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 $\Gamma$-monoids (monoids with an action of an abelian group $\Gamma$ 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 $\Gamma$, a $\Gamma$-graded ring $R$, and non-zero graded finitely generated projective (left) $R$-modules $P$ and $Q$, we construct a universal $\Gamma$-graded ring extension $S$ such that $S\otimes_R P\cong S\otimes_R Q$ 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 $\Gamma$-monoid $M$, we construct a $\Gamma$-graded ring $S$ such that $\mathcal V^{gr}(S)$ is $\Gamma$-isomorphic to $M$. In fact we show that any finitely generated $\Gamma$-monoid can be realised as the non-stable graded $K$-theory of a hyper Leavitt path algebra. Here $\mathcal V^{gr}(S)$ is the monoid of isomorphism classes of graded finitely generated projective $S$-modules and the action of $\Gamma$ on $\mathcal V^{gr}(S)$ is by shift of degrees. Thus the group completion of $M$ can be realised as the graded Grothendieck group $K^{\gr}_0(S)$. We use this machinery to provide a short proof to the fullness of the graded Grothendieck functor $K^{gr}_0$ for the class of Leavitt path algebras (i.e., Graded Classification Conjecture II).