例外模型结构与诱导的外三角范畴

中文摘要 AI 辅助

若Hovey三元组$(\u2702\uf4c6, \uf4c7, \uf4c8)$满足$(\u2702\uf4c6 \u2229 \uf4c7, \u2702\uf4c6 \u2229 \uf4c7 \u2229 \uf4c8)$不是Frobenius对，则称其为**例外的**。Hovey三元组全体可分解为不交并：$\u2702{\text{Hovey triple}} = \u2702{\text{Hereditary Hovey triple}} \uf0ea \u2702{\text{Exceptional Hovey triple}} \uf0ea \u2702{\text{non-hereditary and non-exceptional Hovey triple}}$。例外Hovey三元组广泛存在：对自内射Nakayama代数$A= kC_n/J^t$，$A\text{-mod}$存在例外Hovey三元组当且仅当$\u2785\u2785\u2785(n, t) \u2a7e 2$且$t \u2a7e 3$，其同伦范畴呈现出新现象。Nakaoka-Palu定理表明商范畴$\frac{\u2702\uf4c6\u2229\uf4c7}{\u2702\uf4c6\u2229\uf4c7\u2229\uf4c8}$上存在三角结构。本文证明：弱幂等完备外三角范畴$(\uf4a9, \u211e, \uf4a6)$中的Hovey三元组是例外的，当且仅当诱导外三角结构$\bigl(\frac{\u2702\uf4c6\u2229\uf4c7}{\u2702\uf4c6\u2229\uf4c7\u2229\uf4c8}, \u203e{\u211e}, \u203e{\uf4a6}\bigr)$不是**典范三角化的**。在该外三角结构与由三角结构诱导的外三角结构之间，中间外三角结构与$\u2786\u2786((\frac{\uf4a7(\u2702\uf4c6\u2229\uf4c7)}{\u2702\uf4c6\u2229\uf4c7\u2229\uf4c8})^{\u2787\u2787}, \u2786\u2785)$的Serre子范畴一一对应。

英文摘要

A Hovey triple $(\mathcal{C}, \mathcal{F}, \mathcal{W})$ is {\it exceptional}, if $(\mathcal{C} \cap \mathcal{F}, \ \mathcal{C} \cap \mathcal{F} \cap \mathcal{W})$ is not a Frobenius pair. One has a disjoint union $\{\text{Hovey triple}\} = \{\text{Hereditary Hovey triple}\} \ \dot\bigcup \ \{\text{Exceptional Hovey triple}\}$ \ $\dot\bigcup \ \{\text{non-hereditary and non-exceptional Hovey triple}\}.$ Exceptional Hovey triples appear widely. For a selfinjective Nakayama algebra $A= kC_n/J^t$, $A\mbox{-}{\rm mod}$ admits exceptional Hovey triples if and only if $\gcd (n, t) \ge 2$ and $t \geq 3$. Their homotopy categories reveal new phenomena. Nakaoka-Palu's Theorem implies that there is a triangulation on $\frac{\mathcal C\cap\mathcal F}{\mathcal C\cap\mathcal F\cap\mathcal W}.$ It is proved that a Hovey triple in a weakly idempotent complete extriangulated category $(\mathcal A, \mathbb E, \mathfrak s)$ is exceptional if and only if the induced extriangulated structure $\bigl(\frac{\mathcal C\cap\mathcal F}{\mathcal C\cap\mathcal F\cap\mathcal W}, \ \overline{\mathbb E}, \ \overline{\mathfrak s}\bigr)$ is not {\it canonically triangulated}. Between this extriangulated structure and the one arising from the triangulation, the intermediate extriangulated structures are in one-to-one correspondence with Serre subcategories of $\mathrm{fp}((\frac{\mathcal P(\mathcal C\cap\mathcal F)}{\mathcal C\cap\mathcal F\cap\mathcal W})^{\mathrm{op}}, \mathrm{Ab})$.

发表机构

• Shanghai Jiao Tong University（上海交通大学）

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