Let $X$ be an algebraic set, i.e., a closed subset of $\mathbb{A}^n$. Then $X$ can be written as a finite union of irreducible subsets, its irreducible components. Generalizing affine varieties as locally ringed spaces, and defining prevarieties as locally ringed spaces that

  1. are irreducible/connected
  2. admit a finite open cover of affine varieties
  3. have a sheaf of $k$-valued functions (alg closed)

can we say that coproducts of prevarieties generalize (intrinsically) the notion of algebraic set? If so, can we accomplish the same thing by removing condition 1. above?

I ask because Gathmann says the following as one of several motivations for schemes

5.1. Affine schemes. We now come to the definition of schemes, which are the main objects of study in algebraic geometry. The notion of schemes extends that of prevarieties in a number of ways. We have already met several instances where an extension of the category of prevarieties could be useful:

• We defined a prevariety to be irreducible. Obviously, it makes sense to also consider reducible spaces. In the case of affine and projective varieties we called them algebraic sets, but we did not give them any further structure or defined regular functions and morphisms of them. Now we want to make reducible spaces into full-featured objects of our category.

A second (third?) question is about the follow-up motivation:

At present we have no geometric objects corresponding to non-radical ideals in $k[x_1,\ldots,x_n]$, or in other words to coordinate rings with nilpotent elements. These non-radical ideals pop up naturally however: e. g. we have seen in exercise 1.4.1 that intersections of affine varieties correspond to sums of their ideals, modulo taking the radical. It would seem more natural to define the intersection $X_1 \cap X_2$ of two affine varieties $X_1, X2 \subset \mathbb{A}^n$ to be a geometric object associated to the ideal $I(X_1) + I(X_2) ⊂ k[x_1,\ldots,x_n]$.

Couldn't we just "forget" Nullstellensatz and recover some non-radical theory in varieties? However we get around not having Nullstellensatz for schemes, couldn't we do the same with varieties?


1 Answer 1


Since I can't yet comment, I'll leave this here as an answer.

Both of these cases can be handled with the variety technology over an algebraically closed field. I think Weil handles these things, but as a modern reference, see chapter 11 of version 5.22 of Milne's AG notes. He briefly develops algebraic spaces over an arbitrary field, and in the case $k$ is algebraically closed, you recover the typical theory of varieties, with the generalizations you note.

Probably the cleanest way to generalize things (without schemes!) is to focus on the commutative algebra as in these notes from Pete Clark. Work with affine k-algebras in place of affine varieties. Allowing affine k-algebras to be neither integral nor reduced gets you a more general theory as you you can take limits along localizations (e.g., the limit of the system $k[x], k[y], k[t, 1/t]$ with the identity maps and $x \mapsto 1/y, x \mapsto t, y \mapsto 1/t$) to produce the $k$-algebras associated to algebraic spaces (in the sense of Milne above). If the affine $k$-algebras are integral (as in my example), you recover prevarieties as defined by Gathmann.

After seeing things this way, you're not too far from working schematically anyway, so I suppose it's natural for authors to skip to schemes after generating "just enough" intuition about the geometry from just the basics of variety theory. The schematic theory, though more general, is simply way cleaner and more consistently developed.


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