Intuition for Nagata's altitude formula? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:30:11Z http://mathoverflow.net/feeds/question/5982 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5982/intuition-for-nagatas-altitude-formula Intuition for Nagata's altitude formula? Ho Chung Siu 2009-11-18T15:51:39Z 2009-11-25T14:11:14Z <p>This is theorem 14.C on p.84 of Matsumura's commutative algebra.</p> <blockquote> <p>Let $A$ be a noetherian domain, and let $B$ be a finitely generated overdomain of $A$. Let $P \in Spec(B)$ and $p = P \cap A$. Then we have <code>$ht(P) \leq ht(p) + tr.d._{A} B - tr.d._{K(p)} K(P)$</code> with equality holds when $A$ is universally catenary or if $B$ is a polynomial ring over $A$.</p> </blockquote> <p>Question: How should one understand this formula? I'm hazarding a guess that this factor, <code>$tr.d._{A} B - tr.d._{K(p)}K(P)$</code>, can somehow measure how primes of $B$ will be identified when they are restricted back to $A$. But this sounds woefully wrong and I just want to know how I should view this result or whether there is any (geometric) intuition behind the result.</p> <p>Thanks!</p> http://mathoverflow.net/questions/5982/intuition-for-nagatas-altitude-formula/6816#6816 Answer by Francisco Perdomo for Intuition for Nagata's altitude formula? Francisco Perdomo 2009-11-25T14:11:14Z 2009-11-25T14:11:14Z <p>Put dim B=n for the dimension of the variety with coordinates ring B. Then <br> <br> n-ht P &#8805; ((n-tr deg <sub>A</sub> B)- ht p)+ tr deg <sub><i>k</i>(p)</sub> <i>k</i>(P) <br> <br> The first member of the inequality indicates the dimension of the subvariety definited by P. <br> The term (n-tr deg <sub>A</sub> B) in the second member is the dimension of the variety with coordinates ring A: it looses tr deg <sub>A</sub> B dimensions with respect the other variety with coordinate ring B. <br> Then ((n-tr deg <sub>A</sub> B)- ht p) represent the dimension of the subvariety definited by p. <br> The term tr deg <sub><i>k</i>(p)</sub> <i>k</i>(P) is a corrector term because blow up can occur. </p>