Is $k[x^4,x^3y,xy^3,y^4]$ a local ring?

Question

I noticed that a system of parameters is defined in local rings and some books say that $\{x^4,y^4\}$ is a system of parameters for $R=k[x^4,x^3y,xy^3,y^4]$. Is $R$ a local ring or we use it refers to $R_{\mathfrak m}$? (Here $\mathfrak m=(x^4,x^3y,xy^3,y^4)$.)

Answer 1

$R$ is a graded $k$-algebra and for such algebras we can talk about homogeneous systems of parameters which are in fact homogeneous elements generating an ideal whose radical is the maximal irrelevant ideal $\mathfrak m$.
It's a matter of fact that a homogeneous system of parameters is a system of parameters in $R_m$.
Now you can chose between the two cases which one want to consider.

Edit. It seems that you want to be convinced that $R$ is not local. For this let $\mathfrak p=(x^4,x^3y,xy^3)$ be a prime ideal of $R$, and add to this ideal another generator different from $y^4$, e.g. $y^4+1$. The ideal $\mathfrak n=(x^4,x^3y,xy^3,y^4+1)$ is maximal. (To have all these as a clear as possible I suggest you to look at the isomorphism proved in this topic.)