Anyway, please continue helping out by finding mistakes, etc, thank you very much. Also, we are going to have a party(!) when we reach 5000 pages, so please help: submit a new chapter, add an omitted proof, provide an alternative argument, or just randomly submit a piece of material you think may be useful, so we can get there more quickly. Thanks!
Just a heads up for those people who are taking parts or all of the Stacks project and doing new and exciting things with it. Please make sure you comply with the license that the Stacks project is under. Thanks!
Also, if you receive texts based on the Stacks project in your inbox, send us an email. Maybe in the future we can have a hall of shame or something.
Over the summer I wrote up a bit of material laying out a (very general) theory of formal algebraic spaces for the Stacks project. The idea is to work initially with very general objects and then for later results impose those conditions that make the arguments work (similarly to what is done for algebraic spaces and algebraic stacks in the Stacks project). As is often the case when you work through a new subject some natural very basic questions arise which I am unable to answer.
This paragraph is for motivation only and you can skip it. Let X be a scheme and let Z ⊂ X be a closed subset. The completion of X along Z is the functor which associates to a scheme T the set of morphisms f : T —> X such that f(T) ⊂ Z set theoretically. My question is whether one has “countably indexed => adic*” for such a completion.
In terms of algebra this means the following. Let A be a ring and let I ⊂ A be a radical ideal. Assume there is a countable family
I ⊃ J_1 ⊃ J_2 ⊃ J_3 ⊃ …
of ideals with V(I) = V(J_n) such that for every ideal I ⊃ J with V(I) = V(J) we have J ⊃ J_n for some n. In other words, the partially ordered set of closed subschemes of \Spec(A) supported on Z = V(I) has a countable cofinal subset. Let’s write A^* = \lim A/J_n as a topological ring endowed with the limit topology.
Is there a finitely generated ideal 𐌹 ⊂ A^* such that the powers of 𐌹 form a fundamental system of open neighbourhoods of 0? In other words, is A^* an adic topological ring which has a finitely generated ideal of definition?
Now that I state it like this, it seems this cannot possibly be true. But I haven’t found a counter example. Have you?
PS: I love gothic letters… 𐌰 𐌱 𐌲 𐌳 𐌴 𐌵 𐌶 𐌷 𐌸 𐌹 𐌺 𐌻 𐌼 𐌽 𐌾 𐌿 𐍀 𐍁 𐍂 𐍃 𐍄 𐍅 𐍆 𐍇 𐍈 𐍉 𐍊
Edit Sept 12, 2014. Just got a note from Gabber where he shows that the answer is yes when I is the radical of a countably generated ideal and that there is a counter example in general.