Rasool Hafezi

Let $\mathcal{C}$ be an abelian category. A contravariant functor $F$ from $\mathcal{C}$ to the category of abelian groups $\mathcal{A}b$ is called finitely presented, or coherent \cite{As}, if there exists an exact sequence
$$ \mathrm{Hom}_{\mathcal{C}}(-, X) \longrightarrow \mathrm{Hom}_{\mathcal{C}}(-,Y) \longrightarrow F \longrightarrow 0$$
of functors.
Let ${\rm{{mod\mbox{-}}}} \mathcal{C}$ denote the category of all coherent functors. The systematic study of ${\rm{{mod\mbox{-}}}} \mathcal{C}$ is initiated by Auslander \cite{As}. He not only showed that ${\rm{{mod\mbox{-}}}}\mathcal{C}$ is an abelian category of global dimension less than or equal to two but also provided a nice connection between ${\rm{{mod\mbox{-}}}} \mathcal{C}$ and $\mathcal{C}$. This connection, which is known as Auslander's formula \cite{L,K}, suggests that one way of studying $\mathcal{C}$ is to study ${\rm{{mod\mbox{-}}}} \mathcal{C}$, that has nicer homological properties than $\mathcal{C}$, and then translate the results back to $\mathcal{C}$. In particular if we let $\mathcal{C}$ to be ${\rm{{mod\mbox{-}}}} \Lambda$, where $\Lambda$ is an artin algebra, Auslander's formula translates to the equivalence
$$\frac{{\rm{{mod\mbox{-}}}} ({\rm{{mod\mbox{-}}}} \Lambda)}{\{F\mid F(\Lambda)=0\}} \simeq {\rm{{mod\mbox{-}}}} \Lambda$$
of abelian categories. As it is mentioned in \cite{L}, `a considerable part of Auslander's work on the representation theory of finite dimensional, or more general artin, algebras can be connected to this formula'.

Recently, Krause \cite{K} established a derived version of this equivalence. In my talk, some different (relative and derived)  versions of this formula will be explained. Then I will give some applications of our results.
Especially,  by using a relative version of Auslander's formula, we show that bounded derived
category of every artin algebra admits a categorical resolution. This, in particular, implies that
bounded derived categories of artin algebras of fnite global dimension determine bounded derived categories of all artin algebras.