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Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, equipped with the induced topology.

This is the ring-theoretic analogue of the Gromov--Grigorchuk space of marked groups, which can be used to give nice proofs of simple facts about algebraic geometry over groups (cf. this paper by Champetier and Guirardel). My question is:

Is the 'ideal space' of a ring a standard construction in commutative algebra or algebraic geometry? If so, what's it called and where can I read more about it?

My motivation is to strengthen the analogy between algebraic geometry over groups and classical algebraic geometry. It would be nice, when doing algebraic geometry over groups using the space of marked groups, to be able to say 'This is analogous to the foobar widget in classical algebraic geometry.'

To demonstrate that this concept has some use, I'll give a very simple application. But first, here are a few basic facts.

  • $\mathcal{I}(R)$ is compact (because it is a closed subset of $2^R$, which is itself compact by Tychonoff's Theorem), Hausdorff and totally disconnected.
  • Each point $I\in \mathcal{I}(R)$ is contained in a canonical closed subset

    $U_I=\{J\in \mathcal{I}(R)\mid I\subseteq J \}$

    (which is in fact isomorphic to $\mathcal{I}(R/I)$).

  • If $R$ is Noetherian then each $U_I$ is equal to the set of ideals that contain a (finite) generating set for $I$, and hence is open.

  • The subset of prime ideals in $\mathcal{I}(R)$ is closed: indeed, for each pair of non-units $x,y$, the subset $N(x,y)=\{x\notin I, y\notin I, xy\in I\}$ is open, and the union of these sets is the complement of the set of prime ideals.

Now here's the promised application - a proof of a well known lemma.

Lemma: If $R$ is a Noetherian ring then there is a finite set of prime ideals $\mathfrak{p}_1,\ldots,\mathfrak{p}_k\subseteq R$ with the property that every prime ideal contains one of the $\mathfrak{p}_i$.

Proof: The set of $U_{\mathfrak{p}}$ for $\mathfrak{p}$ prime is an open cover of the set of prime ideals. Since the set of prime ideals is compact, there is a finite subcover. QED

By the way, my research concerns, among other things, algebraic geometry over groups, but I have never seriously studied algebraic geometry or commutative algebra. This question was first posted at math.stackexchange.

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2 Answers

up vote 8 down vote accepted

On the prime ideals this induces what is I believe called the constructible topology because the Boolean algebra of clopens is the constructible sets. This is the natural topology for model theory. See here for the constructible topology on the spectrum of a ring.

This is a special case of the classical Lawson topology on an algebraic lattice.

Added. The Grigorchuk-Gromov topology is also a special case of the Lawson topology on an algebraic lattice. There are many ways to describe what is an algebraic lattice. The simplest is that there is a set $X$ and a closure operator $c$ on $X$ such that $c$ commutes with directed unions. The closed sets form a lattice $L$ with meet the intersection and the join is determined.

For example, taking the ideal generated by a set of elements of a ring or taking the normal closure of a set of elements of a group is such a closure operator. There is also an abstract characterization that it is a complete lattice in which each element is a join of compact elements.

The Lawson topology is the induced topology on $L$ from $2^X$ with the product topology. This can be defined intrinsically if the lattice is not given in terms of a closure operator. This is a profinite meet semilattice with identity. Conversely every profinite meet semilattice with identity is an algebraic lattice with its Lawson topology (which is its unique compact semilattice topology).

So for a commutative ring with 1 and the ideal closure operator, the Lawson topology on ideals is what you discuss above and for a group with the normal closure operator you get the Grigrochuk-Gromov space. If you use the closure operator of generating a subgroup (not normal subgroup) you get the space of marked Schreier graphs considered by Grigorchuk and others.

Details can be found in the famous compendium of continuous lattices book or in Johnstone's Stone Spaces book.

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Thanks very much for this answer, Benjamin. It sounds like 'constructible topology' is the foobar widget I was looking for. –  HJRW Mar 29 '12 at 19:15
    
Glad that it helped. –  Benjamin Steinberg Mar 29 '12 at 20:49
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I wrote a paper about the space of ideals in a commutative ring (or more generally submodules of a module over a commutative ring):

http://arxiv.org/abs/0904.4216

I focused on the case of finitely generated commutative rings, because, in the general case, the pointwise convergence seems not to be the most natural one, see the discussion in Section 5 of the paper, and also because my motivation was to deal with f.g. metabelian groups, that I consider in

http://arxiv.org/abs/0904.4230

As Benjamin mentions, these constructions are particular cases of general abstract constructions known by people in model theory and others, and the contribution of the above papers is to obtain precise description of the homeomorphism type of certain of these objects.

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