A characterisation of local schemes

Let X be a scheme. The following are equivalent:

  1. X is a local scheme, i.e. of the form Spec A where A is a local ring.
  2. X is a scheme with a unique closed point x and the only (open) neighbourhood of x is X itself.
  3. Every open covering of X must contain X itself.

It is well known that the first condition implies the second, and it is an easy exercise (in point set topology) to show that the second condition implies the third. To complete the proof of the claim, it is enough to show that the third condition implies the first.

Suppose X is a scheme satisfying the third condition. Any scheme can be covered by open affine subschemes, so the condition on X implies it is affine. In particular, it has a closed point x. The condition also implies that the complement of x is the unique maximal open proper subset of X, so x must be the unique closed point of X, as required.

We should observe that the third condition can be expressed purely in terms of the Zariski topology on the category of schemes: it says that a scheme X is local if and only if every Zariski-covering family of X contains a split epimorphism, or equivalently, if and only if there is a unique Zariski-covering sieve on X (namely, the maximal sieve).

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