Tilted string algebras

Huard, François; Liu, Shiping

doi:10.1016/s0022-4049(99)00101-2

Cited by 15 publications

(25 citation statements)

References 15 publications

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“…The proof of the following lemma is similar to those of [13] (2.3) and [9] (2.6), which are done in the contexts of gentle and special biserial algebras respectively. Lemma 3.2.4.…”

Section: Tilting Modulementioning

confidence: 94%

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Nakayama oriented pullbacks and stably hereditary algebras

Lévesque

2008

Journal of Pure and Applied Algebra

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We introduce a new class of algebras, the Nakayama oriented pullbacks, obtained from pullbacks of surjective morphisms of algebras A C and B C. We prove that such a pullback is tilted when A and B are hereditary. We also show that stably hereditary algebras respecting the clock condition are Nakayama oriented pullbacks, and we use results about these pullbacks to show when a stably hereditary algebra is tilted or iterated tilted. MSC: 16G20; 16G70; 16S50 IntroductionModule categories and pullbacks of rings and algebras have been studied from many points of view (see, for instance, [1,6,11,12,15,18]), but not from the tilting point of view. In this paper, we introduce a particular class of pullbacks of algebras over an algebraically closed field K , which we call Nakayama oriented pullbacks. These are pullbacks of surjective morphisms of algebras A C and B C with C a hereditary Nakayama algebra. We construct a tilting module T over this kind of pullback, and we compute the endomorphism algebra End T of this module when B is hereditary (see 2.4.6). A first consequence of this result is the principal theorem of this paper:Theorem. Let R be a Nakayama oriented pullback of K -algebra surjective morphisms A C and B C. Suppose that A and B are hereditary. Then R is tilted.We next show that a stably hereditary algebra respecting the clock condition, that is, such that the number of clockwise oriented relations on each cycle of its bound quiver equals the number of counterclockwise oriented relations, can be expressed as a Nakayama oriented pullback. As a second consequence of the main theorem above we give a new proof of Theorem 2.6 of [14] which characterizes iterated tilted and tilted stably hereditary algebras.This paper consists of three sections. The first is devoted to preliminaries, the second to Nakayama oriented pullbacks, and the third to stably hereditary algebras. * Tel.

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“…The proof of the following lemma is similar to those of [13] (2.3) and [9] (2.6), which are done in the contexts of gentle and special biserial algebras respectively. Lemma 3.2.4.…”

Section: Tilting Modulementioning

confidence: 94%

“…Finally, a reduced walk is a double-zero if it contains exactly two zero-relations which point in the same direction in w. Double-zeros have been used for the classification of tilted and quasi-tilted special biserial algebras [2,9,10,13].…”

Section: Notationsmentioning

confidence: 99%

Nakayama oriented pullbacks and stably hereditary algebras

Lévesque

2008

Journal of Pure and Applied Algebra

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“…Proof (⇐) From the characterization of quasi-tilted string algebras given in [15], we get that is quasi-tilted if and only if there are no consecutive zero-relations in (Q, I).…”

Section: Theorem 4 Let Kq/i Be a Gentle Algebra And K An Algebraicallmentioning

confidence: 99%

On the Derived Categories and Quasitilted Algebras

Coelho

Tosar²

2008

Algebr Represent Theor

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In this paper, we present some results on the bounded derived category of Artin algebras, and in particular on the indecomposable objects in these categories, using homological properties. Given a complex X * , we consider the set J X * = {i ∈ Z | H i (X * ) = 0} and we define the application l(X * ) = maxJ X * − minJ X * + 1. We give relationships between some homological properties of an algebra and the respective application l. On the other hand, using homological properties again, we determine two subcategories of the bounded derived category of an algebra, which turn out to be the bounded derived categories of quasi-tilted algebras. As a consequence of these results we obtain new characterizations of quasi-tilted and quasi-tilted gentle algebras.

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“…Among the main recent results in the fast-growing theory of cluster algebras is the paper of Fomin, Shapiro and Thurston [Fomin et al 2008], relating triangulations of oriented surfaces to cluster algebras. This approach, which existed since the beginning of the theory [Caldero et al 2006], was followed in [Labardini-Fragoso 2009;Schiffler 2008], among others. In the same spirit, we consider in the present paper an unpunctured oriented surface S and a finite set of points M, lying on the boundary of S and intersecting every boundary component of S. We then associate to a triangulation of the marked surface (S, M) a quiver Q( ), and a potential on Q( ) (in the sense of [Derksen et al 2008]), thus defining an algebra A( ), namely the (noncompleted) Jacobian algebra defined by Q( ) and the associated potential.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2010

ANT

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Let A be the Néron model of an abelian variety A K over the fraction field K of a discrete valuation ring R. By work of Mazur and Messing, there is a functorial way to prolong the universal extension of A K by a vector group to a smooth and separated group scheme over R, called the canonical extension of A. Here we study the canonical extension when A K = J K is the Jacobian of a smooth, proper and geometrically connected curve X K over K. Assuming that X K admits a proper flat regular model X over R that has generically smooth closed fiber, our main result identifies the identity component of the canonical extension with a certain functor Pic ,0 X/R classifying line bundles on X that have partial degree zero on all components of geometric fibers and are equipped with a regular connection. This result is a natural extension of a theorem of Raynaud, which identifies the identity component of the Néron model J of J K with the functor Pic 0 X/R. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of X K .

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Tilted string algebras

Cited by 15 publications

References 15 publications

Nakayama oriented pullbacks and stably hereditary algebras

Nakayama oriented pullbacks and stably hereditary algebras

On the Derived Categories and Quasitilted Algebras

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