On derived equivalences of lines, rectangles and triangles

Ladkani, Sefi

doi:10.1112/jlms/jds034

Cited by 19 publications

(16 citation statements)

References 18 publications

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“…have been considered, from a representation theoretic perspective, in [11]. Thus, having all of this in mind, in the present paper we are interested in the algebras…”

Section: Introduction and Resultsmentioning

confidence: 99%

On tensor products of path algebras of type A

Hille¹,

Müller²

2014

Linear Algebra and its Applications

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We derive a formula for the Coxeter polynomial of the s-fold tensor productof path algebras of linearly oriented quivers of Dynkin type An i −1, in terms of the weights n1, . . . , ns 2, and show that conversely the weights can be recovered from the Coxeter polynomial of the tensor product. Our results have applications in singularity theory, in particular these algebras occur as endomorphism algebras of tilting objects in certain stable categories of vector bundles.

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“…have been considered, from a representation theoretic perspective, in [11]. Thus, having all of this in mind, in the present paper we are interested in the algebras…”

Section: Introduction and Resultsmentioning

confidence: 99%

On tensor products of path algebras of type A

Hille¹,

Müller²

2014

Linear Algebra and its Applications

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“…We mention a systematic tool to construct new tilting complexes from existing ones using tensor products. This construction is due to Ladkani (see ) and generalizes a result of Rickard . Here, we follow the approach in .…”

Section: Constructions Of Tilting Complexes and Derived Equivalences mentioning

confidence: 99%

“…Also, we extend derived equivalences between smaller algebras of the form

e A e

with

e

an idempotent in an algebra

A

to derived equivalences between the whole algebras themselves. Finally, we mention constructions of derived equivalences for tensor products given by Ladkani in and for Milnor squares by Hu‐Xi in . In Section 6, we mention three applications of derived equivalences of algebras and rings.…”

Section: Introductionmentioning

confidence: 99%

Derived equivalences of algebras

2018

Bull. London Math. Soc.

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Derived categories and equivalences between them are the pièce de résistance of modern homological algebra. They are widely used in many branches of mathematics, especially in algebraic geometry and representation theory. In this note, we shall survey some recently developed construction methods of derived equivalences for algebras and rings, with applications to homological conjectures, such as Broué's abelian defect group conjecture and the finitistic dimension conjecture, and to computation of higher algebraic K‐groups of algebras and rings.

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“…It seems that Leszcynski [49] was the first to take an interest in properties of tensor products of path algebras of Dynkin quivers. The Calabi-Yau property of tensor products is investigated by Ladkani [35]; further [34] is of relevance for the topics of this Section.…”

Section: Negative Euler Characteristicmentioning

confidence: 99%

Triangle singularities, ADE-chains, and weighted projective lines

Kussin

Lenzing

Meltzer

2013

Advances in Mathematics

120

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We investigate the triangle singularity f = x a + y b + z c , or S = k[x, y, z]/(f ), attached to the weighted projective line X given by a weight triple (a, b, c). We investigate the stable category vect-X of vector bundles on X obtained from the vector bundles by factoring out all the line bundles. This category is triangulated and has Serre duality. It is, moreover, naturally equivalent to the stable category of graded maximal Cohen-Macaulay modules over S (or matrix factorizations of f ), and then by results of Buchweitz and Orlov to the singularity category of f .We show that vect-X is fractional Calabi-Yau whose CY-dimension is a function of the Euler characteristic of X. We show the existence of a tilting object which has the shape of an (a − 1) × (b − 1) × (c − 1)-cuboid. Particular attention is given to the weight types (2, a, b) yielding an explanation of Happel-Seidel symmetry for a class of important Nakayama algebras. In particular, the weight sequence (2, 3, p) corresponds to an ADE-chain, the Enchain, extrapolating the exceptional Dynkin cases E 6 , E 7 and E 8 to a whole sequence of triangulated categories.

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On derived equivalences of lines, rectangles and triangles

Cited by 19 publications

References 18 publications

On tensor products of path algebras of type A

On tensor products of path algebras of type A

Derived equivalences of algebras

Triangle singularities, ADE-chains, and weighted projective lines

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