Today, I was on and off wondering about idempotents in Z-graded associative algebras with a unit (which is assumed homogeneous). In my googling of this, I have found the terminology graded idempotents which refers to idempotents which are homogeneous of degree 1. This suggests that there exist others. And indeed, it is easy to make examples of non-homogeneous idempotents by conjugation with units. But we can ask for more.
- Is there an example of an idempotent which is not conjugate to a graded idempotent?
- Is there an example of a Z-graded associative algebra with a nontrivial idempotent but no nontrivial graded idempotents?
Hmm…?
Some more searching and google finally turned up the paper Idempotents in ring extensions by Kanwar, Leroy, and Matczuk which provides the answer to 1. There’s probably tons of papers that make this observation. Namely, suppose that R is a (commutative) domain such that R[x, x^{-1}] and R don’t have the same Picard group. For example R = k[t^2, t^3] with k a field (details omitted). Let L be an invertible module over R[x, x^{-1}] which is not isomorphic to the pullback of an invertible module from R. Pick a surjection
R[x, x^{-1}]⊕ n —> L
As L is a projective R[x, x^{-1}]-module we obtain an idempotent e in the Z-graded ring M_n(R[x, x^{-1}]) = M_n(R)[x, x^{-1}]. And this idempotent is not conjugate to an element of M_n(R) as that would mean L does come from R.
So this answers 1. I do not know the answer to 2.