Relative derived equivalences and relative Igusa-Todorov dimensions

TL;DR

This paper develops a relative framework for derived categories and Igusa-Todorov dimensions in the setting of Artin algebras. It characterizes the relative $\phi$-dimension $\phi\text{-dim}_F(A)$ via the bi-functor $\Ext^1_F(-,-)$ and shows that finiteness of this invariant is preserved under relative derived equivalences. Central to the results is a relative tilting theory: if $T^{\bullet}$ is a relative tilting complex with endomorphism algebra $B=\End(T^{\bullet})$, then the inequality $\phi\text{-dim}_F(A)-t(T^{\bullet})\leq \phi\text{-dim}(B)\leq \phi\text{-dim}_F(A)+t(T^{\bullet})+2$ bounds the relative Igusa-Todorov dimension of $B$ in terms of that of $A$ and the term length $t(T^{\bullet})$. In particular, finiteness is invariant under relative derived equivalence, and the framework recovers known results in the classical case when $F=\Ext^1(-,-)$, extending them to the relative homological setting with a broad class of subfunctors $F$.

Abstract

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 $\phd_{F}(A)-t(T^{\bullet})\leq \phd(B)\leq\phd_{F}(A)+t(T^{\bullet})+2$.

Theorems & Definitions (30)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Lemma 4.1
• Definition 4.2
• Lemma 4.3
• Proposition 4.4
• proof