Generalised BGP Reflection Functors via the Grothendieck Construction

Inspired by work of Ladkani and Groth–Šťovíček, we explain how to construct generalisations of the classical reflection functors of Bernšteĭn, Gel′fand, and Ponomarev by means of the Grothendieck construction.

C-RINGS, COPRODUCTS, AND REFLECTION FUNCTORS

The paper begins with a detailed study of the category of modules over two different rings using a coproduct construction. If C is a commutative ring and R and S are rings together with ring homomorphisms from C to R and C to S, then we show that the category of C-modules that are also left R-modules and right S-modules is equivalent to the category of left modules over the coproduct of R and S op in an appropriate category. Letting K denote a field, we apply this to show that the category of K-representations of a quiver that is reflection equivalent to a path algebra KΓ is equivalent to a full subcategory of left and right K-representations of KΓ.