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?

  • $\begingroup$ Oh! and I am assuming we are over an algebraically closed field :) $\endgroup$ – 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.

  • $\begingroup$ Dear @Andrew, $\mathrm{Spec}(A)$ consists of two points: the closed point $(x)$ and the generic point $(0)$. $\endgroup$ – Keenan Kidwell Jul 18 '13 at 19:31
  • $\begingroup$ @KeenanKidwell, whoops, thanks for catching that! $\endgroup$ – Andrew Jul 18 '13 at 19:34
  • $\begingroup$ hmm... Okay... Then basically what I was wondering is: can all Pre-Varieties be considered as the closed points of Noetherian Schemes? (Sorry If I'm missing something :0 ) $\endgroup$ – AIM_BLB Jul 20 '13 at 17:00
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    $\begingroup$ I would say yes, since we can consider the associated schemes to the affine varieties forming the covering, and glue in the category of schemes. The covering consists of special types of noetherian schemes, and any closed point is contained in one of them. $\endgroup$ – Andrew Jul 20 '13 at 17:23

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