Is a prevariety the same as a notherian scheme?

Question

By a pre-variety I mean a quasi-compact locally ringed space which can be covered by (a finite # of) affine varieties.
I was wondering, this seems to be the same in scheme language as a Noetherian scheme, or am I overseeing something?

Answer 1

The typical example of a pre-variety that is not a variety is the "affine line with doubled origin." This pre-variety is covered by two copies of $\mathbb A^1$ that are glued along $\mathbb A^1\setminus \{0\}$ via the identity map $x\mapsto x.$ This differs only slightly from the construction of $\mathbb P^1$ as a gluing of the same open sets via $x\mapsto 1/x.$

A noetherian scheme is a different beast in general, for we can consider as examples any spectrum of a noetherian ring. In particular, let $A = K[[x]]$ be a power series ring in a variable $x$ over our field $K.$ This ring is noetherian, but not of finite type, and hence $\operatorname{Spec}(A),$ which contains a single closed point, is not covered by an affine variety.