Discreteness of silting objects and t‐structures in triangulated categories

We introduce the notion of ST-pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite-dimensional algebra. For an ST-pair $(\C,\D)$, we construct an order-preserving map from silting objects in $\C$ to bounded $t$-structures on $\D$ and show that the map is bijective if and only if $\C$ is silting-discrete if and only if $\D$ is $t$-discrete. Based on a work of Qiu and Woolf, the above result is applied to show that if $\C$ is silting-discrete then the stability space of $\D$ is contractible. This is used to obtain the contractibility of the stability spaces of some Calabi--Yau triangulated categories associated to Dynkin quivers.

Related papers:

title: On t-structures and torsion theories induced by compact objects
abstract: First, we show that a compact object C in a triangulated category, which satisfies suitable conditions, induces a t-structure. Second, in an abelian category we show that a complex P · of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for P · to be a tilting complex. Finally, in the case of artin algebras, we give a one-to-one correspondence between tilting complexes of term length two and torsion theories with certain conditions.

title: Homological and Homotopical Aspects of Torsion Theories
abstract: 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.

title: Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)
abstract: In this paper we introduce a new notion of a weight structure (w) for a triangulated category C; this notion is an important natural counterpart to the notion of t-structures. This allows to generalize certain results of the previous paper  to a large class of triangulated categories and functors. The heart of w is an additive Hw C; there are no non-trivial C-distinguished triangles in Hw. We prove that a weight structure defines Postnikov towers for any X2 ObjC (whose "factors" are X i 2 ObjHw); these towers are canonical and functorial "up to zero cohomology". For any (co)homological functor H : C! A (A is abelian) we construct a weight spectral sequence T : H(X i [j]) =) H(X[i+j]); it is canonical and functorial starting from E2. This spectral sequences specializes to the usual weight spectral sequences for "classical" realizations of (Voevodsky’s) motives and to the Atiyah-Hirzebruch sequence in topology. We prove that K0(C) = K0(Hw) in the bounded case if Hw is idempotent complete. Under certain restrictions, we prove a similar equality for K0(EndC). We define a canonical conservative weakly exact functor t from C to a certain weak category of complexes Kw(Hw).

title: Auslander–Buchweitz Context and Co-t-structures
abstract: We show that the relative Auslander–Buchweitz context on a triangulated category $\mathcal{T}$ coincides with the notion of co-t-structure on certain triangulated subcategory of $\mathcal{T}$ (see Theorem 3.8). In the Krull–Schmidt case, we establish a bijective correspondence between co-t-structures and cosuspended, precovering subcategories (see Theorem 3.11). We also give a characterization of bounded co-t-structures in terms of relative homological algebra. The relationship between silting classes and co-t-structures is also studied. We prove that a silting class ω induces a bounded non-degenerated co-t-structure on the smallest thick triangulated subcategory of $\mathcal{T}$ containing ω. We also give a description of the bounded co-t-structures on $\mathcal{T}$ (see Theorem 5.10). Finally, as an application to the particular case of the bounded derived category ${\mathbf{D}^b}(\mathcal{H}),$ where $\mathcal{H}$ is an abelian hereditary category which is Hom-finite, Ext-finite and has a tilting object (see Happel and Reiten,  1999), we give a bijective correspondence between finite silting generator sets ω = add (ω) and bounded co-t-structures (see Theorem 6.7).

title: Realisation functors in tilting theory
abstract: Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (non-compact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Rickard on the shape of derived equivalences. Finally, we apply the techniques of this new derived Morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t-structures glueing to a tilting or a cotilting t-structure. As a further application, we answer a question by Xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras.

title: Almost complete cluster tilting objects in generalized higher cluster categories
abstract: We study higher cluster tilting objects in generalized higher cluster categories arising from dg algebras of higher Calabi-Yau dimension. Taking advantage of silting mutations of Aihara-Iyama, we obtain a class of $m$-cluster tilting objects in generalized $m$-cluster categories. For generalized $m$-cluster categories arising from strongly ($m+2$)-Calabi-Yau dg algebras, by using truncations of minimal cofibrant resolutions of simple modules, we prove that each almost complete $m$-cluster tilting $P$-object has exactly $m+1$ complements with periodicity property. This leads us to the conjecture that each liftable almost complete $m$-cluster tilting object has exactly $m+1$ complements in generalized $m$-cluster categories arising from $m$-rigid good completed deformed preprojective dg algebras.

