Better than excellent

MH once pointed out the “linguistic trap” Grothendieck created when he defined the notion of an excellent ring: “Suppose somebody finds an even better class of rings? Then what?”

It turns out there IS an even better class of rings/schemes, which occurs naturally in some contexts.

Definition. A scheme X is marvelous if it is Noetherian and excellent, and if \dim \mathcal{O}_{Y,y} = \dim Y for every irreducible component Y \subset X and every closed point y \in Y. A ring A is marvelous if \mathrm{Spec}(A) is marvelous.

You can easily check that any marvelous scheme is finite-dimensional. Moreover, it turns out that a Noetherian quasi-excellent scheme is marvelous if and only if the function x \in |X| \mapsto \dim \overline{ \{ x \} } is a true dimension function for X (in a certain technical sense). This function is of course the most naive and clean possibility for a dimension function on any given scheme, but it doesn’t always have the right properties.

Unfortunately, marvelous schemes are so marvelous that, unlike excellent schemes, they aren’t stable under many natural operations, not even under passing to an open subscheme! In fact, X is marvelous if it is covered by marvelous open affines, but the converse fails. You can check that a scheme as simple as \mathrm{Spec}\mathbf{Z}_p[x] isn’t marvelous, even though \mathrm{Spec}\mathbf{Z}_p is marvelous. So regular excellent schemes aren’t always marvelous, and adjoining a polynomial variable can kill marvelousity. I briefly entertained the hope that any Jacobson excellent scheme is marvelous, but this fails too (the scheme S considered in EGAIV3 (10.7.3) is a counterexample).

It’s not all bad news, though:

  1. anything of finite type over \mathbf{Z} or a field is marvelous,
  2. any excellent local ring is marvelous,
  3. any ring of finite type over an affinoid K-algebra in the sense of rigid geometry is marvelous,
  4. any scheme proper over a marvelous scheme is marvelous; more generally, if X is marvelous and f: Y \to X is a finite type morphism which sends closed points to closed points, then Y is marvelous,
  5. if A is a marvelous domain, then the dimension formula holds: \dim (A/\mathfrak{p}) + \mathrm{ht}\,\mathfrak{p} = \dim A for all prime ideals \mathfrak{p} \subset A. (Recall that the dimension formula can fail, even for excellent regular domains.)

You might be wondering why I would care about such a stupid and delicate property. The reason is the following. Fix any marvelous scheme X and any n invertible on X. Then there is a canonical potential dualizing complex \omega_{X} \in D^{b}_{c}(X,\mathbf{Z}/n) (in the sense of Gabber) which restricts to \mathbf{Z}/n[2\dim ](\dim) on the regular locus of X. Here \dim is the (locally constant) dimension of the regular locus, so this numerology is the same as in the case of varieties. Moreover, for any prime \ell invertible on X, there is a good theory of \ell-adic perverse sheaves on X with the same numerology as in the case of varieties; in particular, the IC complex restricts to \mathbf{Q}_{\ell}[\dim] on the regular locus. (See sections 2.2 and 2.4 of Morel’s paper for more. Note in particular the hypothesis on X in the first sentence of section 2.2: it is exactly the condition that X is marvelous.) This discussion all applies, in particular, when X=\mathrm{Spec}(A) for any K-affinoid ring A. This turns out to be an important ingredient in my forthcoming paper with Bhargav…

(One more comment: Most real-life examples of marvelous schemes, e.g. examples 1. and 3. above, are also Jacobson. It might be more reasonable to consider the class of marvelous Jacobson schemes, because these are permanent under finite type maps. But on the other hand we lose excellent local rings when we do this.)

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