User phillip williams - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:21:45Z http://mathoverflow.net/feeds/user/9960 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103025/when-does-dimension-behave-nicely-for-quotients-of-affine-algebraic-varieties-by When does dimension behave nicely for quotients of affine algebraic varieties by the action of a group? Phillip Williams 2012-07-24T17:21:08Z 2012-07-26T15:38:46Z <p>Let $X$ be an affine algebraic variety over an algebraically closed field $k$ of characteristic zero. Let $G$ be a reductive algebraic group acting on $X$. In this setting, there exists a categorical quotient variety $X / G$. Are there nice conditions (involving $X$ and/or $G$) that imply that $\mathrm{dim}(X / G) = \mathrm{dim}(X) - \mathrm{dim}(G)$? As pointed out in the comments, at a minimum one would have to require the action to be faithful.</p> http://mathoverflow.net/questions/56591/what-are-the-monomorphisms-in-the-category-of-schemes What are the monomorphisms in the category of schemes? Phillip Williams 2011-02-25T03:23:36Z 2012-07-06T09:33:53Z <p>Someone recently <a href="http://mathoverflow.net/questions/56564/what-are-the-epimorphisms-in-the-category-of-schemes" rel="nofollow">asked</a> what the epimorphisms in the category of schemes are; the other day I had been wondering about the similar question: what are the monomorphisms in the category of schemes? I am often frustrated working with schemes because, unlike a lot of other categories, it is not immediate that you have left cancellation of morphisms when you know the underlying map on sets is injective--and I think it must not be true in general, though I don't have an example in mind. Are there nice situations or additional conditions that guarantee that one may safely cancel morphisms of schemes on the left? </p> http://mathoverflow.net/questions/60457/elementaryshortuseful/60646#60646 Answer by Phillip Williams for Elementary+Short+Useful Phillip Williams 2011-04-05T04:48:27Z 2011-04-05T04:48:27Z <p>I've always been thrilled by the fact that the coefficients of a (monic) polynomial are obtained by taking the elementary symmetric functions in (minus) the roots of that polynomial:</p> <p>$$\prod_{i=1}^n (X+\alpha_i) = \sum_{k=0}^n (\sum_{i_1 &lt; \cdots &lt; i_k} \alpha_{i_1}\cdots \alpha_{i_k})X^{n-k}$$ A lot is built on this, I think. I'd like to explain the connection to automorphisms and fixed fields and how the roots of a polynomial are permuted by an automorphism that fixes the coefficient field of that polynomial. Then maybe mention the beginnings of Galois theory.</p> http://mathoverflow.net/questions/51905/how-to-picture-mathbbc-p How to picture $\mathbb{C}_p$? Phillip Williams 2011-01-13T03:31:47Z 2011-02-25T15:41:04Z <p>I hope this is appropriate for mathoverflow. Understanding $\mathbb{C}_p$ has always been something of a stumbling block for me. A standard thing to do in number theory is to take the completion $\mathbb{Q}_p$ of the rationals with respect to a $p$-adic absolute value. The resulting field is then complete, but has no good reason to be algebraically closed. You can take its algebraic closure, but that is not complete, so then you take the completion of that, and get a field which is both complete, and algebraically closed, denoted by $\mathbb{C}_p$.</p> <p>I understand that it is a reasonable desire to have a field extension of $\mathbb{Q}_p$ that is both complete and algebraically closed; my trouble, however, is getting some sort of grasp on how to picture this object, and to develop any intuition about how it is used. Here are my questions; I'd imagine the answers are related:</p> <ol> <li>Am I even supposed to be able to picture it? </li> <li>Is there some way I ought to think of a typical element?</li> <li>Is it worth it, in terms of these goals, to look at the proofs of the assertions in my first paragraph?</li> <li>How is $\mathbb{C}_p$ typically used? (this question may be too vague, feel free to ignore it!)</li> </ol> <p>Please feel free to answer any or all of these questions.</p> http://mathoverflow.net/questions/41836/elementary-proof-of-nakayamas-lemma Elementary Proof of Nakayama's Lemma? Phillip Williams 2010-10-11T21:10:05Z 2010-10-12T08:13:24Z <p>Nakayama's Lemma is as follows: Let $A$ be a ring, and $\frak{a}$ an ideal such that $\frak{a}$ is contained in every maximal ideal. Let $M$ be a finitely generated $A$-module. Then if $\frak{a}$$M=M, we have that M = 0.