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Let P n \textbf {P}^n be the projective n n -space over the complex numbers. In this note we show that an indecomposable rigid coherent sheaf on P n \textbf {P}^n has a trivial endomorphism algebra. This generalizes a result of Drézet for n = 2. n=2.

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field $k$. We will investigate homological properties of piecewise hereditary algebras $\Lambda$. In particular we give lower and upper bounds of the strong global dimension, show the behavior of the strong global under one point extensions and tilting. Moreover we show that the pieces of mod $\Lambda$ have Auslander-Reiten sequence., Oberwolfach Preprints;2009,11

Let \({{\mathcal G}}\) be a group, Λ a \({{\mathcal G}}\)-graded Artin algebra and gr(Λ) denote the category of finitely generated \({{\mathcal G}}\)-graded Λ-modules. This paper provides a framework that allows an extension of tilting theory to \({{\mathcal D}}^b(\rm gr(\Lambda))\) and to study connections between the tilting theories of \({{\mathcal D}}^b(\Lambda)\) and \({{\mathcal D}}^b(\rm gr(\Lambda))\). In particular, using that if T is a gradable Λ-module, then a grading of T induces a \({{\mathcal G}}\)-grading on EndΛ(T), we obtain conditions under which a derived equivalence induced by a gradable Λ-tilting module T can be lifted to a derived equivalence between the derived categories \({{\mathcal D}}^b(\rm gr(\Lambda))\) and \({{\mathcal D}}^b(\rm gr(\rm End_{\Lambda}(\textit T)))\).

Let k be an algebraically closed field. Let Λ be the path algebra over k of the linearly oriented quiver \(\mathbb A_n\) for n ≥ 3. For r ≥ 2 and n > r we consider the finite dimensional k −algebra Λ(n,r) which is defined as the quotient algebra of Λ by the two sided ideal generated by all paths of length r. We will determine for which pairs (n,r) the algebra Λ(n,r) is piecewise hereditary, so the bounded derived category Db(Λ(n,r)) is equivalent to the bounded derived category of a hereditary abelian category \(\mathcal H\) as triangulated category.

For a basic, hereditary, finite dimensional algebra Λ over an algebraically closed field k we consider the quiver \({\overrightarrow{\mathcal K}\!_{\Lambda}}\) of tilting modules and the subquivers of \({\overrightarrow{\mathcal K}\!_{\Lambda}}\) which are links \({\overrightarrow{{\text{lk}}}(M)}\) to partial tilting modules M and show that \({\overrightarrow{{\text{lk}}}(M)}\) is connected if M is faithful.

Let Λ be a finite-dimensional algebra over an algebraically closed field k of finite global dimension. Let M be a finitely generated Λ-module and let Γ=Λ[M] be the one point extension algebra. We show how to compute the Coxeter polynomial for Γ from the Coxeter polynomial of Λ and homological invariants of M.

For a finite dimensional hereditary algebra Λ local properties of the quiver [Formula: see text] of tilting modules are investigated. The existence of special neighbors of a given tilting module is shown. If Λ has more than 3 simple modules it is shown as an application that Λ is of wild representation type if and only if [Formula: see text] is a subquiver of [Formula: see text].

Let Λ be a finite dimensional algebra over a field k. We will show here that Λ is piecewise hereditary if and only if its strong global dimension is finite.

Let k be an algebraically closed field, B be a finite dimensional k-algebra and A be the one-point extension of B by the projective B-module P 0. We compare the posets 𝒯 A and 𝒯 B of tilting A-modules and B-modules, respectively. We prove that the restriction and the extension functors define morphisms of posets r: 𝒯 A → 𝒯 B and e: 𝒯 B → 𝒯 A such that re = id. Moreover, e induces a full embedding of the quiver of 𝒯 B into that of 𝒯 A , whose image is closed under successors, and mapping distinct connected components of the first into distinct connected components of the second.

We consider algebras Λ which satisfy the property that for each indecomposable module Λ X , either its projective dimension pd Λ X is at most one or its injective dimension id Λ X is at most one and that the global dimension gl dim Λ is three. We will show that this class is in bijective correspondence with a class of algebras of global dimension two admitting a special tilting torsion pair.

