Timeline for Derived equivalences of Dyck paths
Current License: CC BY-SA 4.0
19 events
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| Sep 28, 2018 at 17:15 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 28, 2018 at 17:10 | vote | accept | Mare | ||
| Sep 28, 2018 at 17:08 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 25, 2018 at 21:26 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 25, 2018 at 21:23 | answer | added | Mare | timeline score: 6 | |
| Sep 25, 2018 at 20:13 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 25, 2018 at 18:54 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 25, 2018 at 18:42 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 25, 2018 at 18:21 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 25, 2018 at 18:11 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 25, 2018 at 18:06 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 25, 2018 at 13:04 | comment | added | Jeremy Rickard | I think, but am not sure, that I once knew of an example of derived equivalent algebras with acyclic/non-acyclic quivers. I’ll try to remember what it is or where I saw it. | |
| Sep 25, 2018 at 13:01 | comment | added | Jeremy Rickard | I think Happel showed, by producing an explicit non-inner derivation, that the path algebra of a non-acyclic quiver by an admissible and homogeneous (in terms of lengths of paths) ideal has non-zero $HH^1$. | |
| Sep 25, 2018 at 12:57 | comment | added | Mare | Is the question whether an acyclic quiver algebra can be derived equivalent to a quiver algebra with non-acyclic quiver really non-trivial? If yes, I'd be very suprised if that question has not been considered in the literature before. | |
| Sep 25, 2018 at 12:55 | comment | added | Mare | @JeremyRickard Here what my computer suggests: Let $A$ be a Nakayama algebra with a cyclic quiver and Kupisch series $[c_1,...,c_n]$ with minimal entry $c=s(n+1)+k \geq 2$ for $k \leq n$ and $s \geq 0$. Then the vector space dimension of the first Hochschild cohomology is equal to $s+1$. So the first Hochschild cohomology counts in a way how often you walk around the circle completely +1 it seems. | |
| Sep 25, 2018 at 12:53 | comment | added | Mare | @JeremyRickard That is a good idea. That the Hochschild cohomology is zero for the linear quiver case is by a result of Happel. I cant remembers seeing the first Hochschild cohomology calculated for Nakayama algebras with a cyclic quiver but I guess you are right. | |
| Sep 25, 2018 at 10:05 | comment | added | Jeremy Rickard | In answer to the second part of the parenthesized Question 3, I think it's the case that the first Hochschild cohomology $HH^1(A,A)$ (which is an invariant of the derived category) is zero if $A$ is a Nakayama algebra with acyclic quiver, but non-zero if $A$ is a Nakayama algebra with non-acyclic quiver. | |
| Sep 25, 2018 at 7:20 | history | edited | Mare | CC BY-SA 4.0 |
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| Sep 25, 2018 at 7:12 | history | asked | Mare | CC BY-SA 4.0 |