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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.20588 |
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| _version_ | 1866911079565623296 |
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| author | Wu, Mawei |
| author_facet | Wu, Mawei |
| contents | Let $\mathcal{C}$ be a small category, $\mathfrak{A}$ be a precosheaf of unital $k$-algebras on $\mathcal{C}$ and $\mathfrak{M}$ be an $\mathfrak{A}$-bimodule. We introduce two new notions, namely, the Grothendieck construction $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$ of $\mathfrak{A}$ and $\mathfrak{M}$, as well as the extension category algebra $\mathfrak{A} \ltimes \mathfrak{M}$. The extension category algebra contains the trivial extension algebra and the skew category algebra as special cases. If $\mathcal{C}$ is object-finite, we prove that the category of modules of $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$ is equivalent to the category of modules over $\mathfrak{A} \ltimes \mathfrak{M}$. Finally, we obtain two LHS-spectral sequences about $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{N})$ for a right $\mathfrak{A}$-module $\mathfrak{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extension category algebras and LHS--spectral sequences Wu, Mawei Representation Theory Let $\mathcal{C}$ be a small category, $\mathfrak{A}$ be a precosheaf of unital $k$-algebras on $\mathcal{C}$ and $\mathfrak{M}$ be an $\mathfrak{A}$-bimodule. We introduce two new notions, namely, the Grothendieck construction $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$ of $\mathfrak{A}$ and $\mathfrak{M}$, as well as the extension category algebra $\mathfrak{A} \ltimes \mathfrak{M}$. The extension category algebra contains the trivial extension algebra and the skew category algebra as special cases. If $\mathcal{C}$ is object-finite, we prove that the category of modules of $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$ is equivalent to the category of modules over $\mathfrak{A} \ltimes \mathfrak{M}$. Finally, we obtain two LHS-spectral sequences about $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{N})$ for a right $\mathfrak{A}$-module $\mathfrak{N}$. |
| title | Extension category algebras and LHS--spectral sequences |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2507.20588 |