Extending faithfully flat covers of closed subschemes to open neighborhoods - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:08:22Z http://mathoverflow.net/feeds/question/84296 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84296/extending-faithfully-flat-covers-of-closed-subschemes-to-open-neighborhoods Extending faithfully flat covers of closed subschemes to open neighborhoods Ryan Reich 2011-12-26T03:17:33Z 2011-12-27T10:24:40Z <p>I am curious about the analogue of <a href="http://mathoverflow.net/questions/56865/the-etale-site-of-a-closed-subscheme-and-its-etale-grothendieck-subtopology" rel="nofollow">this question</a>, as stated in the title. Namely,</p> <blockquote> <p>If $Z \subset X$ is a closed subscheme and $Y \to Z$ is faithfully flat (let's also say of finite presentation), can we find a map $Y' \to X$ which is flat, whose image contains $Z$ (and is thus a neighborhood of $Z$), and whose restriction to $Z$ is the given map?</p> </blockquote> <p>This may be too strong; i.e. just as for &eacute;tale maps, in the other question, this may be true only Zariski locally on $X$. That's fine too. In that case, in its most basic form it becomes a question of commutative algebra:</p> <blockquote> <p>If $R$ is a ring, $I \subset R$ an ideal, and $S$ a faithfully flat $R/I$-algebra, is there a set of elements $f \in R$ such that $\sum (R/I) f = R/I$ and a corresponding set of faithfully flat <code>$R_f$</code>-algebras <code>${}_f\widetilde{S}$</code> such that <code>${}_f \widetilde{S}/I_f \cong S_f$</code> for each $f$?</p> </blockquote> <p>Also, it would be nice to know that, in the event that this is true, it preserves finiteness hypotheses like "finitely presented".</p> <p>For those who are curious about my motivations: I want to show that "<code>$!$</code> pushforwards (resp. pullbacks) commute with <code>$*$</code> pushforwards (resp. pullbacks)" under appropriate circumstances, those being when the <code>$!$</code>'s are along closed immersions and the <code>$*$</code>'s are along faithfully flat covers or vice-versa. At the very least I need to be able to switch the order in which maps of these types appear; that is, rewrite a cover of a closed immersion as a closed immersion into a cover, as my question asks.</p> http://mathoverflow.net/questions/84296/extending-faithfully-flat-covers-of-closed-subschemes-to-open-neighborhoods/84372#84372 Answer by Bhargav for Extending faithfully flat covers of closed subschemes to open neighborhoods Bhargav 2011-12-27T09:54:03Z 2011-12-27T10:24:40Z <p>(Essentially copied from the comments as requested.)</p> <p>I think this question is asking for something too strong; here's an example. Let $R = \mathbf{Z}_p$, and let $S_0$ be a finite flat $\mathbf{F}_p$-algebra that does not lift to a finite flat $R$-algebra; an explicit example can be found <a href="http://mathoverflow.net/questions/63969/what-is-an-explicit-example-of-a-variety-x-which-is-finite-over-spec-f-p-but-whic" rel="nofollow">here</a>. Since $R$ is local, the present question asks: does there exist a faithfully flat $R$-algebra $S$ lifting $S_0$? If there was such an $S$, then the $p$-adic completion $\widehat{S}$ of S would be a finite flat R-algebra lifting $S_0$ (the flatness is clear, and the finiteness comes from Nakayama, completeness, and finiteness modulo $p$), which cannot exist. Hence, there is no such $S$ either.</p>