Bergman algebras: The graded universal algebra constructions

• DOI: 10.48550/arxiv.2403.01703
• Publication Date: 2024-03-03
• Authors: Roozbeh Hazrat; Huanhuan Li; Raimund Preusser
• Keywords: Universal algebra; Differential graded algebra

A half a century ago, George Bergman introduced stunning machinery which would realise any commutative conical monoid as the non-stable $K$-theory of ring. The ring constructed is ``minimal" or ``universal". Given success graded in classification algebras and its connections to dynamics operator algebras, realisation $\Gamma$-monoids (monoids with an action abelian group $\Gamma$ on them) rings becomes vital. In this paper, we revisit Bergman's work develop version universal construction. For $\Gamma$, $\Gamma$-graded $R$, non-zero finitely generated projective (left) $R$-modules $P$ $Q$, construct extension $S$ such that $S\otimes_R P\cong S\otimes_R Q$ $S$-modules. This makes it possible bring techniques, smash products Zhang twists into machinery. $\Gamma$-monoid $M$, $\mathcal V^{gr}(S)$ $\Gamma$-isomorphic $M$. fact show can be realised hyper Leavitt path algebra. Here isomorphism classes $S$-modules by shift degrees. Thus completion $M$ Grothendieck $K^{\gr}_0(S)$. We use provide short proof fullness functor $K^{gr}_0$ for class (i.e., Graded Classification Conjecture II).