Non-commutative graded algebras with restricted growth

Graded algebras play a major role in many topics, including algebraic geometry, topology, and homological algebra, besides classical ring theory. These are algebras which admit a decomposition into a sum of homogeneous components which 'behave well' with respect to multiplication.

In this talk we present several structure-theoretic results concerning affine (that is, finitely generated) Z-graded algebras which grow 'not too fast'.

In particular, we bound the classical Krull dimension both for algebras with quadratic growth and for domains with cubic growth, which live in the heart of Artin's proposed classification of non-commutative projective surfaces. We also prove a dichotomy result between primitive and PI-algebras, relating a graded version of a question of Small.

From a radical-theoretic point of view, we prove that unless a graded affine algebra has infinitely many zero homogeneous components, its Jacobson radical vanishes. Under a suitable growth restriction, we prove a stability result for graded Brown-McCoy radicals of Koethe conjecture type: they remain Brown-McCoy even after being tensored with some arbitrary algebra.

Finally, we pose several open questions which could be seen as graded versions of the Kurosh and Koethe conjectures.

The talk is based on joint work with A. Leroy, A. Smoktunowicz and M. Ziembowski.