Auslander's Formula: Variations and Applications

Javad Asadollahi, Najmeh Asadollahi, Rasool Hafezi, Razieh Vahed

Published 2016-05-16Version 1

According to the Auslander's formula one way of studying an abelian category $\mathcal{C}$ is to study ${\rm{{mod\mbox{-}}}} \mathcal{C}$, that has nicer homological properties than $\mathcal{C}$, and then translate the results back to $\mathcal{C}$. Recently Krause established a derived version of this formula. In this paper, some different versions of this formula will be established. One point is to replace special flat resolutions with projective resolutions in Auslander's argument to prove the existence of a fundamental four terms exact sequence. This will lead us to prove some interesting results. In particular, we show that $\mathbb{D}({\rm{Mod\mbox{-}}} R)$ can be expressed as two new Verdier quotients, there exists a duality between triangulated categories $\mathbb{K}^{{\rm{b}}}({\rm{{mod\mbox{-}}}} R^{{\rm{op}}})$ and $\mathbb{K}^{{\rm{b}}}({\rm{{mod\mbox{-}}}} R)$, where $R$ is a (left and right) coherent ring that induces a duality between triangulated categories $\mathbb{K}^{{\rm{b}}}_{\rm{ac}}({\rm{{mod\mbox{-}}}} R^{{\rm{op}}})$ and $\mathbb{K}^{{\rm{b}}}_{\rm{ac}}({\rm{{mod\mbox{-}}}} R),$ whenever $R$ is an artin algebra of finite global dimension and show that in this case $\mathbb{K}_{\rm{ac}}^{{\rm{b}}}({\rm{{mod\mbox{-}}}} R)$ admits Serre duality.