119169401
Bounded t-Structures on the Bounded Derived Category of Coherent Sheaves over a Weighted Projective Line
We use recollement and HRS-tilt to describe bounded t-structures on the bounded derived category $\mathcal{D}^b(\mathbb{X})$ of coherent sheaves over a weighted projective line $\mathbb{X}$ of virtual genus $\leq 1$. We will see from our description that the combinatorics in classification of bounded t-structures on $\mathcal{D}^b(\mathbb{X})$ can be reduced to that in classification of bounded t-structures on bounded derived categories of finite dimensional right modules over representation-finite finite dimensional hereditary algebras.
118584112
Tame Algebras and Integral Quadratic Forms
Integral quadratic forms.- Quivers, module categories, subspace categories (Notation, results, some proofs).- Construction of stable separating tubular families.- Tilting functors and tubular extensions (Notation, results, some proofs).- Tubular algebras.- Directed algebras.
1215706
Stability conditions on triangulated categories
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.
15519035
SILTING OBJECTS, SIMPLE-MINDED COLLECTIONS, t-STRUCTURES AND CO-t-STRUCTURES FOR FINITE-DIMENSIONAL ALGEBRAS
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. MSC 2010 classification: 16E35, 16E45, 18E30.
121549520
Almost split sequences in subcategories
Let R be a commutative artin ring and let /i be an R-algebra which is a finitely generated R-module. In [5] Auslander and Reiten introduced the notion of an almost split sequence in mod A, the category of finitely generated modules over A, and the existence and uniqueness of such short exact sequences were established there. The theory of almost split sequences developed further, and similar results for certain subcategories of mod/i were proved by Bautista and Martinez [8] and Roggenkamp [9]. The main purpose of this paper is to develop a more general theory for subcategories of mod/i having almost split sequences, which has the previous examples as special cases. Notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms. These notions were introduced in [ 61 for mod/i and further developed in [7] for subcategories of mod II. This paper is based upon [ 7] and we therefore use the same notations and conventions as used there. We now give some definitions and then proceed to describe the content of the paper section by section. Let C be a subcategory of mod II. Then a morphism g: B + C in C is said to be a right almost split morphism in C if (i) g is not a splittable epimorphism and (ii) whenever there is a nonsplittable epimorphism h: C’ + C in C, there exists an h’: C’-+ B such that gh’ = h. Dually, a morphismfi A --f B in C is said to be left almost split if (i) f is not a splittable monomorphism and (ii) whenever there is a nonsplittable monomorphism h: A + A’ there exists an h’: B + A’ such that h’f= h. We say that C has right almost split morphisms if for all indecomposable objects
120353300
Ext-projectives in suspended subcategories
Abstract Let H be a hereditary abelian k -category with tilting object and D b ( H ) denote the bounded derived category of H . This paper is devoted to a study of suspended subcategories of D b ( H ) by means of their Ext-projectives.
120681708
Gluing Silting Objects
Abstract Recent results by Keller and Nicolás and by Koenig and Yang have shown bijective correspondences between suitable classes of t-structures and cot-structures with certain objects of the derived category: silting objects. On the other hand, the techniques of gluing (co-)t-structures along a recollement play an important role in the understanding of derived module categories. Using the above correspondence with silting objects, we present explicit constructions of gluing of silting objects, and, furthermore, we answer the question of when the glued silting is tilting.
119172342
Derived equivalences for hereditary Artin algebras
We study the role of the Serre functor in the theory of derived equivalences. Let A A be an abelian category and let (U,V) ( U , V ) be a t -structure on the bounded derived category D b A D b A with heart H H . We investigate when the natural embedding H→D b A H → D b A can be extended to a triangle equivalence D b H→D b A D b H → D b A . Our focus of study is the case where A A is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exists if and only if the t -structure is bounded and the aisle U U of the t -structure is closed under the Serre functor.
123032434
Perpendicular Categories with Applications to Representations and Sheaves
This paper is concerned with the omnipresence of the formation of the subcategories right (left) perpendicular to a subcategory of objects in an abelian category. We encounter these subcategories in various contexts: 
• 
• the formation of quotient categories with respect to localizing subcategories (cf. Section 2); 
 
• 
• the deletion of vertices and shrinking of arrows (see [37]) in the representation theory of finite dimensional algebras (cf. Section 5); 
 
• 
• the comparison of the representation theories of different extended Dynkin quivers (cf. Section 10); 
 
• 
• the theory of tilting (cf. Sections 4 and 6); 
 
• 
• the study of homological epimorphisms of rings (cf. Section 4); 
 
• 
• the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme (cf. Section 7 and [21]); 
 
• 
• the study of (maximal) Cohen-Macaulay modules over surface singularities (cf. Sections); 
 
• 
• the comparison of weighted projective lines for different weight sequences (cf. Section 9); 
 
• 
• the formation of affine and local algebras attached to path algebras of extended Dynkin quivers, canonical algebras, and weighted projective lines (cf. Section 11 and [21] and the concept of universal localization in [40]). 
 
