Let $X$ be a noetherian separated scheme over the spectrum of a field $K$. By definition, I can find an open covering of $X$ by finitely many affine schemes $Spec(R_i)$ where each $R_i$ is a $K$-algebra. A scheme is a bunch of rings glued together. Is a scheme also a bunch of local rings glued together? More precisely:
Can every such $X$ be covered by finitely many spectra of local rings (which are also $K$-algebras) such that the intersection of any number of them is either empty or again the spectrum of a local ring?
My first try was to cover $X$ by finitely many $D(f)$ where each $f$ is an element of some $R_i$ and $D(f)\cong Spec((R_i)_f)$. However, the prime ideals of $(R_i)_f$ correspond one-to-one to those of $R_i$ not containing a power of $f$. In particular, $(R_i)_f$ does not have to be a local ring.