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

TL;DR

This paper investigates tilting theory in the stable category of graded Cohen-Macaulay modules over finite dimensional 1-Iwanaga-Gorenstein algebras, providing conditions for the existence of tilting objects and analyzing their endomorphism algebras.

Contribution

It offers a complete answer to the finiteness of global dimension of endomorphism algebras of tilting objects and introduces an invariant to identify tilting modules in the stable category.

Findings

Endomorphism algebras of tilting objects have finite global dimension.

A new invariant $g(A)$ helps determine tilting objects.

Tilting objects exist in stable categories of certain truncated preprojective algebras when the underlying graph is a tree.

Abstract

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 $\underline{\mathsf{CM}}^{\mathbb{Z}}\Pi(Q)_w$ where $\Pi(Q)_w$ 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 $\underline{\mathsf{CM}}^{\mathbb{Z}}\Pi(Q)_w$ has a tilting object.