A note on standard equivalences (Q2830652): Difference between revisions
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scientific article | scientific article; zbMATH DE number 6645451 | ||
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A finite dimensional algebra \(A\) over a field is said to be triangular if its Gabriel quiver has no oriented cycles. If \(A\) is derived equivalent to a triangular algebra, it is proved that any derived equivalence (of bounded derived categories of finitely generated modules) between \(A\) and another finite dimensional algebra is standard. This means that the equivalence is obtained by tensoring with a two-sided tilting complex. Examples of triangular algebras are piecewise hereditary ones. However, it should be noted that the triangular property is not invariant under derived equivalence. | |||
| Property / review text: A finite dimensional algebra \(A\) over a field is said to be triangular if its Gabriel quiver has no oriented cycles. If \(A\) is derived equivalent to a triangular algebra, it is proved that any derived equivalence (of bounded derived categories of finitely generated modules) between \(A\) and another finite dimensional algebra is standard. This means that the equivalence is obtained by tensoring with a two-sided tilting complex. Examples of triangular algebras are piecewise hereditary ones. However, it should be noted that the triangular property is not invariant under derived equivalence. / rank | |||
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| Property / reviewed by: Wolfgang Rump / rank | |||
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Latest revision as of 21:16, 2 June 2025
scientific article; zbMATH DE number 6645451
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A note on standard equivalences |
scientific article; zbMATH DE number 6645451 |
Statements
A note on standard equivalences (English)
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28 October 2016
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A finite dimensional algebra \(A\) over a field is said to be triangular if its Gabriel quiver has no oriented cycles. If \(A\) is derived equivalent to a triangular algebra, it is proved that any derived equivalence (of bounded derived categories of finitely generated modules) between \(A\) and another finite dimensional algebra is standard. This means that the equivalence is obtained by tensoring with a two-sided tilting complex. Examples of triangular algebras are piecewise hereditary ones. However, it should be noted that the triangular property is not invariant under derived equivalence.
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