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.<\/p>\n
What is fun about this functor\u00a0 is that it is a natural candidate for a 1-point compactification of A^1, as the following discussion shows.<\/p>\n
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>\n
P_1 —> P_2 —> P_3 —> …<\/p>\n
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<\/p>\n
colim P_n —> F.<\/p>\n
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.<\/p>\n
Does anybody have a reference? What about the same thing for A^2?<\/p>\n
[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.]<\/p>\n","protected":false},"excerpt":{"rendered":"
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 … Continue reading