The following is an Exercise 1.1.11 of Hartshorne's Algebraic Geometry.

Let $Y\subset \mathbb{A}^3$ be the curve given parametrically by $x=t^3, y=t^4, z=t^5$. Show that $I(Y)$ is a prime ideal of height $2$ in $k[x,y,z]$ which can not be generated by $2$ elements. We say $Y$ is "not a local complete intersection".

My question is why is it called "local"? Should not we look some localization of $k[x,y,z]/I(Y)$?