Which Canonical Algebras Are Derived Equivalent to Incidence Algebras of Posets?

نویسنده

  • SEFI LADKANI
چکیده

We give a full description of all the canonical algebras over an algebraically closed field that are derived equivalent to incidence algebras of finite posets. These are the canonical algebras of weight type other than (1, p) whose number of weights does not exceed 3. This note concerns the characterization of the canonical algebras over an algebraically closed field that are derived equivalent to incidence algebras of finite partially ordered sets (posets), expressed in the following theorem. Theorem. Let Λ be a canonical algebra of type (p,λ) over an algebraically closed field. Then Λ is derived equivalent to an incidence algebra of a poset if and only if the number of weights of p does not exceed 3 and p 6= (1, p) for p ≥ 1. This theorem can be interpreted both geometrically and algebraically. From a geometric viewpoint, by considering modules over incidence algebras as sheaves over finite spaces [8] and using the derived equivalence between the categories of modules over a canonical algebra and coherent sheaves over a weighted projective line [3], we are able to obtain explicit derived equivalences between the categories of sheaves of finite dimensional vector spaces over certain finite T0 topological spaces and the categories of coherent sheaves over certain weighted projective lines. From an algebraic viewpoint, in an attempt to classify all piecewise hereditary incidence algebras over an algebraically closed field, one first asks which types of piecewise hereditary categories can actually occur. Happel’s classification [6] tells us that we only need to consider the canonical algebras and path algebras of quivers. For the canonical algebras the theorem above gives a complete answer, while for path algebras, see the remarks in Section 2.2. We finally note that for the constructions of incidence algebras derived equivalent to canonical algebras, the assumption that the base field is algebraically closed can be omitted. Acknowledgement. I would like to thank Helmut Lenzing for useful discussions related to this paper. 0. Notations The canonical algebras were introduced in [11]. Let k be a field, p = (p1, . . . , pt) be a sequence of t positive integers (weights), and λ = (λ3, . . . , λt) be a sequence of pairwise distinct elements of k \ {0}. The canonical algebra of type (p,λ), is the algebra Λ(p,λ) = kQ/I where Q is 1

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Universal Derived Equivalences of Posets of Tilting Modules

We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their tilting modules are related by a simple combinatorial construction, which we call flip-flop. We deduce that the posets of tilting modules of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent.

متن کامل

Universal Derived Equivalences of Posets of Cluster Tilting Objects

We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flip-flop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian latt...

متن کامل

CLUSTER ALGEBRAS AND CLUSTER CATEGORIES

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...

متن کامل

On derived equivalences of lines, rectangles and triangles

We present a method to construct new tilting complexes from existing ones using tensor products, generalizing a result of Rickard. The endomorphism rings of these complexes are generalized matrix rings that are “componentwise” tensor products, allowing us to obtain many derived equivalences that have not been observed by using previous techniques. Particular examples include algebras generalizi...

متن کامل

STONE DUALITY FOR R0-ALGEBRAS WITH INTERNAL STATES

$Rsb{0}$-algebras, which were proved to be equivalent to Esteva and Godo's NM-algebras modelled by Fodor's nilpotent minimum t-norm, are the equivalent algebraic semantics of the left-continuous t-norm based fuzzy logic firstly introduced by Guo-jun Wang in the mid 1990s.In this paper, we first establish a Stone duality for the category of MV-skeletons of $Rsb{0}$-algebras and the category of t...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2008