$2$-Periodic complexes over regular local rings
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929276235808768 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$. Let $\mathcal{D}^2_{fg}(A)$ denote the derived category of $2$-periodic complexes with finitely generated cohomology modules. Let $\mathcal{K}^2(\proj A) $ denote the homotopy category of $2$-periodic complexes of finitely generated free $A$-modules.
We show the natural map $\mathcal{K}^2(\ proj \ A) \longrightarrow \mathcal{D}^2(A)$ is an equivalence of categories. When $A$ is complete we show that $\mathcal{K}^2_f(\ proj \ A)$ ($2$-periodic complexes with finite length cohomology) is Krull-Schmidt with Auslander-Reiten (AR) triangles. We also compute the AR-quiver of $\mathcal{K}^2_f(\ proj \ A)$ when $\ dim \ A = 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09149 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $2$-Periodic complexes over regular local rings Puthenpurakal, Tony J. Commutative Algebra Primary 13D09, 16G70, Secondary 13H05, 13H10 Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$. Let $\mathcal{D}^2_{fg}(A)$ denote the derived category of $2$-periodic complexes with finitely generated cohomology modules. Let $\mathcal{K}^2(\proj A) $ denote the homotopy category of $2$-periodic complexes of finitely generated free $A$-modules. We show the natural map $\mathcal{K}^2(\ proj \ A) \longrightarrow \mathcal{D}^2(A)$ is an equivalence of categories. When $A$ is complete we show that $\mathcal{K}^2_f(\ proj \ A)$ ($2$-periodic complexes with finite length cohomology) is Krull-Schmidt with Auslander-Reiten (AR) triangles. We also compute the AR-quiver of $\mathcal{K}^2_f(\ proj \ A)$ when $\ dim \ A = 1$. |
| title | $2$-Periodic complexes over regular local rings |
| topic | Commutative Algebra Primary 13D09, 16G70, Secondary 13H05, 13H10 |
| url | https://arxiv.org/abs/2403.09149 |