Rational maps

As a kind of secondary goal for the stacks project, I would like the terminology to be as “standard” as possible. What this means exactly may not be clear in all instances, but to start off with I decided to make all definitions logically equivalent to their counterparts in EGA I, II, III, IV. In only one case sofar have I changed the definition: namely David Rydh convinced me that we should change unramified to the notion used in Raynaud’s book on henselian rings (i.e., only require locally of finite type and not require locally of finite presentation).

A good example of the kind of confusion that happens over definitions is the case of rational maps. In EGA I (both the original version and the new edition) a rational map from a scheme X to a scheme Y is defined to be an equivalence class of pairs (U, f) where U is a dense open of X and f : U —> Y is a morphism of schemes. In my opinion this is a very handy notion which in almost all situations does exactly what you want, and is quite easy to explain to students, etc. Next, one defines a rational function on X to be a rational map from X to the affine line. You can also define a sheaf of rational functions on X which is denoted by a calligraphic R.

Next, one can define the sheaf of meromorphic functions. Kleiman has a nice paper “Misconceptions about K_X” which corrects the construction of the sheaf of meromorphic functions on X in EGA IV 20.1. Note how the symbol used here is a K and not an R. Basically one inverts the multiplicative subsheaf of O_X consisting of sections which are nonzero divisors in each stalk. A meromorphic function on X is then defined to be a global section of the sheaf of meromorphic functions. An (easy but not completely trivial) argument shows that a meromorphic function f on X actually gives rise to a regular function on a schematically dense open part of X.

Some people conclude that EGA’s definition of rational functions is wrong and that we should replace the notion of a rational map by something that has a chance of recovering meromorphic functions when applied to rational maps from X to A^1. To do this sometimes people redefine a rational map as an equivalence class of pairs (U, f) where U is a schematically dense open of X…

… but this notion also exists in EGA where these maps are called pseudo-morphisms or strict rational maps from X to Y, see EGA IV 20.2. A pseudo-function is a pseudo morphism from X to A^1. It is not at all clear to me that a pseudo-function is the same thing as a meromorphic function (hopefully Brian Conrad will chime in here and tell us, but the point I am trying to make is that it is not a triviality).

My approach in the stacks project has been to use the notion of rational maps as defined in EGA I (i.e. not pseudo-morphisms). Also we define the sheaf of meromorphic functions as in Kleiman’s paper (i.e. not using pseudo-functions). Only if absolutely necessary will we work through the material in EGA IV about pseudo-morphisms and introduce it.

Of course a definition cannot be wrong. What is great about having good definitions is that they allow you to make very precise statements about the relationships between objects. My tendency is to go with the definitions as stated in EGA; it appears that Grothendieck and Dieudonne tried their best to make sure the definitions are good in the sense above.

1 thought on “Rational maps

  1. Johan, I suppose it’s worth noting the result in 20.2.11 in EGA IV_4 that the sheaf of meromorphic functions is always a subsheaf of the sheaf of pseudo-functions, with equality when the scheme is either locally noetherian or is reduced with locally finite set of irreducible components. One merit of the concept of pseudo-morphism (or pseudo-function) is that by focusing on schematically dense opens it has a natural “relative” version over a base $S$ using $S$-morphisms defined on opens which are universally schematically dense relative to $S$; see 20.5 and 20.6 in EGA IV_4. In contrast, the notion of “rational map” is not relative.

    It does seem very reasonable to distinguish (as EGA does) these diverse concepts (rational map, pseudo-morphism, pseudo-morphism relative to $S$, etc.) I agree that it would be good to have an example of a scheme on which there is a pseudo-function that is not a meromorphic function. There doesn’t seem to be an example given in EGA, but I wonder if Example 2.5/4 in “Neron Models” may be an example (comparing its “domain of definition” as a rational function, which has nothing to do with $S$, and its “domain of definition” as a pseudo-morphism relative to $S$); could be a silly suggestion (I have given it almost zero thought). If this doesn’t work, then the obvious route to follow is to ask Raynaud or Gabber if they know an example offhand.

    This stuff is all tied up with the notion of $S$-rational map used in the “Neron models” book (section 2.5), which is what EGA calls a pseudo-morphism relative to $S$ (except that “Neron models” imposes $S$-smoothness conditions to simplify matters; e.g., geometric fibers are reduced). They prove some nice properties of this concept (I think including some things not in EGA, but using their $S$-smoothness hypothesis all over the place). The $S$-rational maps are a key ingredient in their definition of an $S$-birational group law, used in their statement and proof of Weil’s general theorem on promoting such structures to actual $S$-groups in reasonable situations (smooth and separated of finite type, with $S$ the spectrum of a field or Dedekind…or more general too). Weil’s theorem is useful not only in the theory of Neron models, but also in the SGA3 approach to proving the Existence Theorem over Spec(Z) for general root data.

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