title: Three kinds of mutation
abstract: For a finite dimensional hereditary algebra, we consider: exceptional sequences in the category of finite dimensional modules, silting objects in the bounded derived category, and m-cluster tilting objects in the m-cluster category. There are mutation operations on both the set of m-cluster tilting objects and the set of exceptional sequences. It is also possible to define a mutation operation for silting objects. We compare these three different notions of mutation.

title: Silting reduction and Calabi--Yau reduction of triangulated categories
abstract: It is shown that the silting reduction $\ct/\thick\cp$ of a triangulated category $\ct$ with respect to a presilting subcategory $\cp$ can be realized as a certain subfactor category of $\ct$, and that there is a one-to-one correspondence between the set of (pre)silting subcategories of $\ct$ containing $\cp$ and the set of (pre)silting subcategories of $\ct/\thick\cp$. This is analogous to a result for Calabi-Yau reduction. This result is applied to show that Amiot-Guo-Keller's construction of $d$-Calabi-Yau triangulated categories with $d$-cluster-tilting objects takes silting reduction to Calabi-Yau reduction.

title: Discrete derived categories II: the silting pairs CW complex and the stability manifold
abstract: Discrete derived categories were studied initially by Vossieck [‘The algebras with discrete derived category’,  (2001) 168–176] and later by Bobinski, Geis and Skowronski [‘Classification of discrete derived categories’, Cent.  (2004) 19–49]. In this article, we define the CW complex of silting pairs for a triangulated category and show that it is contractible in the case of discrete derived categories. We provide an explicit embedding from the silting CW complex into the stability manifold. By work of Qiu and Woolf [‘Contractible stability spaces and faithful braid group actions’,  2014, ], there is a deformation retract of the stability manifold onto the silting pairs CW complex. We obtain that the space of stability conditions of discrete derived categories is contractible.

title: Universal localizations via silting
abstract: We show that silting modules are closely related with localizations of rings. More precisely, every partial silting module gives rise to a localization at a set of maps between countably generated projective modules and, conversely, every universal localization, in the sense of Cohn and Schofield, arises in this way. To establish these results, we further explore the finite-type classification of tilting classes and we use the morphism category to translate silting modules into tilting objects. In particular, we prove that silting modules are of finite type.

title: SILTING OBJECTS, SIMPLE-MINDED COLLECTIONS, t-STRUCTURES AND CO-t-STRUCTURES FOR FINITE-DIMENSIONAL ALGEBRAS
abstract: Bijective correspondences are established between (1) silting objects, (2) simple- minded collections, (3) bounded t-structures with length heart and (4) bounded co-t-structures. These correspondences are shown to commute with mutations. The results are valid for finite- dimensional non-positive differential graded algebras, in particular for finite dimensional alge- bras. A concrete example is given to illustrate how these correspondences help to compute the space of Bridgeland's stability conditions.

title: Stability conditions on triangulated categories
abstract: This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's notion of $\Pi$-stability. From a mathematical point of view, the most interesting feature of the definition is that the set of stability conditions $\Stab(\T)$ on a fixed category $\T$ has a natural topology, thus defining a new invariant of triangulated categories. After setting up the necessary definitions I prove a deformation result which shows that the space $\Stab(\T)$ with its natural topology is a manifold, possibly infinite-dimensional.

title: STABILITY CONDITIONS AND QUANTUM DILOGARITHM IDENTITIES FOR DYNKIN QUIVERS
abstract: We study the fundamental groups of the exchange graphs for the bounded derived category D ( Q ) of a Dynkin quiver Q and the finite-dimensional derived category D ( Γ N Q ) of the Calabi–Yau-N Ginzburg algebra associated to Q. In the case of D ( Q ) , we prove that its space of stability conditions (in the sense of Bridgeland) is simply connected. As an application, we show that the Donaldson–Thomas type invariant associated to Q can be calculated as a quantum dilogarithm function on its exchange graph. In the case of D ( Γ N Q ) , we show that the faithfulness of the Seidel–Thomas braid group action (which is known for Q of type A or N = 2 ) implies the simply connectedness of the space of stability conditions.

