Being reduced is local-global; an existence theorem for closed points of a Noetherian scheme; reduced at closed points of an affine scheme implies reduced

In commutative algebra there are several arguments which follow some local to global principle. In the following we will prove one of these arguments: “the local to global property of being reduced” (this last phrase, being reduced, stems from the awkward adjective ‘reduced’; to this extent maybe reducedness would be a more appropriate term).

Depending on whose definition we use, a reduced scheme is one where the ring of sections of the structure sheaf over an open subset is reduced as a ring, or, that the structure sheaf will have reduced stalks. These definitions are equivalent (without any further hypothesis), and in some sense this is enough. This says that a scheme is globally reduced if and only if it is at every point. However, there should be good reason to reduce this to the case of knowing whether it is also equivalent to being reduced at its closed points — if only because we typically have a more clear description of the maximal ideals of a ring than general prime ideals (which is a concrete description of closed points in the affine case). For example, in any finitely generated algebra over an algebraically closed field the maximal ideals are in one to one correspondence with points in a finite dimensional vectorspace over this field. For non algebraically closed fields, we can say the same thing except we need to include points which may live in a finite extension of the base field as well.

The difficulty with generalizing this idea to general schemes is the lack of (or my lack of) a precise description of its closed points. As an explanation to what I mean: given a closed point of a scheme, it is a closed point in any affine open set which contains it, so it is therefore equivalent to a maximal ideal of this scheme however, given a closed point of an affine open subset of a scheme does not imply the point is closed in the scheme under consideration. As to why this is: intuitively, we are only given that we have a closed subset of an open subset which does not in general imply the closed subset is closed in the whole space (we omit examples due to their complicated nature; see this paper by Karl Schwede).

What follows is a brief treatment of the ideas mentioned above. In particular, we show the equivalence of the definition of a reduced scheme, give an existence theorem for a scheme to have closed points (in any closed subset of the scheme), and show the equivalence for affine schemes of being reduced to being reduced at closed points.

1. Definition: A scheme X will be called reduced if for every open set U\subset X we have \mathcal{O}_X(U) is a reduced ring — here \mathcal{O}_X is the structure sheaf of X.

2. Proposition: A scheme X is reduced if and only if for every point x\in X, the ring \mathcal{O}_{X,x} is reduced.

\implies: In constructing the ring \mathcal{O}_{X,x} we may first restrict to an affine subset U=Spec(A)\subset X containing x. Then \mathcal{O}_{X,x}\cong (\mathcal{O}_X|_U)_x\cong A_x and the latter ring is the localization of a reduced ring.

\impliedby: Suppose every stalk of the structure sheaf is reduced. Suppose further f\in \mathcal{O}_X(U) is a section over the open subset U\subset X with f^n=0 as germs at every point x\in U. Then f=0 on an open subset W\subset U which contains x\in U. This gives a collection of open sets, one for each point x\in U, W_x which cover U. Since \mathcal{O}_X is a sheaf we may glue these together to realize f is the 0 section on U.\square

Of course, we must touch on some aspects of this proof. In the initial paragraph, we made an explicit choice of affine subset to restrict to. Naturally we should check that the result is independent of choice. But this shouldn’t matter, up to isomorphism, as the stalk is unique, up to isomorphism. Also, we made direct use of our definition of reduced. It is perhaps useful to use a different definition, that a scheme is reduced if it admits a cover by open affine subsets which are reduced, but this turns out to imply that the sections over any subset form a reduced ring. This is nice, in philosophy, if our aim is to show that a scheme is in fact reduced (to see its truth, consider the stalk at any point, after reducing to one of these affine subsets, and then conclude by prop. 2 that the scheme is reduced; this will be used in theorem 5 below).

