Relative derived equivalences and relative Igusa-Todorov dimensions
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913802742661120 |
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| author | Guo, Peizheng Pan, Shengyong |
| author_facet | Guo, Peizheng Pan, Shengyong |
| contents | Let $A$ be an Artin algebra and $F$ a non-zero subfunctor of $\Ext_A^{1}(-,-)$. In this paper, we characterize the relative $ϕ$-dimension of $A$ by the bi-functor $\Ext_F^1(-,-)$. Furthermore, we show that the finiteness of relative $ϕ$-dimension of an Artin algebra is invariant under relative derived equivalence. More precisely, for an Artin algebra $A$, assume that $F$ has enough projectives and injectives, such that there exists $G\in \modcat{A}$ such that $\add G=\mathcal {P}(F)$, where $\mathcal {P}(F)$ is the category of all $F$-projecitve $A$-modules. If $\cpx{T}$ is a relative tilting complex over $A$ with term length $t(\cpx{T})$ such that $B=\End(\cpx{T})$, then we have $\phd_{F}(A)-t(T^{\bullet})\leq \phd(B)\leq\phd_{F}(A)+t(T^{\bullet})+2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relative derived equivalences and relative Igusa-Todorov dimensions Guo, Peizheng Pan, Shengyong Representation Theory Let $A$ be an Artin algebra and $F$ a non-zero subfunctor of $\Ext_A^{1}(-,-)$. In this paper, we characterize the relative $ϕ$-dimension of $A$ by the bi-functor $\Ext_F^1(-,-)$. Furthermore, we show that the finiteness of relative $ϕ$-dimension of an Artin algebra is invariant under relative derived equivalence. More precisely, for an Artin algebra $A$, assume that $F$ has enough projectives and injectives, such that there exists $G\in \modcat{A}$ such that $\add G=\mathcal {P}(F)$, where $\mathcal {P}(F)$ is the category of all $F$-projecitve $A$-modules. If $\cpx{T}$ is a relative tilting complex over $A$ with term length $t(\cpx{T})$ such that $B=\End(\cpx{T})$, then we have $\phd_{F}(A)-t(T^{\bullet})\leq \phd(B)\leq\phd_{F}(A)+t(T^{\bullet})+2$. |
| title | Relative derived equivalences and relative Igusa-Todorov dimensions |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2504.15500 |