Folding derived categories with Frobenius functors.
In a previous paper [Trans. Am. Math. Soc. 358, No. 8, 3591-3622 (2006; Zbl 1095.16007)] the authors introduced the concept of Frobenius morphism and studied relationships between representation theoretical properties of an algebra \(A\) and the fixed point algebra \(B=A^F\), where \(F\) is a Frobenius morphism on \(A\). This means that \(A\) is a \(k\)-algebra, \(k\) is the algebraic closure of a finite field \(\mathbb{F}_q\) and \(F\colon A\to A\) is a ring homomorphism such that \(F(\lambda a)=\lambda^qF(a)\), for any \(a\in A\), \(\lambda\in k\), and for any \(a\in A\) there is \(n>0\) such that \(F^n(a)=a\). By \(A^F\) we denote the \(\mathbb{F}_q\)-subalgebra of \(A\) consisting of the elements \(a\) such that \(f(a)=a\). Given an algebra \(C\) let \({\mathcal D}^b(C)\) be the bounded derived category of the category of left finite dimensional \(C\)-modules. The main aim of the present paper is to investigate the relationships between \({\mathcal D}^b(A)\) and \({\mathcal D}^b(A^F)\). The morphism \(F\) induces an endofunctor of \({\mathcal D}^b(A)\) which is an equivalence of triangulated categories. It is shown that \({\mathcal D}^b(A^F)\) can be identified with the triangulated subcategory of \({\mathcal D}^b(A)\) consisting of \(F\)-stable objects of \({\mathcal D}^b(A)\). If \(A\) is finite dimensional of finite global dimension then there is a ``folding relation'' between \({\mathcal D}^b(A)\) and \({\mathcal D}^b(A^F)\) and the Auslander-Reiten quiver of \({\mathcal D}^b(A^F)\) can be obtained by folding the Auslander-Reiten quiver of \({\mathcal D}^b(A)\).
scientific article; zbMATH DE number 5083178