# Is connected graded $k$-algebra local?

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?

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.