Let $\Lambda$ be an artin algebra over a commutative artinian ring $R$. $\Lambda$ is said to be $n$-Gorenstein, for some natural number $n$, provided it have finite self-injective dimension at most $n$ from both sides.

Let $T$ be a $\Lambda$-module, $T$ is called a tilting module if the following conditions are satisfied:

- $pd T_{\Lambda} \leq 1$;
- $Ext_{\Lambda}^1(T,T)=0$;
- There exists a short exact sequence $0 \rightarrow \Lambda \rightarrow T' \rightarrow T'' \rightarrow 0$ with $T', T'' \in addT$

Since selfinjective algebras are preserved under derived equivalences and selfinjective algebras are 0-Gorenstein. We know that $\Lambda$ is 0-Gorenstein iff $End_{\Lambda}(T)$ is 0-Gorenstein for a tilting $\Lambda$-module $T$. Now I want to know consider the relationships between $n$-Gorenstein algebras for larger $n$ and tilting modules. Are there any researches and results about it? Thank you for any help and references.