 
 
 
Formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization. This applies in particular to applications in the domain of finite dimensional algebras and their representations. Several applications of the methods presented in this paper are already in existence, partly published, or appearing in print in the near future (see, for instance, [40, 39, 4, 45, 26, 49, 46]) and have show the versatility of the notion of a perpendicular category. 
 
It seems that (right) perpendicular categories first appeared—as the subcategories of so-called closed objects—in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing Serre subcategory (see [18, 47, 34]). Another natural occurrence is encountered in Commutative Algebra, forming the possibly infinitely generated modules of depth ⩾2 (cf. Section 7). The concept and some of the central applications were first presented in a talk given by the first author at the Honnef meeting in January 1985. 
 
We also note that the perfectly matching nomination “perpendicular category” was coined by A. Schofield, who discovered independently the usefulness of this concept in dealing with hereditary algebras (see [39], cf. also Section 7). The authors further acknowledge the support of the Deutsche Forschungsgemeinschaft (SPP “Darstellungstheorie von endlichen Gruppen und endlichdimensionalen Algebren”). 
 
Throughout this paper rings are associative with unit and modules are unitary right modules. Mod(R) (respectively mod(R)) denotes the category of all (respectively all finitely presented) right R-modules.
44551953
Silting mutation in triangulated categories
In representation theory of algebras the notion of 'mutation' often plays important roles, and two cases are well known, i.e. 'cluster tilting mutation' and 'exceptional mutation'. In this paper we focus on 'tilting mutation', which has a disadvantage that it is often impossible, i.e. some of summands of a tilting object can not be replaced to get a new tilting object. The aim of this paper is to take away this disadvantage by introducing 'silting mutation' for silting objects as a generalization of 'tilting mutation'. We shall develop a basic theory of silting mutation. In particular, we introduce a partial order on the set of silting objects and establish the relationship with 'silting mutation' by generalizing the theory of Riedtmann-Schofield and Happel-Unger. We show that iterated silting mutation act transitively on the set of silting objects for local, hereditary or canonical algebras. Finally we give a bijection between silting subcategories and certain t-structures.
121044188
Representation Theory of Artin Algebras: Contents
1. Artin rings 2. Artin algebras 3. Examples of algebras and modules 4. The transpose and the dual 5. Almost split sequences 6. Finite representation type 7. The Auslander-Reiten-quiver 8. Hereditary algebras 9. Short chains and cycles 10. Stable equivalence 11. Modules determining morphisms.
119636880
Triangle singularities, ADE-chains, and weighted projective lines
We investigate the triangle singularity $f=x^a+y^b+z^c$, or $S=k[x,y,z]/(f)$, attached to a weighted projective line $X$ given by the weight triple $(a,b,c)$. We investigate the stable category of vector bundles on $X$ obtained from the vector bundles by factoring out all line bundles. This category is triangulated and has Serre duality. It is, moreover, naturally equivalent to the stable category of graded maximal Cohen-Macaulay modules over $S$ (or matrix factorizations of $f$), and then by results of Buchweitz and Orlov to the graded singularity category of $f$. We show that this category is fractional Calabi-Yau with a CY-dimension that is a function of the Euler characteristic of $X$. We show the existence of a tilting object which has the shape of an $(a-1)(b-1)(c-1)$-cuboid. Particular attention is given to the weight types $(2,a,b)$, yielding an explanation of Happel-Seidel symmetry for a class of important Nakayama algebras. In particular, the weight sequence $(2,3,p)$ corresponds to an ADE-chain, the $E_n$-chain, extrapolating the exceptional Dynkin cases $E_6$, $E_7$ and $E_8$ to a whole sequence of triangulated categories.
122755509
Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines
Background Summary Weighted projective lines Mutations of exceptional sequences Tubular mutations Twisted mutations On the number of exceptional vector bundles Tilting sheaves Tilting complexes Hyperelliptic weighted projective lines Bibliography Index.
62874
Hereditary triangulated categories
We call a triangulated category \emph{hereditary} provided that it is equivalent to the bounded derived category of a hereditary abelian category, where the equivalence is required to commute with the translation functors. If the triangulated category is algebraical, we may replace the equivalence by a triangle equivalence. We give two intrinsic characterizations of hereditary triangulated categories using a certain full subcategory and the non-existence of certain paths. We apply them to piecewise hereditary algebras.
10682282
Generators and representability of functors in commutative and noncommutative geometry
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.
9612880
A short proof of HRS-tilting
We give a short proof to the following tilting theorem by Happel, Reiten and Smal{\o} via an explicit construction: given two abelian categories $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{B}$ is tilted from $\mathcal{A}$, then $\mathcal{A}$ and $\mathcal{B}$ are derived equivalent.
119166723
Derived equivalences via HRS-tilting
Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be the Happel-Reiten-Smal{\o} tilt of $\mathcal{A}$ with respect to a torsion pair. We give necessary and sufficient conditions for the existence of a derived equivalence between $\mathcal{B}$ and $\mathcal{A}$, which is compatible with the inclusion of $\mathcal{B}$ into the derived category of $\mathcal{A}$. In particular, any splitting torsion pair induces a derived equivalence. We prove that for the realization functor of any bounded $t$-structure, its denseness implies its fully-faithfulness.