In the following, we will be employing the technique I have recently learned and wanted to test out, that of the local to global arguments of commutative algebra. In totality our proposition will read X reduced \implies X_p reduced for all points p\in X \implies X_m reduced for all closed points m\in X \implies  X is reduced. We make note the last implication is only shown to be true for affine schemes in particular. It should be stated some proofs provided for general schemes will be false (in fact vacuous!) if there are, for example, are no closed points at all. In this regard, we include at least one sufficient criterion for a scheme to have closed points before continuing to the main work I wanted to write.

3. Lemma: Noetherian schemes have closed points.

Proof. Given a Noetherian scheme X, we will use the fact X satisfies the Noetherian condition on topological spaces. That is to say, given any descending chain of closed subsets, say V_1\supset V_2\supset \cdots \supset V_n, there is a minimal, closed, nonempty subset V that continues this chain.

Since V is nonempty, there is a point \zeta\in V. Since V is minimal, we have V\subset \overline{\{\zeta\}} where the line indicates closure in X. But, it also follows \overline{\{\zeta\}}\subset \overline{V}=V, showing \overline{\{\zeta\}}=V. If \zeta is a closed point, we are done, so we may assume \zeta is not closed in X.

Let m be a point in \overline{\{\zeta\}}=V distinct from \zeta. Again, by the minimality of V, we have \overline{\{m\}}=V. But, since V is minimally closed, it is irreducible, and irreducible closed subsets of schemes have unique generic points. This means there is no point m\in \overline{\{\zeta\}} other than \zeta, so \zeta is a closed point of the Noetherian scheme X.\square

4. Corollary: Any closed subset of a Noetherian scheme has a closed point.

Proof. Careful reading of the above proof actually finds this as a direct implication.

We will use the following technical lemma to prove our main theorem. It is encouraged to first read the theorem, and then the lemma as needed.

5. Lemma: The natural map of commutative rings

\varphi:A\longrightarrow \prod\limits_{\mathfrak{m}\in maxSpec(A)} A_{\mathfrak{m}}

is injective.

Proof. Suppose f\in A is such that \varphi(f)=0 or, equivalently, for every maximal ideal of A, \mathfrak{m}, the image of f under the localization A\rightarrow A_{\mathfrak{m}} is 0. This means for every maximal ideal, indexed by i say, there is an element b_i\notin \mathfrak{m}_i such that b_if=0. Consider the ideal I:=\sum_i (b_i), the sum of all the principal ideals generated by a b_i. If I were proper, then I would be contained in some maximal ideal \mathfrak{m}_i. However, by  construction, there is a b_i\notin\mathfrak{m}_i so that I\not\subset \mathfrak{m}_i. Therefore I=A is the whole ring. But f\sum_i (b_i)=(0) so that f\cdot 1 =0 or, equivalently, f=0. From this we may conclude the kernel of this map is trivial, and injectivity follows.\square

6. Theorem: We may reduce (prop. 2) further: iX is an affine scheme, then it is reduced if and only if it is reduced at its closed points.

Proof: By proposition 2, it is clear X being reduced implies X is reduced at every point, which implies X is reduced at every closed point. Assume then X is reduced at all of its closed points.

Since X is assumed affine, its closed points are exactly the maximal ideals of the ring defining X. Call this ring A. By lemma 5, A is a subring of the product of the localizations of A at its maximal ideals. As this last ring is reduced at each component by assumption, A is reduced as well.\square

Unfortunately, this theorem seems rather underwhelming after dealing with the complications of closed points in general schemes. The lesson I gained from this post was to be more cautious when approaching the closed points of a scheme. For example, any closed point is certainly a maximal ideal in any affine open subset, as mentioned in the beginning of this article. However, the converse is not, in general, true. After trying, and failing, for some time to come up with a good description of the closed points of a (at least Noetherian) scheme, I’ve decided to stop. Of course, the final statement of the theorem can be modified to a result for general schemes by introducing the following

(7. Definition: A point x in the scheme X is affine closed if it is closed in some affine open subset of X.)

This immediately generalizes from the affine case to the general case of schemes but, not every affine closed point is closed and we would have to face the consequences of such a definition if it were made. I’ll end here as I can not see a natural stopping point soon.


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