So I am gearing up to write a bit about Grothendieck’s existence theorem.<\/p>\n
Let R be a Noetherian ring complete with respect to an ideal I. Let X be a proper scheme over R. Let O_n = O_X\/I^nO_X. Consider an inverse system (F_n) of sheaves on X, such that F_n is a coherent O_n-module and such that the maps F_{n + 1} —> F_n induce isomorphisms F_n = F_{n + 1} ⊗_{O_{n + 1}} O_n. The statement of the theorem is that given any such system there exists a coherent O_X-module F such that F_n ≅ F\/I^nF (compatible with transition maps and module structure).<\/p>\n
Mike Artin told me Grothendieck was proud of this result.<\/p>\n
Because it is all the rage, let’s try to construct F directly from the system via category theory. So consider the functor<\/p>\n
G |—-> lim_n Hom_{O_X}(G, F_n)<\/p>\n
on QCoh(O_X). Since QCoh(O_X) is a Grothendieck abelian category (see Akhil Mathew’s post<\/a>) and since this functor transforms colimits into limits, we can apply the folklore result Lemma Tag 07D7<\/a>. Thus there exists a quasi-coherent sheaf F such that<\/p>\n Hom_{O_X}(G, F) = lim_n Hom_{O_X}(G, F_n)<\/p>\n The existence of F comes for free. (A formula for F is F = Q(lim F_n) where Q is the coherator as in Lemma Tag 077P<\/a>).<\/p>\n Of course, now the real problem is to show that F is coherent and that F\/I^nF = F_n, and I don’t see how proving this is any easier than attacking the original problem. Do you?<\/p>\n","protected":false},"excerpt":{"rendered":" So I am gearing up to write a bit about Grothendieck’s existence theorem. Let R be a Noetherian ring complete with respect to an ideal I. Let X be a proper scheme over R. Let O_n = O_X\/I^nO_X. Consider an … Continue reading