Abstract We prove that if the Auslander–Reiten triangles generate relations for Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with (co)generator, then has only finitely many isomorphism classes indecomposable objects up to translation. This gives converse theorem Butler and on groups. Our approach applications in context Frobenius categories.

It is proved that for a ring R either an affine algebra over field, or equicharacteristic complete local ring, some power of the Jacobian ideal annihilates ExtRd+1(−,−), where d Krull dimension R. Sufficient conditions are identified under which itself these Ext-modules, and examples provided show this not always case. The crucial new idea to consider derived version Noether different algebra.

We establish Green equivalences for all Mackey 2-functors, without assuming Krull-Schmidt. By running through the examples of we recover variants equivalence and correspondence known in representation theory obtain new ones several other contexts. Such applications include equivariant stable homotopy topology sheaves geometry.