Here is a question I have been struggling with for the last couple of weeks.<\/p>\n
Question: Let A be a Noetherian henselian local ring. Let A —> B be (a) local ring map of local rings, (b) essentially of finite type, (c) the residue field extension is trivial, and (d) injective. Is the map A^ —> B^ of completions injective?<\/p>\n
If the answer is “yes”, then this somehow tells us that (very very roughly) “taking scheme theoretic image commutes with completion”.<\/p>\n
The answer is yes if A is in addition excellent. But I would like to know if it is also true in general. It is very possible that there exists a simple counter example, it is also possible that it is true for trivial reasons. The most vexing aspect of this question to me is that I cannot even decide whether it should be true or not. Please leave a comment if you have any references, comments, or suggestions. Thanks!<\/p>\n
[Edit on August 22, 2010: I finally figured out that this is wrong. Namely, take A to be the example of Ogoma. It is a normal henselian Noetherian local domain whose completion is k[[x, y, z, w]]\/(yz, yw). So the completion is the union of a nonsingular 3 dimensional component and a nonsingular 2 dimensional component. Let C be an affine chart of the blow up of A at its maximal ideal. The special fiber of C has two irreducible components (a plane and a line). Let B be the localization of C at a maximal ideal which is a point on one of them but not the other. Then clearly the completion of B “picks out” one of the irreducible components of the completion of A.]<\/p>\n","protected":false},"excerpt":{"rendered":"
Here is a question I have been struggling with for the last couple of weeks. Question: Let A be a Noetherian henselian local ring. Let A —> B be (a) local ring map of local rings, (b) essentially of finite … Continue reading