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Tilting theory for finite dimensional $1$-Iwanaga-Gorenstein algebras

In representation theory of graded Iwanaga-Gorenstein algebras, tilting theory of the stable category $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ of graded Cohen-Macaulay modules plays a prominent role. In this paper we study the following two central problems of tilting theory of $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ in the case where $A$ is finite dimensional: (1) Does $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ have a tilting object? (2) Does the endomorphism algebras of tilting objects in $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ have finite global dimension? To the problem (2) we give the complete answer. We show that the endomorphism algebra of any tilting object in $\underline{\mathsf{CM}}^{\mathbb{Z}}A$ has finite global dimension. To the problem (1) we give a partial answer. For this purpose, first we introduce an invariant $g(A)$ for a finite dimensional graded algebra $A$. Then, we prove that in the case where $A$ is 1-Iwanaga-Gorenstein, an inequality for $g(A)$ gives a sufficient condition that a specific Cohen-Macaulay module $V$ becomes a tilting object in the stable category. As an application, we study the existence of tilting objects in $\underline{\mathsf{CM}}^{\mathbb{Z}}Π(Q)_w$ where $Π(Q)_w$ is the truncated preprojective algebra of a quiver $Q$ associated to $w\in W_Q$. We prove that if the underling graph of $Q$ is tree, then $\underline{\mathsf{CM}}^{\mathbb{Z}}Π(Q)_w$ has a tilting object.