# Is a prevariety the same as a notherian scheme?

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?

• Oh! and I am assuming we are over an algebraically closed field :) – AIM_BLB Jul 18 '13 at 19:03

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.
• Dear @Andrew, $\mathrm{Spec}(A)$ consists of two points: the closed point $(x)$ and the generic point $(0)$. – Keenan Kidwell Jul 18 '13 at 19:31