Tilted Algebras of Type 𝔸˜<sub><i>n</i></sub>

Lévesque, Jessica

doi:10.1081/agb-120017325

Cited by 2 publications

(4 citation statements)

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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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“…For instance, it is shown in [3] that an iterated tilted algebra of type A n satisfies the clock condition, that is, on the unique cycle of its bound quiver, the number of clockwise oriented relations equals the number of counterclockwise oriented relations. Furthermore, it is shown in [15,20] that if such an algebra is tilted, then its bound quiver cannot contain a double-zero, that is, two consecutive monomial relations pointing in the same direction. In this paper, we prove the following result:…”

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confidence: 99%

“…i+1 for all i such that 1 ≤ i < n. It is called a non-zero walk if it does not contain any zerorelation. Finally, a reduced walk is called a double-zero if it contains exactly two zero-relations that point in the same direction in w. The double-zero has been used for the classification of tilted and quasi-tilted special biserial algebras, string algebras and gentle algebras [1,[12][13][14][15].…”

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confidence: 99%