At most one point

Consider the functor F which to a scheme T associates the set of closed subschemes Z of T \times A^1 such that the projection Z —> T is an open immersion. In other words F is the functor of flat families of closed sub schemes of degree <= 1 on A^1, whence the title of this post. We note that F is a sheaf for the fppf topology.

What is fun about this functorĀ  is that it is a natural candidate for a 1-point compactification of A^1, as the following discussion shows.

Namely, consider for each integer n >= 1 the scheme P_n which is P^1 but crimped at infinity to order n. What I mean is this: If y = x^{-1} denotes the usual coordinate on the standard affine of P^1 which contains infinity, then the local ring of P_n at infinity is the Z-algebra generated by y^n, y^{n + 1}, y^{n + 2}, … Note that there are morphisms

P_1 —> P_2 —> P_3 —> …

and that for each n there is a natural map P_n —> F compatible with the transition maps of the system. Hence we obtain a transformation of sheaves

colim P_n —> F.

It seems likely that this map is an isomorphism (we take the colimit in the category of fppf sheaves), but I did not write out all the details.

Does anybody have a reference? What about the same thing for A^2?

[Edit 18:57 July 31 2010: Original definition of F omitted the condition that the fibres are a point or empty. Replaced by the open immersion condition. This make sense because a morphism Z —> T which is flat and locally of finite presentation whose fibres Z_t are either empty or Z_t = t (scheme theoretically) is an open immersion.]

3 thoughts on “At most one point

  1. I don’t have a reference, per se. But I do know that Ben-Zvi and Nevins considered the limit of the schemes P_n in their work on a “cuspidal Riemann-Hilbert correspondence”.

  2. Pingback: One point compactification « Stacks Project Blog

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