An isomorphism lemma for graded rings

TL;DR

This paper proves that for connected graded algebras finitely generated in degree one, an ungraded algebra isomorphism implies a graded algebra isomorphism, highlighting a strong structural rigidity.

Contribution

It establishes an isomorphism lemma showing that ungraded isomorphisms between such graded algebras imply graded isomorphisms, revealing a key structural property.

Findings

Ungraded isomorphism implies graded isomorphism for the class of algebras considered.

The result applies to connected graded algebras finitely generated in degree one.

Provides a foundational lemma for understanding algebraic isomorphisms in graded contexts.

Abstract

Let $A$ and $B$ be two connected graded algebras finitely generated in degree one. If $A$ is isomorphic to $B$ as ungraded algebras, then they are also isomorphic to each other as graded algebras.