On compatibility of Koszul- and higher preprojective gradings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917019053457408 |
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| author | Dramburg, Darius Sandøy, Mads Hustad |
| author_facet | Dramburg, Darius Sandøy, Mads Hustad |
| contents | We investigate compatibility of gradings for an almost Koszul or Koszul algebra $R$ that is also the higher preprojective algebra $Π_{n+1}(A)$ of an $n$-hereditary algebra $A$. For an $n$-representation finite algebra $A$, we show that $A$ must be Koszul if $Π_{n+1}(A)$ can be endowed with an almost Koszul grading. For an acyclic basic $n$-representation infinite algebra $A$, we show that $A$ must be Koszul if $Π_{n+1}(A)$ can be endowed with a Koszul grading. From this we deduce that a higher preprojective grading of an (almost) Koszul algebra $R = Π_{n+1}(A)$ is, in both cases, isomorphic to a cut of the (almost) Koszul grading. Up to a further assumption on the tops of the degree $0$ subalgebras for the different gradings, we also show a similar result without the basic assumption in the $n$-representation infinite case. As an application, we show that $n$-APR tilting preserves the property of being Koszul for $n$-representation infinite algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13283 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On compatibility of Koszul- and higher preprojective gradings Dramburg, Darius Sandøy, Mads Hustad Representation Theory Rings and Algebras 16E65, 16G70, 16S37, 16W50 We investigate compatibility of gradings for an almost Koszul or Koszul algebra $R$ that is also the higher preprojective algebra $Π_{n+1}(A)$ of an $n$-hereditary algebra $A$. For an $n$-representation finite algebra $A$, we show that $A$ must be Koszul if $Π_{n+1}(A)$ can be endowed with an almost Koszul grading. For an acyclic basic $n$-representation infinite algebra $A$, we show that $A$ must be Koszul if $Π_{n+1}(A)$ can be endowed with a Koszul grading. From this we deduce that a higher preprojective grading of an (almost) Koszul algebra $R = Π_{n+1}(A)$ is, in both cases, isomorphic to a cut of the (almost) Koszul grading. Up to a further assumption on the tops of the degree $0$ subalgebras for the different gradings, we also show a similar result without the basic assumption in the $n$-representation infinite case. As an application, we show that $n$-APR tilting preserves the property of being Koszul for $n$-representation infinite algebras. |
| title | On compatibility of Koszul- and higher preprojective gradings |
| topic | Representation Theory Rings and Algebras 16E65, 16G70, 16S37, 16W50 |
| url | https://arxiv.org/abs/2411.13283 |