Let H be a connected hereditary abelian category over an algebraically closed field k, with finite dimensional homomorphism and extension spaces. There are two main known types of such categories: those derived equivalent to mod λ for some finite dimensional hereditary k-algebra λ and those derived equivalent to some category cohX of coherent sheaves on a weighted projective line X in the sense of Geigle and Lenzing (1987). The aim of this paper is to give a characterization of the second class in terms of some properties known to hold for these hereditary categories.

The concept of the characteristic tilting module and of the Ringel dual for quasi-hereditary algebras is generalized for the setting of standardly stratified algebras.

We prove that the projectively and the injectively defined finitistic dimensions of a standardly stratified algebra are always finite by giving the optimal bound for these numbers in terms of the number of simple modules.

Let k be an algebraically closed field and H a connected abelian k-category which is hereditary, that is Ext2( , ) vanishes on H. Assume also that Hom(X, Y ) and Ext1(X, Y ) are finite dimensional vector spaces over k for all X and Y in H. We consider such hereditary categories H which in addition have a tilting object T , that is, an object T such that {X; Ext1(T, X) = 0} = FacT , the factors of finite direct sums of copies of T . Hereditary categories H with a tilting object T are of special interest in connection with the construction of the class of algebras called quasitilted algbras, which was introduced in [HRS1]. They are by definition the algebras of the form EndH(T )op. An important property is that H and EndH(T )op have equivalent bounded derived categories. The main examples of such hereditary categories are the category modH of finitely generated modules over a finite dimensional hereditary k-algebra H and the category cohX of coherent sheaves on a weighted projective line in the sense of [GL1]. There are also others derived equivalent to them. Because of the simple description of the corresponding bounded derived category Db(H), it is possible to give a description of those in the same derived equivalence class. (See [LS] [H2]). It follows from [HRe] that they automatically have a tilting object. The quasitilted algebras provide common generalization for the tilted and the canonical finite dimensional algebras. Note that the tilted algebras are those coming from modH using an arbitrary

We develop criteria for deciding the contravariant finiteness status of a subcategory $A \subseteq \Lambda\text{-mod}$, where $\Lambda$ is a finite dimensional algebra. In particular, given a finite dimensional $\Lambda$-module $X$, we introduce a certain class of modules -- we call them $A$-phantoms of $X$ -- which indicate whether or not $X$ has a right $A$-approximation: We prove that $X$ fails to have such an approximation if and only if $X$ has infinite-dimensional $A$-phantoms. Moreover, we demonstrate that large phantoms encode a great deal of additional information about $X$ and $A$ and that they are highly accessible, due to the fact that the class of all $A$-phantoms of $X$ is closed under subfactors and direct limits.

For a basic finite-dimensional k-algebra Λ of finite global dimension over an algebraically closed field k the Coxeter matrix ΦΛ is defined. The following formula is shown: trM Θ Λ = ∑ i⩾ (−1) i dim k H i (Λ) where tr is the trace and Hi(Λ) is the ith Hochschild cohomology space.