title: Wakamatsu tilting modules
abstract: We study a generalization of tilting modules to modules of possibly infinite projective dimension, introduced by Wakamatsu in . In particular, we characterize these modules in terms of suitable subcategories of finitely generated modules and in terms of cotorsion theories.

title: Discrete triangulated categories
abstract: We introduce and study several homological notions which generalise the discrete derived categories of D. Vossieck. As an application, we show that Vossieck discrete algebras have this property with respect to all bounded t-structures. We give many examples of triangulated categories regarding these notions.

title: On differential graded categories
abstract: Differential graded categories enhance our understanding of triangulated categories
appearing in algebra and geometry. In this survey, we review their foundations and report on
recent work by Drinfeld, Dugger�Shipley, Toen, Toen�Vaquie, and others.

title: Derived equivalences from mutations of quivers with potential
abstract: We show that Derksen–Weyman–Zelevinsky's mutations of quivers with potential yield equivalences of suitable 3-Calabi–Yau triangulated categories. Our approach is related to that of Iyama–Reiten and ‘Koszul dual’ to that of Kontsevich–Soibelman. It improves on previous work by Vitoria.  A, the first-named author studies pseudo-compact derived categories of certain pseudo-compact dg algebras.

title: Generators and representability of functors in commutative and noncommutative geometry
abstract: We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.

title: Dimensions of triangulated categories
abstract: We 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.

title: Weight structures and simple dg modules for positive dg algebras
abstract: Using techniques due to Dwyer-Greenlees-Iyengar we construct weight structures in triangulated categories generated by compact objects. We apply our result to show that, for a dg category whose homology vanishes in negative degrees and is semi-simple in degree 0, each simple module over the homology lifts to a dg module which is unique up to isomorphism in the derived category. This allows us, in certain situations, to deduce the existence of a canonical t-structure on the perfect derived category of a dg algebra. From this, we can obtain a bijection between hearts of t-structures and sets of so-called simple-minded objects for some dg algebras (including Ginzburg algebras associated to quivers with potentials). In three appendices, we elucidate the relation between Milnor colimits and homotopy colimits and clarify the construction of t-structures from sets of compact objects in triangulated categories as well as the construction of a canonical weight structure on the unbonded derived category of a non positive dg category.

title: Relative singularity categories I: Auslander resolutions
abstract: Let R be an isolated Gorenstein singularity with a non-commutative resolution A = End R ( R ⊕ M ) . In this paper, we show that the relative singularity category Δ R ( A ) of A has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category D s g ( R ) of Buchweitz and Orlov as a certain canonical quotient category. If R has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that D s g ( R ) determines Δ R ( Aus ( R ) ) , where Aus ( R ) is the corresponding Auslander algebra. The proofs of these results use dg algebras, A ∞ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.

title: Categorical resolution of singularities
abstract: Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We then propose the definition of a categorical resolution of singularities. Our main example is the derived category D ( X ) of quasi-coherent sheaves on a scheme X. We prove that D ( X ) has a canonical categorical resolution if the base field is perfect and X is a separated scheme of finite type with a dualizing complex.

title: Cluster algebras II: Finite type classification
abstract: This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. 
The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.

title: Stability conditions on CYN categories associated to An-quivers and period maps
abstract: In this paper, we study the space of stability conditions on a certain N- Calabi-Yau (CYN) category associated to an An-quiver. Recently, Bridgeland and Smith constructed stability conditions on some CY3 categories from meromorphic quadratic differentials with simple zeros. Generalizing their results to higher dimen- sional Calabi-Yau categories, we describe the space of stability conditions as the universal cover of the space of polynomials of degree n + 1 with simple zeros. In particular, central charges of stability conditions on CYN categories are constructed as the periods of quadratic differentials with zeros of order N −2 which are associated to polynomials.