Homogeneous prime ideals of two isomorphic rings

This question is motivated from here

Let $S=\Bbb{C}[x,y]$. Consider the degree $0$ homogeneous elements of the localization at $x$, denoted $S_{(x)}.$ Then it is alleged that

$S_{(x)}\cong \Bbb{C}[y/x] \cong \Bbb{C}[s]$

Where I imagine the second isomorphism takes $y/x\mapsto s$.

Now I have two concerns (that I suspect are the one concern)

• The first concern is that $y/x$ has degree $0$, and is sent to $s$ which has degree $1$.

• The second concern is that $\Bbb{C}[s]$ has two homogeneous prime ideals at all, the irrelevant ideal $(s)$ and the trivial ideal $(0)$ (which is generic), the latter being the only homogeneous prime ideal. Whereas $\Bbb{C}[y/x]$ has numerous homogeneous prime degree $0$ ideals right, say $\frac{y+3x}{x}$?

I imagine that the homogeneous prime ideals should be in correspondence, between two isomorphic rings.

• When you take degree zero elements, there is no more grading left. So, your worry about degree zero element going to a degree one element is moot. If I understand you correctly, a typical generator of your ring is of the form $f(x,y)/x^d$ where $f$ is homogeneous of degree $d$. Then, it is same $f(1,s)\in\mathbb{C}[s]$. No homogeneous issues come up. – Mohan Oct 4 '17 at 15:04

To expand on my previous answer, say we take a graded ring $S$. Then $Proj(S)$ is a scheme covered by the open affines $D(f)$. It is a theorem that the homogeneous primes in $D(f)$ are in bijection with the primes of degree zero in the localization $S_f$.
If $S=k[x,y]$, then the homogeneous primes of $k[x,y]_x$ that lie in degree $0$ are those of the form $(\frac{ay+bx}{x})$. These are clearly in bijection with primes of $k[y/x] \cong k[s]$. So we are not looking for homogeneous primes in $k[s]$, but rather primes generated by elements of degree zero in the localized ring $S_x$. Does this make sense?