Counterexamples without local finiteness

Construct or rule out a counterexample to the graded-isomorphism conclusion for standard graded semiconnected algebras when local finiteness is removed.

Background

Local finiteness is used repeatedly to turn surjections between graded pieces into bijections by comparing dimensions. Without finite-dimensional graded components, equality of cardinal dimensions does not imply that the constructed maps are bijective. The authors state that the proof fails in this setting but do not know a counterexample.

References

However, we are not aware of a counterexample in this case either.

— Isomorphisms of graded semiconnected algebras  (2609.03288 - Dramburg, 3 Sep 2026) in Section 4, Further questions and observations, Remark 4.3(2)

Let $A_\bullet$ and $B_\bullet$ be locally finite and standard graded. If $A \simeq B$, does there exist an isomorphism $\psi \colon A \to B$ such that the induced grading $B = \bigoplus_{i \geq 0} \psi(A_i)$ and $B_\bullet$ form a $\mathbb{Z}2$-grading?

— Isomorphisms of graded semiconnected algebras  (2609.03288 - Dramburg, 3 Sep 2026) in Section 4, Further questions and observations, Question 4.5