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In this paper we construct non-negative gradings on a basic Brauer tree algebra AΓ corresponding to an arbitrary Brauer tree Γ of type (m,e). We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra AS, whose tree is a star with the exceptional vertex in the middle, to AΓ. The grading on AS comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green’s walk around Γ (cf. Schaps and Zakay-Illouz (2001) [17]). By computing endomorphism rings of these tilting complexes we get graded algebras.We also compute , the group of outer automorphisms that fix the isomorphism classes of simple AΓ-modules, where Γ is an arbitrary Brauer tree, and we prove that there is unique grading on AΓ up to graded Morita equivalence and rescaling.  相似文献   

3.
Let A be an Artin algebra. A pair (C,T) of A-modules is tilting provided that C and T are (finitely generated) self-orthogonal, and . Particularly, T is a tilting module if and only if (A,T) is a tilting pair. In the note, we will extend the Auslander-Reiten correspondence for tilting modules to the context of tilting pairs.  相似文献   

4.
Following the work [B. Deng, J. Du, Frobenius morphisms and representations of algebras, Trans. Amer. Math. Soc. 358 (2006) 3591-3622], we show that a Frobenius morphism F on an algebra A induces naturally a functor F on the (bounded) derived category Db(A) of , and we further prove that the derived category Db(AF) of for the F-fixed point algebra AF is naturally embedded as the triangulated subcategory Db(A)F of F-stable objects in Db(A). When applying the theory to an algebra with finite global dimension, we discover a folding relation between the Auslander-Reiten triangles in Db(AF) and those in Db(A). Thus, the AR-quiver of Db(AF) can be obtained by folding the AR-quiver of Db(A). Finally, we further extend this relation to the root categories ?(AF) of AF and ?(A) of A, and show that, when A is hereditary, this folding relation over the indecomposable objects in ?(AF) and ?(A) results in the same relation on the associated root systems as induced from the graph folding relation.  相似文献   

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A morphism of left R-modules is a phantom morphism if for any morphism , with A finitely presented, the composition fg factors through a projective module. Equivalently, Tor1(X,f)=0 for every right R-module X. It is proved that every R-module possesses a phantom cover, whose kernel is pure injective.If is the category of finitely presented right R-modules modulo projectives, then the association M?Tor1(−,M) is a functor from the category of left R-modules to that of the flat functors on . The phantom cover is used to characterize when this functor is faithful or full. It is faithful if and only if the flat cover of every module has a pure injective kernel; this is equivalent to the flat cover being the phantom cover. The question of fullness is only reasonable when the functor is restricted to the subcategory of cotorsion modules. This restriction is full if and only if every phantom cover of a cotorsion module is pure injective.  相似文献   

7.
We study irreducible morphisms in the bounded derived category of finitely generated modules over an Artin algebra Λ, denoted , by means of the underlying category of complexes showing that, in fact, we can restrict to the study of certain subcategories of finite complexes. We prove that as in the case of modules there are no irreducible morphisms from X to X if X is an indecomposable complex. In case Λ is a selfinjective Artin algebra we show that for every irreducible morphism f in either fj is split monomorphism for all jZ or split epimorphism, for all jZ. Moreover, we prove that all the non-trivial components of the Auslander-Reiten quiver of are of the form ZA.  相似文献   

8.
A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by Db(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if Db(X) and Db(Y) are equivalent as triangulated categories.We give explicit combinatorial properties of X which are invariant under derived equivalence; among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence.For any closed subset YX, we construct a strongly exceptional collection in Db(X) and use it to show an equivalence Db(X)?Db(A) for a finite dimensional algebra A (depending on Y). We give conditions on X and Y under which A becomes an incidence algebra of a poset.We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite .This construction produces many new derived equivalences of posets and generalizes other well-known ones.As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands.  相似文献   

9.
In this paper we obtain a general version of Gabriel-Popescu theorem representing any Grothendieck category as a quotient category of the category of modules over a ring (not necessarily with unit) with enough idempotents to right using a family of generators (Ui)iI of where Ui are not supposed to be small. Applications to locally finite categories are obtained. In particular, for a coalgebra C (over a field) we prove that C is right semiperfect if and only if the category has the AB4∗ condition.  相似文献   

10.
In [H. Krause, O. Solberg, Applications of cotorsion pairs, J. London Math. Soc. 68 (2003) 631-650], the Telescope Conjecture was formulated for the module category of an artin algebra R as follows: “If C=(A,B) is a complete hereditary cotorsion pair in with A and B closed under direct limits, then ”. We extend this conjecture to arbitrary rings R, and show that it holds true if and only if the cotorsion pair C is of finite type. Then we prove the conjecture in the case when R is right noetherian and B has bounded injective dimension (thus, in particular, when C is any cotilting cotorsion pair). We also focus on the assumptions that A and B are closed under direct limits and on related closure properties, and detect several asymmetries in the properties of A and B.  相似文献   

