singularity.
We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.
We prove 2-out-of-3 property for rationality of motivic zeta function in distinguished triangles in Voevodsky's category DM. As an application, we show rationality of motivic zeta functions for all varieties whose motives are in the thick triangulated monoidal subcategory generated by motives of quasi-projective curves in DM. Joint with a result of P.O'Sullivan it also gives an example of a variety whose motive is not finite-dimensional while the motivic zeta function is rational.
In this paper we establish an equivalence between the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W and the triangulated category of singularities of the fiber of W over zero. The main result is a theorem that shows that the graded triangulated category of singularities of the cone over a projective variety is connected via a fully faithful functor to the bounded derived category of coherent sheaves on the base of the cone. This implies that the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W …
We develop a point-free approach for constructing the Nakano-Vashaw-Yakimov-Balmer spectrum of a noncommutative tensor triangulated category under some mild assumptions. In particular, we provide a conceptual way of classifying radical thick tensor ideals of a noncommutative tensor triangulated category using frame theoretic methods, recovering the universal support data in the process. We further show that there is a homeomorphism between the spectral space of radical thick tensor ideals of a noncommutative tensor triangulated category and the collection of open subsets of its spectrum in the…
To any triangulated category with tensor product $(K,\otimes)$, we associate a topological space $Spc(K,\otimes)$, by means of thick subcategories of $K$, a la Hopkins-Neeman-Thomason. Moreover, to each open subset $U$ of $Spc(K,\otimes)$, we associate a triangulated category $K(U)$, producing what could be thought of as a presheaf of triangulated categories. Applying this to the derived category $(K,\otimes):=(D^{perf}(X),\otimes^L)$ of perfect complexes on a noetherian scheme $X$, the topological space $Spc(K,\otimes)$ turns out to be the underlying topological space of $X$; moreover, for ea…
We give a sufficient condition for a Verdier quotient $\ct/\cs$ of a triangulated category $\ct$ by a thick subcategory $\cs$ to be realized inside of $\ct$ as an ideal quotient. As applications, we deduce three significant results by Buchweitz, Orlov and Amiot--Guo--Keller.
We introduce invariants, called shifting numbers, that measure the asymptotic amount by which an autoequivalence of a triangulated category translates inside the category. The invariants are analogous to Poincare translation numbers that are widely used in dynamical systems. We additionally establish that in some examples the shifting numbers provide a quasimorphism on the group of autoequivalences. Additionally, we relate our shifting numbers to the entropy function introduced by Dimitrov, Haiden, Katzarkov, and Kontsevich.
Given an action of a finite group on a triangulated category, we investigate under which conditions one can construct a linearised triangulated category using DG-enhancements. In particular, if the group is a finite group of automorphisms of a smooth projective variety and the category is the bounded derived category of coherent sheaves, then our construction produces the bounded derived category of coherent sheaves on the smooth quotient variety resp. stack. We also consider the action given by the tensor product with a torsion canonical bundle and the action of a finite group on the category…
We construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a non-degenerate regular system of weights whose smallest exponents are equal to -1. In the associated Grothendieck group, the strongly exceptional collection defines a root basis of a generalized root system of sign (l,0,2) and a Coxeter element of finite order, whose primitive eigenvector is a regular element in the expanded symmetric domain of type IV with respect to the Weyl group.
We give a definition of the action of a tensor triangulated category T on a triangulated category K. In the case that T is rigidly-compactly generated and K is compactly generated we show this gives rise to a notion of supports which categorifies work of Benson, Iyengar, and Krause and extends work of Balmer and Favi. We prove that a suitable version of the local-to-global principle holds very generally. A relative version of the telescope conjecture is formulated and we give a sufficient condition for it to hold.
We introduce the notion of a "baric structure" on a triangulated category, as an abstraction of S. Morel's weight truncation formalism for mixed l-adic sheaves. We study these structures on the derived category D_G(X) of G-equivariant coherent sheaves on a G-scheme X. Our main result shows how to endow this derived category with a family of nontrivial baric structures when G acts on X with finitely many orbits. We also describe a general construction for producing a new t-structure on a triangulated category equipped with given t- and baric structures, and we prove that the staggered t-structu…
We define a dimension for a triangulated category. We prove a representabilityTheorem for a certain class of functors on finite dimensional triangulatedcategories. We study the dimension of the boundedderived category of an algebra or a scheme and we show in particularthat the bounded derived category of coherent sheaves over avariety has a finite dimension. For a self-injective algebra, a lowerbound for Auslander's representation dimension is given by the dimensionof the stable category. We use this to compute the representationdimension of exterior algebras. This provides the first known exa…
We study singularity categories through Gorenstein objects in triangulated categories and silting theory. Let $ω$ be a semi-selforthogonal (or presilting) subcategory of a triangulated category $\mathcal{T}$. We introduce the notion of $ω$-Gorenstein objects, which is far extended version of Gorenstein projective modules and Gorenstein injective modules in triangulated categories. We prove that the stable category $\underline{\mathcal{G}_ω}$, where $\mathcal{G}_ω$ is the subcategory of all $ω$-Gorenstein objects, is a triangulated category and it is, under some conditions, triangle equivalent …
