monomorphisms and epimorphisms of local rings - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:37:48Z http://mathoverflow.net/feeds/question/24066 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24066/monomorphisms-and-epimorphisms-of-local-rings monomorphisms and epimorphisms of local rings Martin Brandenburg 2010-05-10T07:17:01Z 2010-05-10T10:38:53Z <p>I want to understand the structure of monomorphisms/epimorphisms in the category of local rings (with local homomorphisms), or dually in the category of local schemes. Let $LR$ denote this category.</p> <ul> <li>Every monomorphism is injective.</li> </ul> <p>Proof: Let $R \to S$ be a monomorphism and $a,b \in R$ mapped to the same element. Regard $a,b$ as ring maps $R[t] \to R$. Let $\mathfrak{p},\mathfrak{q}$ the preimages of the maximal ideal $\mathfrak{m}$. Then we get two morphisms of local rings <code>$R[t]_{\mathfrak{p}} \times_S R[t]_{\mathfrak{q}} \to R$</code> which agree when composed with $R \to S$, thus they are equal. Evaluating at $(t,t)$ gives $a=b$. There is a shorter proof which uses $R[t]_{(t)+\mathfrak{m}}$.</p> <ol> <li><p>Is every monomorpism regular? (i.e. the equalizer of some morphism pair)</p></li> <li><p>Is every monomorphism extremal? (i.e. $i=ep$ with $p$ epi implies $p$ iso)</p></li> <li><p>Is every epimorphism, which is also injective, and isomorphism?</p></li> <li><p>Is every epimorphism surjective?</p></li> </ol> <p>Clearly we have 4 => 3 &lt;=> 2 and 1 => 2. Since in $LR$ every morphism can be factorized as $ep$ where $p$ is surjective and $e$ is injective, we also have 3 => 4.</p> <p>In the category of rings, 1,2,3,4 are false (see <a href="http://mathoverflow.net/questions/109/what-do-epimorphisms-of-commutative-rings-look-like" rel="nofollow">this discussion</a>). However, the counterexamples don't carry over to $LR$. In the "category theory bible" (joy of cats!) you can find nothing about local rings. The questions are partially motivated by Anton's problem about <a href="http://mathoverflow.net/questions/63/can-a-coequalizer-of-schemes-fail-to-be-surjective" rel="nofollow">coequalizers</a> in the category of (local) schemes.</p> <p>In Lazards article about flat epimorphisms (<a href="http://archive.numdam.org/ARCHIVE/SAC/SAC_1967-1968__2_/SAC_1" rel="nofollow">pdf link</a>), there is an example (1.6) in which 4 fails. I have not understood it yet. What is $Z$ there? Perhaps $B$ is a localization of $k[T,Z]$ and not $k[T]$? But I don't believe that $f$ is well-defined.</p> <p>If the answers turn out to be "no":</p> <ul> <li>Is there a nice description for the regular monomorphisms? What about field extensions?</li> </ul> <p>If $L/K$ is a finite galois extension, then $K \to L$ is a regular monomorphism iff $L/K$ is cyclic (see Brians comments).</p>