Tilting in $Q$-shaped derived categories
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913658270908416 |
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| author | Gratz, Sira Holm, Henrik Jorgensen, Peter Stevenson, Greg |
| author_facet | Gratz, Sira Holm, Henrik Jorgensen, Peter Stevenson, Greg |
| contents | The main result of this paper is that there is sometimes a triangulated equivalence between $D_Q( A )$, the $Q$-shaped derived category of an algebra $A$, and $D( B )$, the classic derived category of a different algebra $B$. By construction, $D_Q( A )$ consists of $Q$-shaped diagrams of $A$-modules for a suitable small category $Q$. Our result concerns the case where $Q$ consists of shifts of indecomposable projective modules over a self-injective $\mathbb{Z}$-graded algebra $Λ$.
A notable special case is the result by Iyama, Kato, and Miyachi that $D_N( A )$, the $N$-derived category of $A$, is triangulated equivalent to $D( T_{ N-1 }A )$, the classic derived category of $T_{ N-1 }( A )$, which denotes upper diagonal $( N-1 ) \times ( N-1 )$-matrices over $A$. Several other special cases will also be discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11412 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tilting in $Q$-shaped derived categories Gratz, Sira Holm, Henrik Jorgensen, Peter Stevenson, Greg Representation Theory Rings and Algebras 16E35, 18E35, 18G80, 18N40 The main result of this paper is that there is sometimes a triangulated equivalence between $D_Q( A )$, the $Q$-shaped derived category of an algebra $A$, and $D( B )$, the classic derived category of a different algebra $B$. By construction, $D_Q( A )$ consists of $Q$-shaped diagrams of $A$-modules for a suitable small category $Q$. Our result concerns the case where $Q$ consists of shifts of indecomposable projective modules over a self-injective $\mathbb{Z}$-graded algebra $Λ$. A notable special case is the result by Iyama, Kato, and Miyachi that $D_N( A )$, the $N$-derived category of $A$, is triangulated equivalent to $D( T_{ N-1 }A )$, the classic derived category of $T_{ N-1 }( A )$, which denotes upper diagonal $( N-1 ) \times ( N-1 )$-matrices over $A$. Several other special cases will also be discussed. |
| title | Tilting in $Q$-shaped derived categories |
| topic | Representation Theory Rings and Algebras 16E35, 18E35, 18G80, 18N40 |
| url | https://arxiv.org/abs/2411.11412 |