Bergman algebras: The graded universal algebra constructions

A half a century ago, George Bergman introduced stunning machinery which would realise any commutative conical monoid as the non-stable K-theory of a ring. The ring constructed is ``minimal" or ``universal". Given the success of graded K-theory in classification of algebras and its connections to dynamics and operator algebras, the realisation of Γ-monoids (monoids with an action of an abelian group Γ on them) as non-stable graded K-theory of graded rings becomes vital. In this paper, we revisit Bergman's work and develop the graded version of this universal construction. For an abelian group Γ, a Γ-graded ring R, and non-zero graded finitely generated projective (left) R-modules P and Q, we construct a universal Γ-graded ring extension S such that S⊗R P S⊗R Q as graded S-modules. This makes it possible to bring the graded techniques, such as smash products and Zhang twists into Bergman's machinery. Given a commutative conical Γ-monoid M, we construct a Γ-graded ring S such that Vᵍʳ(S) is Γ-isomorphic to M. In fact we show that any finitely generated Γ-monoid can be realised as the non-stable graded K-theory of a hyper Leavitt path algebra. Here Vᵍʳ(S) is the monoid of isomorphism classes of graded finitely generated projective S-modules and the action of Γ on Vᵍʳ(S) is by shift of degrees. Thus the group completion of M can be realised as the graded Grothendieck group K^₀(S). We use this machinery to provide a short proof to the fullness of the graded Grothendieck functor Kᵍʳ₀ for the class of Leavitt path algebras (i.e., Graded Classification Conjecture II).