Tilting theory for finite dimensional 1-Iwanaga-Gorenstein algebras

Abstract

We study tilting objects of the stable category ${\underset{\_}{\mathrm{CM}}}^{Z}A$ of graded Cohen-Macaulay modules over a finite dimensional graded Iwanaga-Gorenstein algebra A. We first show that if there exists a tilting object in ${\underset{\_}{\mathrm{CM}}}^{Z}A$, then its endomorphism algebra always has finite global dimension. Next, to study the existence of a tilting object, we introduce a numerical invariant $g\left(A\right)$. In the case where A is 1-Iwanaga-Gorenstein, we give a sufficient condition on $g\left(A\right)$ for the existence of a tilting object. As an application, for a truncated preprojective algebra $\Pi {\left(Q\right)}_w$ of a tree quiver Q, we prove that ${\underset{\_}{\mathrm{CM}}}^Z\Pi {\left(Q\right)}_w$ always admits a tilting object.