11.
A polynomial f(T)∈Z[T] is represented by q(T)∈Z[T] if ; f(T) is graphically represented if for χM(T) the characteristic polynomial of a symmetric matrix M. Many instances of Coxeter polynomialsfA(T), for A a finite dimensional algebra, are (graphically) representable. We study the case of extended canonical algebras A, see [H. Lenzing, J.A. de la Peña, Extended canonical algebras and Fuchsian singularities, in press], show that the corresponding polynomials fA(T) are representable and satisfy a Chebysheff type recursion formula. We get consequences for the eigenvalues of the Coxeter transformation of A showing, for instance, that at most four eigenvalues may lie outside the unit circle.  相似文献   

12.
Let D be a division algebra finite-dimensional over its center C and let A=Mm(D), the m×m matrix ring over D. By the length of a linear generalized polynomial (GP) ?(X), we mean the least positive integer n such that ?(X) can be represented in the form for some . We denote by L(?)=n the length of ?. By a central linear GP for A we mean a nonzero linear GP with central values on A. In this paper we characterize all central linear GPs for A and determine the lengths of all central linear GPs for A.  相似文献   

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Let A be a noetherian AS-regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and the category of finitely generated graded left A-modules. Following Jørgensen, we define the Castelnuovo-Mumford regularity of a complex in terms of the local cohomologies or the minimal projective resolution of M. Let A! be the quadratic dual ring of A. For the Koszul duality functor , we have . Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules (also called componentwise linear modules) over A!. As an application, refining a result of Herzog and Römer, we show that if J is a monomial ideal of an exterior algebra E=?〈y1,…,yd〉, d≥3, then the (d−2)nd syzygy of E/J is weakly Koszul.  相似文献   

15.
Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpendicular category of P is equivalent to the full module category where and ?R is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism φ:RS. We characterize the case when φ is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225-3237], there exist mutually inverse isomorphisms μ and ν between the interval in the lattice of torsion classes in , and the lattice of all torsion classes in . We provide necessary and sufficient conditions for μ and ν to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when R is a Dedekind domain, and if P is finitely presented and R is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of μ and ν contains all injectives.  相似文献   

16.
Let T and A be two nonnegative regular summability matrices and W(T,p)∩l and cA(b) denote the spaces of all bounded strongly T-summable sequences with index p>0, and bounded summability domain of A, respectively. In this paper we show, among other things, that is a multiplier from W(T,p)∩l into cA(b) if and only if any subset K of positive integers that has T-density zero implies that K has A-density zero. These results are used to characterize the A-statistical comparisons for both bounded as well as arbitrary sequences. Using the concept of A-statistical Tauberian rate, we also show that is not a multiplier from W(T,p)∩l into cA(b) that leads to a Steinhaus type result.  相似文献   

17.
The main objective of this paper is to prove in full generality the following two facts:A. For an operadOinAb, letAbe a simplicialO-algebra such thatAmis generated as anO-ideal by, form>1, and letbe the Moore complex ofA. Then
  相似文献   

18.
In the paper one shows that for two indecomposable non-simple self-injective algebras over a field K we have that if the functor induces a stable equivalence then the bimodule ANB is contained in the frame of a connected component in the Auslander-Reiten quiver ΓAKBop.  相似文献   

19.
Integral quadratic forms q:ZnZ, with n≥1, and the sets Rq(d)={vZn;q(v)=d}, with dZ, of their integral roots are studied by means of mesh translation quivers defined by Z-bilinear morsifications bA:Zn×ZnZ of q, with Z-regular matrices AMn(Z). Mesh geometries of roots of positive definite quadratic forms q:ZnZ are studied in connection with root mesh quivers of forms associated to Dynkin diagrams An,Dn,E6,E7,E8 and the Auslander-Reiten quivers of the derived category Db(R) of path algebras R=KQ of Dynkin quivers Q. We introduce the concepts of a Z-morsification bA of a quadratic form q, a weighted ΦA-mesh of vectors in Zn, and a weighted ΦA-mesh orbit translation quiver Γ(Rq,ΦA) of vectors in Zn, where Rq?Rq(1) and ΦA:ZnZn is the Coxeter isomorphism defined by A. The existence of mesh geometries on Rq is discussed. It is shown that, under some assumptions on the morsification bA:Zn×ZnZ, the set admit a ΦA-orbit mesh quiver , where ΦA:ZnZn is the Coxeter isomorphism defined by A. Moreover, splits into three infinite connected components , , and , where are isomorphic to a translation quiver ZΔ, with Δ an extended Dynkin quiver, and has the shape of a sand-glass tube.  相似文献   

20.
Let H be a hereditary abelian k-category with tilting object and denote the bounded derived category of H. This paper is devoted to a study of suspended subcategories of by means of their Ext-projectives.  相似文献   

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