# n-Gorenstein algebras and tilting modules

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:

1. $pd T_{\Lambda} \leq 1$;
2. $Ext_{\Lambda}^1(T,T)=0$;
3. 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.

• There is alot research about tilting module in connection to Gorenstein algebras so it would be helpful to make more precise questions. One nice result is that an algebra is $n$-GOrenstein for some $n$ if and only if every tilting module is a cotilting module, see eudml.org/doc/165463. Also note that all selfinjective algebras only contain trivial tilting modules so your observation on 0-Gorenstein algebras seems to be trivial. – Mare Sep 7 '18 at 13:16
• @Mare Yes, tilting modules are trivial in selfinjective algebras. One question I am interested is: Suppose $\Lambda$ is $n$-Gorenstein algebra, $T$ is a tilting $\Lambda$-module, is $End_{\Lambda}(T)$ also $n$-Gorenstein? Generally, $\Gamma$ is derived equivalent to $\Lambda$, is $\Gamma$ also $n$-Gorenstein? – Xiaosong Peng Sep 7 '18 at 13:56
• No, this is of course not true: A hereditary algebra is 1-Gorenstein but there are tilted algebras of global dimension two, and for algebras of finite global dimension the global dimension coincides with the Gorenstein dimension(=selfinjective dimension). It should be true however that derived equivalences preserve the Gorenstein property (but not the Gorenstein dimension). – Mare Sep 7 '18 at 14:02