Quasitilted algebras are generalizations of tilted algebras. As a main result we show here that the Auslander-Reiten quiver of such an algebra has a preprojective component Let k be an algebraically closed field and A be a finite-dimensional k-algebra. We denote by mod A the category of finitely generated left A-modules. By 1A we denote the Auslander-Reiten quiver of A. Recall that the vertices of 1A correspond to the isomorphism classes of indecomposable finitely generated A-modules. The number of arrows from an indecomposable A-module X to an indecomposable Amodule Y is the dimension of the k-vector space rad(X,Y)/rad2(X,Y), where rad(-, -) denotes the Jacobson radical of mod A. We denote by TX = DTr X the Auslander-Reiten translate of the indecomposable A-module X. This is defined for each indecomposable module, and in case X is non-projective the translate TX will be indecomposable and non-injective. Dually there is defined T-X = TrDX. A connected component P of 1A is called a preprojective component if P does not contain an oriented cycle and each X c P is of the form T-rp for some r c N and an indecomposable projective A-module P. For details see [ARS]. The existence of preprojective components has been established for various classes of algebras such as tilted algebras [St] or algebras satisfying the separation condition [B]. It is well known that tilted algebras may have several preprojective components. One of the important features of an indecomposable module X lying in a preprojective component is that X is homologically trivial, i.e. ExtX(X, X) = 0 for i > 0 and EndAX = k and that its isomorphism class is uniquely determined by the composition factors. Also these modules can be constructed easily by the socalled knitting procedure. Quasitilted algebras have been introduced and investigated in [HRS2]. Recall that a finite-dimensional k-algebra A is called a quasitilted algebra if there exist a hereditary abelian k-category XH and a tilting object T c XH such that A = End'HT. In this article we will not work with this definition but rather with the homological characterization established in [HRS2]. We will use the following notation. For Received by the editors March 31, 1995 and, in revised form, November 10, 1995. 1991 Mathematics Subject Classification. Primary 16G10, 16E10.

Let H = kzt be the path algebra of a quiver A, where k denotes an arbitrary field. J will always be assumed to be finite without oriented cycles. We denote by mod H the category of finite dimensional left modules over H. An indecomposable H-module X is called a stone provided Ext}~(X, X) = 0. Note that for a stone X the endomorphism ring End X = k (see for example part 1). For a stone X we denote by X l the full subcategory of mod H of the modules Y with Homn(X, Y) = 0 = Ext,(X, Y). Dually, • consists of those H-modules Y satisfying Homn(Y, X) = 0 = Ext,(Y, X). The categories X • and "Xare called the right respectively left perpendicular category of X. They are equivalent to mod Ar respectively mod At, where A r and At are again finite dimensional, hereditary k-algebras, and both have one simple module less than H (compare [GL] and [S1]). We denote by Ar respectively zt t the quivers with kAr = Ar and kJz = A,. Since their introduction to Geigle, Lenzing and Schofield, perpendicular categories have become a powerful tool in the representation theory of finite dimensional algebras. They inherit essential properties from mod H, furnish a reduction procedure and open the possibility for proofs by induction. In general, it is difficult problem to determine the (right or left) perpendicular category of a given stone X. We briefly recall what is known. It is quite easy to see (Proposition 2.1) that the quiver Ar differs from a full subquiver of zl by an admissible change of orientation if and only if X is preprojective or preinjective over H. Moreover, if H is tame, then there are only finitely many regular stones X over H, and the structure of its perpendicular categories is contained in [Rill. Hence we may restrict to the context where H is wild and X is a regular stone. Baer and Straul3 proved [B2], [St] that in this situation the perpendicular categories are wild again, but combining results of Xi IX] and Kerner [K3] it follows that certain wild module categories, namely of those algebras classified in [X], do not arise as perpendicular categories of regular stones over wild algebras. On the other hand, investigations of growth numbers in [K3] prove, that for a fixed hereditary, connected, wild algebra with at least three simple modules, there are infinitely many pairwise non isomorphic algebras Ai such that mod Ai is equivalent to X • for a regular stone X over H.

We survey some results on finite dimensional algebras of finite global dimension and address some open problems.

Let A A be an artin algebra. An indecomposable finitely generated A A -module X X is said to be on a short cycle if there exists an indecomposable finitely generated A A -module Y Y and two nonzero noninvertible maps f : X → Y f:X \to Y and g : Y → X g:Y \to X . If there are no short cycles we show that there exist only finitely many indecomposable A A -modules up to isomorphism.

with M V add M the additive category generated by ,tf, then AW iis called a tzltzng module. Let M be a partial tilting module and let C be a 11-rnod~ile such that J I b; C' is a tilting module and add M n a d d C = 0. Then C' will be called a complement to M. There are a lot of examples known where a partial tilting ~riothlle .1/1 admits a co~nplement. To mention two: if pd,ziVf 0. Auslander and Reiten prove in [AR. 5.121 that M admits a conlplen~ent provided ML is covariantlv finite. Thev ask if also the converse holds. In section one we show in an example that in general the answer to this question is negative

