Selfinjective biserial standard algebras

Pogorzały, Zygmunt; Skowroński, Andrzej

doi:10.1016/0021-8693(91)90183-9

Cited by 42 publications

(28 citation statements)

References 12 publications

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“…2 Proof The fact that the algebra is gentle, follows from Proposition 9 and the definition of the gentle algebras. Moreover, the gentle algebras are derived-tame, see [31,37].…”

Section: The Categories Perf(e) Andmentioning

confidence: 99%

Tilting on non-commutative rational projective curves

Burban

Drozd

2010

Math. Ann.

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In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy 2 = x 3 + x 2 z.

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“…2 Proof The fact that the algebra is gentle, follows from Proposition 9 and the definition of the gentle algebras. Moreover, the gentle algebras are derived-tame, see [31,37].…”

Section: The Categories Perf(e) Andmentioning

confidence: 99%

Tilting on non-commutative rational projective curves

Burban

Drozd

2010

Math. Ann.

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“…It was shown in Assem and Skowroński (1987) and Pogorzały and Skowroński (1991) that a triangular algebra KQ A /I A is gentle if and only if its trivial extension is special biserial.…”

Section: Preliminariesmentioning

confidence: 99%

On Trivial Extensions of 2-Fundamental Algebras^#

Jaworska

Pogorzały

2006

Communications in Algebra

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“…6 We shall now describe the indecomposable modules and Auslander-License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Reiten sequences of self-injective special biserial algebra. Moreover, by [PS,Lemma 8], R = B is a special biserial simply connected self-injective category such that every full finite subcategory is representation-finite. Let A be a representation-infinite selfinjective special biserial algebra.…”

Section: Preliminaries On Biserial Algebrasmentioning

confidence: 99%

“…R(y ,x)^Qi) for some object y of D. It follows from [PS,Lemma 9] that D+ (resp. R(y ,x)^Qi) for some object y of D. It follows from [PS,Lemma 9] that D+ (resp.…”

Section: Auslander-reiten Components Of Self-injective Special Biserimentioning

confidence: 99%