When do primes lift uniquely (provided they lift at all)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:28:38Z http://mathoverflow.net/feeds/question/14210 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14210/when-do-primes-lift-uniquely-provided-they-lift-at-all When do primes lift uniquely (provided they lift at all)? Harry Gindi 2010-02-05T01:49:55Z 2010-02-05T05:57:45Z <p>Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial sufficient conditions for the induced map $Spec(S) \to Spec(R)$ to be injective as a map of topological spaces (completely disregarding any of our sheaf structure)? That is, every prime that lifts lifts to a unique prime? If this problem is intractable as-is, I can add more conditions, so please don't add answers unless they are actual answers. </p> http://mathoverflow.net/questions/14210/when-do-primes-lift-uniquely-provided-they-lift-at-all/14216#14216 Answer by Felipe Voloch for When do primes lift uniquely (provided they lift at all)? Felipe Voloch 2010-02-05T02:32:02Z 2010-02-05T02:32:02Z <p>If $S/R$ is purely inseparable then the primes will lift uniquely.</p> http://mathoverflow.net/questions/14210/when-do-primes-lift-uniquely-provided-they-lift-at-all/14222#14222 Answer by Anton Geraschenko for When do primes lift uniquely (provided they lift at all)? Anton Geraschenko 2010-02-05T03:01:25Z 2010-02-05T03:01:25Z <p>Rather than listing various non-trivial sufficient conditions, let me give you a reference. Search EGA for the word "radiciel" (or start reading at Definition 3.5.4 of EGA I). A morphism of schemes $X\to Y$ is said to be <em>radiciel</em> (or <em>universally injective</em>) if for every field $K$, the induced map on $K$-points $X(K)\to Y(K)$ is injective. This generalizes the notion of a purely inseparable field extension.</p>