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A commutative ring with unity is called pm-ring if every prime ideal is contained in a unique maximal ideal. In [dMO71], it is shown that pm-rings are characterized by the fact that $\operatorname{max Spec} R$ (the set of all maximal ideals under Zariski subspace topology) is a retract of $\operatorname{Spec} R$, and in this case, the unique retraction is given by $u: \operatorname{Spec} R \to \operatorname{max Spec} R$ , $u(P)$ is the unique maximal ideal containing $P\in \operatorname{Spec} R$ .

It can moreover be shown that for pm-rings, this retract is actually also a deformation retract , because $H : \operatorname{Spec} R \times [0,1] \to \operatorname{Spec} R$ given by $H(P,t)=P, \forall t \in [0,1)$ and $H(P,1)=u(P)$ is continuous, so gives a homotopy between $i\circ u$ and $Id_{\operatorname{Spec} R}$, where $i:\operatorname{max Spec} R \to \operatorname{Spec} R$ is the inclusion map.

So the questions I want to ask are the following :

(1) Can we characterize (possibly algebraic characterization) commutative rings (with unity) $R$ such that $\operatorname{max Spec} R$ is homotopy equivalent with $\operatorname{Spec} R$ ?

(2) Can we characterize commutative rings (with unity) $R$, such that $i : \operatorname {max} \operatorname {Spec} R \to \operatorname {Spec} R$ is a homotopy equivalence i.e. there exists a map $g : \operatorname {Spec} R \to \operatorname {max} \operatorname {Spec} R$ such that $i\circ g$ and $g \circ i$ are homotopic to the respective identity maps ?

As noted, pm-rings are definitely in both the class, but what are all such rings ? Even if we can't say what are all such rings, can we atleast find class of rings for each case (1) and (2) which are not necessarily pm-rings ?


References.

[dMO71] De Marco, Giuseppe; Orsatti, Adalberto, Commutative rings in which every prime ideal is contained in a unique maximal ideal, Proc. Am. Math. Soc. 30, 459-466 (1971). ZBL0207.05001.

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This is not a full answer, but some examples to show that the situation is a bit tricky. (This answer was worked out together with Dmitrii Pirozhkov.)

Lemma. Let $R$ be a domain. Then $X = \operatorname{Spec} R$ is contractible.

Proof. In fact, the inclusion of the generic point $\eta$ is a deformation retract, by the map \begin{align*} [0,1] \times X &\to X \\ (t,\mathfrak p) &\mapsto \left\{\begin{array}{ll}\mathfrak p, & t = 0, \\ \eta, & t > 0.\end{array}\right. \end{align*} (One easily shows that this map is continuous.) $\square$

Thus, for a domain, the question is when $\operatorname{Specmax} R$ is contractible. Even for simple examples like $R = k[t]$, the result depends on the base field $k$:

Example. Let $R$ be a $1$-dimensional Noetherian domain. Then $X = \operatorname{Specmax} R$ is a cofinite topological space. But the homotopy type depends on the cardinality of $X$:

  • If $|X| = 1$, then $R$ is local and $X$ is evidently contractible (since it is a point).
  • If $X$ is countable but has more than $1$ element, then $X$ is not path connected (see e.g. this MO post), so in particular it is not contractible.
  • If $|X| \geq |\mathbb R|$, then we can choose a bijection $\phi \colon (0,1) \times X \to X$. Extend $\phi$ to a map $\phi \colon [0,1] \times X \to X$ by setting $\phi(0,-) = \operatorname{id}$ and $\phi(1,-) = \mathfrak m_0$ a constant map. Then each fibre of $\phi$ is closed, hence $\phi$ is continuous. Thus, $\phi$ is a homotopy from the identity to a constant map, so $X$ is contractible.
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  • $\begingroup$ If $R$ has exactly one minimal prime over $0$, then $\operatorname {Spec} R$ is contractible because $\{0\}$ is a generic point of $\operatorname {Spec} R$ then. Do you have any general idea about what happens to $max \operatorname {Spec} R$ then ? ($R$ being a domain is a special case of this). And your $|max Spec R|=1$ implying it is contractible is covered by the context of the question because local rings are pm-rings. $\endgroup$ – user111524 Mar 23 '18 at 18:37
  • $\begingroup$ In the above comment, I meant to say that $\{\mathfrak p\}$ is a generic point for the unique minimal prime ideal $\mathfrak p$ over $0$ $\endgroup$ – user111524 Mar 23 '18 at 18:56
  • $\begingroup$ @users: Note that we may always assume $R$ is reduced, because this doesn't change $\operatorname{Spec} R$. The case of your comment is exactly the point of my post: the spectrum of a domain is always contractible, but the contractibility question for $\operatorname{Specmax} R$ is very subtle, even in the one-dimensional case. $\endgroup$ – R. van Dobben de Bruyn Mar 23 '18 at 19:46
  • $\begingroup$ Wait, for 1-dimensional domains , is $max Spec R$ really co-finite i.e. all proper closed subsets are finite ? i.e. every non-zero ideal is contained in only finitely many maximal ideals ? I don't think that is true ... you need some extra assumptions like Dedekind for that ... $\endgroup$ – user111524 Mar 23 '18 at 20:05
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    $\begingroup$ @users: you might be right that it's not true in general. However, Noetherian should be enough, because then $V(I)$ is also Noetherian, hence has finitely many components, which are necessarily zero-dimensional unless $V(I) = X$. $\endgroup$ – R. van Dobben de Bruyn Mar 23 '18 at 21:10

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