User tim carstens - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:36:34Z http://mathoverflow.net/feeds/user/1301 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence/8740#8740 Answer by Tim Carstens for Describe a topic in one sentence. Tim Carstens 2009-12-13T08:34:25Z 2009-12-13T08:34:25Z <p>Geometric group theory: the large-scale geometry of a group is invariant under quasi-isometry.</p> http://mathoverflow.net/questions/8204/how-can-i-really-motivate-the-zariski-topology-on-a-scheme/8218#8218 Answer by Tim Carstens for How can I really motivate the Zariski topology on a scheme? Tim Carstens 2009-12-08T17:48:43Z 2009-12-08T17:48:43Z <p>If you buy into the idea that you want a topological model for your ring, then right away it becomes sensible to ask that any map Ring -> Top be functorial. Of course, m-Spec -- which is already classically motivated -- doesn't lend itself to this, simply because there isn't an obvious way to use a ring homomorphism $f : A \to B$ to move maximal ideals around.</p> <p>Such a map <em>can</em> move around ideals, both by extension and contraction, and this is a good first thing to investigate. Your choice of whether or not you want to push ideals forward or pull them back will determine if your functor should be co- or contravariant.</p> <p>To decide between these two, look at the initial and terminal objects in Ring, as well as the initial and terminal objects in Top. The ring {0,1} has a single ideal, hence its topological space has (at-most) a single point, hence should probably be sent to the final object in Top. The 0 ring has no ideals, hence its topological space has no points, hence should be sent to the initial object in Top. Your hand has thus been forced: you need a contravariant functor, hence contraction is the thing to look at.</p> <p>Now observe that $f^{-1}(\mathfrak m)$ need not be maximal, even if $\mathfrak m$ is, but it <em>will</em> be prime. You're thus immediately led to seeing if you can put a topology on Spec the same way you did for m-Spec. It works, and you move on.</p> http://mathoverflow.net/questions/3242/canonical-examples-of-algebraic-structures/3417#3417 Answer by Tim Carstens for Canonical examples of algebraic structures Tim Carstens 2009-10-30T06:46:31Z 2009-10-30T06:46:31Z <p>Lots of good answers. I figured I'd throw in a list of non-examples, since these are pretty handy as well. (These are all standard non-examples, nothing fancy.)</p> <p>A non-Noetherian ring with only one prime ideal: <code>(k[x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>, ...]/(x<sub>i</sub> x<sub>j</sub> : 1 &lt;= i,j), (x<sub>1</sub>,x<sub>2</sub>,...))</code>.</p> <p>A non-Cohen-Macaulay ring: <code>k[x, y]/(x<sup>2</sup>, xy)</code>.</p> <p>A category that doesn't have products: the category of fields with field homomorphisms.</p> <p>A ring which isn't flat over another ring: <code>A = k[x<sup>2</sup>, x<sup>3</sup>]</code> and <code>B = k[x]</code>.</p> <p>Two non-zero rings whose tensor product is zero: <code>Z<sub>2</sub></code> and <code>Z<sub>3</sub></code></p> http://mathoverflow.net/questions/8204/how-can-i-really-motivate-the-zariski-topology-on-a-scheme/8218#8218 Comment by Tim Carstens Tim Carstens 2009-12-08T18:15:17Z 2009-12-08T18:15:17Z Contravariance is also sensible in light of the ideal correspondence for quotients and localization, which suggest that the topological spaces associated to A/p and A_p should both be subspaces of the space for A, a fact which runs in the opposite direction of the canonical maps A-&gt;A/p and A-&gt;A_p.