Dimension filtration of the bounded Derived category of a Noetherian ring
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929348873814016 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $A$ be a Noetherian ring of dimension $d$ and let $\mathcal{D}^b(A)$ be the bounded derived category of $A$. Let $\mathcal{D}_i^b(A)$ denote the thick subcategory of $\mathcal{D}^b(A)$ consisting of complexes $\mathbf{X}_\bullet$ with $\dim H^n(\mathbf{X}_\bullet) \leq i$ for all $n$. Set $\mathcal{D}_{-1}^b(A) = 0$. Consider the Verdier quotients $\mathcal{C}_i(A) = \mathcal{D}_i^b(A)/\mathcal{D}_{i-1}^b(A)$. We show for $i = 0, \ldots, d$, $\mathcal{C}_i(A)$ is a Krull-Remak-Schmidt triangulated category with a bounded $t$-structure. We identify its heart. We also prove that if $A$ is regular then $\mathcal{C}_i(A)$ has AR-triangles. We also prove that $$ \mathcal{C}_i(A) \cong \bigoplus_{\stackrel{P}{\dim A/P = i}} \mathcal{D}_0^b(A_P). $$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_11991 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dimension filtration of the bounded Derived category of a Noetherian ring Puthenpurakal, Tony J. Commutative Algebra Primary 13D09, Secondary 13D02 Let $A$ be a Noetherian ring of dimension $d$ and let $\mathcal{D}^b(A)$ be the bounded derived category of $A$. Let $\mathcal{D}_i^b(A)$ denote the thick subcategory of $\mathcal{D}^b(A)$ consisting of complexes $\mathbf{X}_\bullet$ with $\dim H^n(\mathbf{X}_\bullet) \leq i$ for all $n$. Set $\mathcal{D}_{-1}^b(A) = 0$. Consider the Verdier quotients $\mathcal{C}_i(A) = \mathcal{D}_i^b(A)/\mathcal{D}_{i-1}^b(A)$. We show for $i = 0, \ldots, d$, $\mathcal{C}_i(A)$ is a Krull-Remak-Schmidt triangulated category with a bounded $t$-structure. We identify its heart. We also prove that if $A$ is regular then $\mathcal{C}_i(A)$ has AR-triangles. We also prove that $$ \mathcal{C}_i(A) \cong \bigoplus_{\stackrel{P}{\dim A/P = i}} \mathcal{D}_0^b(A_P). $$ |
| title | Dimension filtration of the bounded Derived category of a Noetherian ring |
| topic | Commutative Algebra Primary 13D09, Secondary 13D02 |
| url | https://arxiv.org/abs/2405.11991 |