Support $\tau$-tilting modules and semibricks over group graded algebras

TL;DR

This paper investigates the behavior of support τ-tilting modules and semibricks over strongly G-graded algebras, establishing conditions for induction and restriction that preserve support τ-tilting pairs and exploring applications to semibricks.

Contribution

It provides new Clifford and Maschke type theorems for support τ-tilting modules over G-graded algebras, linking G-invariance to tilting properties.

Findings

Induction of support τ-tilting pairs preserves support τ-tilting if and only if modules are G-invariant.

Restriction from the algebra to the 1-component preserves support τ-tilting under G-invariance.

Applications to semibricks and wide subcategories extend the theoretical framework.

Abstract

We consider a finite dimensional strongly $G$-graded algebra $A$ with { self-injective} $1$-component $B$, and in our main result we prove that the induction from $B$ to $A$ of a basic support $\tau$-tilting pair of $B$-modules is a support $\tau$-tilting pair $(M,P)$ of $A$-modules if and only if $M$ is $G$-invariant. A similar statement holds for the restriction from $A$ to $B$, so our results may be viewed as Clifford and Maschke type theorems for $2$-term silting complexes. We also give applications to semibricks and the associated wide subcategories.