Let A be a finite-dimensional^-algebra over an algebraicallyclosedfieldk. We denote by mod A the category of finitelygenerated left ^-modules. For an A-module AX we denote by pd^Z (resp.id^Z) the projective(resp.injective) dimension of X. With D=Homk(―, k) we denote the standard duality with respect to the ground field.Then AD{AA) isan injectivecogenerator formod A. To formulate some of the homological conjectures we need some more notation. Let AJcmod A be the fullsubcategory containing the finitelygenerated injective A-modules. Let Kb{AS) be the homotopy category of bounded complexes over AS. Let Db(A) be the derived category of bounded complexes over mod A. We consider Kb(AJ) as a fullsubcategory of Db(A). We define. KbUJV= {X(=-Db(A)＼Hom(I, X) = 0 for allI^Kb(AJ)}.

Let Λ \Lambda be an Artin algebra. This paper presents a condition that is necessary for a subcategory of mod Λ \bmod \Lambda to be contravariantly finite. Several examples are explored in the case that Λ \Lambda is the group algebra of a finite group over a field of finite characteristic.

Let A be an iterated tilted algebra. We will construct an Auslander generator M in order to show that the representation dimension of A is three in case A is representation infinite.

Let A A be a finite-dimensional algebra. The category b m o d A bmod A of finitely generated left A A -modules canonically embeds into the derived category D b ( A ) {D^b}\left ( A \right ) of bounded complexes over b m o d A bmod A and the stable category mod _ Z T ( A ) {\underline {\bmod } ^\mathbb {Z}}T\left ( A \right ) of Z \mathbb {Z} -graded modules over the trivial extension algebra of A A by the minimal injective cogenerator. This embedding can be extended to a full and faithful functor from D b ( A ) {D^b}\left ( A \right ) to mod _ Z T ( A ) \underline {\bmod }^{\mathbb {Z}}T\left ( A \right ) . Using the concept of Auslander-Reiten triangles it is shown that both categories are equivalent only if A A has finite global dimension.

By a theorem due to the first author, the bounded derived category of a finite-dimensional algebra over a field embeds fully faithfully into the stable category over its repetitive algebra. This embedding is an equivalence iff the algebra is of finite global dimension. The purpose of this paper is to investigate the relationship between the derived category and the stable category over the repetitive algebra from various points of view for algebras of infinite global dimension. The most satisfactory results are obtained for Gorenstein algebras, especially for selfinjective algebras., 24 pages

1. Introduction 2. Basic results of classic tilting theory L. Angeleri Hugel, D. Happel and H. Krause 3. Classification of representation-finite algebras and their modules T. Brustle 4. A spectral sequence analysis of classical tilting functors S. Brenner and M. C. R. Butler 5. Derived categories and tilting B. Keller 6. Fourier-Mukai transforms L. Hille and M. Van den Bergh 7. Tilting theory and homologically finite subcategories with applications to quasihereditary algebras I. Reiten 8. Tilting modules for algebraic groups and finite dimensional algebras S. Donkin 9. Combinatorial aspects of the set of tilting modules L. Unger 10. Cotilting dualities R. Colpi and K. R. Fuller 11. Infinite dimensional tilting modules and cotorsion pairs J. Trlifaj 12. Infinite dimensional tilting modules over finite dimensional algebras O. Solberg 13. Representations of finite groups and tilting J. Chuang and J. Rickard 14. Morita theory in stable homotopy theory B. Shipley.

The aim of this article is to survey aspects of tilting theory. In the last 25 years the idea of tilting has emerged from rather special situations in terms of reflection functors [BGP] to a thorough investigation of derived categories of abelian categories admitting a tilting complex. It is impossible to survey here a full account of these developments. For some historic remarks see [H1] or [K]. Instead we will focus after a general introduction on one particular aspect. This is the theory of quasitilted algebras or equivalently that of hereditary abelian categories with tilting object as introduced in the work with Reiten and Smalo [HRS1].