Is connected graded $k$-algebra local?

Question

Let $k$ be a field and $A=\bigoplus_{n \in \mathbb{N}_0}A_n$ a connected $\mathbb{N}_0$-graded $k$-algebra, i.e. $A_0=k$. I am pretty sure that $A^+:=\bigoplus_{n \in \mathbb{N}}A_n$ is a unique maximal ideal of $A$, but I'm afraid that I overlooked something. If already this is wrong, you don't have to read further :)

My caution comes from the following considerations: if $(A,\epsilon)$ is an augmented algebra, then $A \cong k \oplus A^+$ as $k$-vector space, where $A^+$ is the augmentation ideal. Thus, we can see $A$ as a connected $(\{0,1\},\times)$-graded $k$-algebra. But above, it shouldn't make a difference if we consider an $\mathbb{N}_0$-graded or $(\{0,1\},\times)$-graded algebras, right?

Answer 1

The polynomial ring $A=k[x]$ is a connected $\Bbb N_0$-graded $k$-algebra with the grading given by $A_n$ being the homogenous polynomials of degree $n$: $A_n=k \cdot x^n$. Then $A^+=(x)$ is indeed a maximal ideal, but it is not unique, because $(x-1)$ is another maximal ideal.