</p> <p>Most proofs of this result that I've seen in books use some non-trivial linear algebra results (like Cramer's rule), and I had come to believe that these were certainly necessary. However, in Lang's Algebraic Number Theory book, I came across a quick proof using only the definitions and induction. I felt initially like something must be wrong--I thought perhaps the proof is simpler because Lang is assuming throughout that all rings are integral domains, but he doesn't use this in the proof he gives, as far as I can see.</p> <p>Here is the proof, verbatim: We do induction on the number of generators of M. Say M is generated by w_1, \cdots, w_m. There exists an expression$$w_1 = a_1w_1 + \cdots + a_mw_m$$with a_i \in \frak{a}. Hence$$(1-a_1)w_1 = a_2w_2 + \cdots +a_mw_m$$If$(1-a_1)$is not a unit in A, then it is contained in a maximal ideal$\frak{p}$. Since$a_1 \in \frak{p}$by hypothesis, we have a contradiction. Hence$1-a_1$is a unit, and dividing by it shows that$M$can be generated by$m-1$elements, thereby concluding the proof.</p> <p>Is the fact that$A\$ is assumed to be a domain being smuggled in here in some way that I missed? Or is this really an elementary proof of Nakayama's lemma, in full generality?</p> http://mathoverflow.net/questions/111004/good-reduction-for-singular-varieties Comment by Phillip Williams Phillip Williams 2012-10-29T16:14:15Z 2012-10-29T16:14:15Z I think in the curves setting one could consider stability or semi-stability as a property preserved by reduction. I don't know if this generalizes. http://mathoverflow.net/questions/103025/when-does-dimension-behave-nicely-for-quotients-of-affine-algebraic-varieties-by/103048#103048 Comment by Phillip Williams Phillip Williams 2012-07-26T15:37:10Z 2012-07-26T15:37:10Z OK, thanks! I had been looking at Mumford but this directs my reading a lot more. http://mathoverflow.net/questions/103025/when-does-dimension-behave-nicely-for-quotients-of-affine-algebraic-varieties-by/103048#103048 Comment by Phillip Williams Phillip Williams 2012-07-25T16:41:50Z 2012-07-25T16:41:50Z This criterion will certainly be useful; thank you. Can you give me a reference for the statements in your first paragraph? http://mathoverflow.net/questions/103025/when-does-dimension-behave-nicely-for-quotients-of-affine-algebraic-varieties-by Comment by Phillip Williams Phillip Williams 2012-07-25T16:40:24Z 2012-07-25T16:40:24Z Yes Oliver I believe you are probably right. I am not an expert on this stuff. Forgive me for the naive question regarding your terminology: is &quot;good quotient&quot; the same thing as &quot;categorical quotient&quot;? http://mathoverflow.net/questions/97538/inductive-proof-of-a-version-of-nakayamas-lemma Comment by Phillip Williams Phillip Williams 2012-05-21T22:09:58Z 2012-05-21T22:09:58Z I believe Martin is right and said so here: <a href="http://mathoverflow.net/questions/41836/elementary-proof-of-nakayamas-lemma" rel="nofollow" title="elementary proof of nakayamas lemma">mathoverflow.net/questions/41836/&hellip;</a> http://mathoverflow.net/questions/61515/life-after-hartshorne-the-book-not-the-person/61525#61525 Comment by Phillip Williams Phillip Williams 2011-04-13T16:00:54Z 2011-04-13T16:00:54Z For GIT, there is also Dolgachev's &quot;Lectures on Invariant theory.&quot; http://mathoverflow.net/questions/56591/what-are-the-monomorphisms-in-the-category-of-schemes/56608#56608 Comment by Phillip Williams Phillip Williams 2011-02-25T16:32:50Z 2011-02-25T16:32:50Z Thanks, this is definitely helpful! http://mathoverflow.net/questions/53316/why-are-extensions-so-heavily-emphasized-in-valuation-theory/53318#53318 Comment by Phillip Williams Phillip Williams 2011-01-27T04:04:49Z 2011-01-27T04:04:49Z Ok cool, thank you. http://mathoverflow.net/questions/53316/why-are-extensions-so-heavily-emphasized-in-valuation-theory/53318#53318 Comment by Phillip Williams Phillip Williams 2011-01-26T15:42:40Z 2011-01-26T15:42:40Z Pete, can you recommend a reference that develops the stuff you talked about in your second to last paragraph? http://mathoverflow.net/questions/51905/how-to-picture-mathbbc-p/51912#51912 Comment by Phillip Williams Phillip Williams 2011-01-14T05:10:37Z 2011-01-14T05:10:37Z Wow I like your new picture even more...I appreciate the effort in making it too, and the explanation. http://mathoverflow.net/questions/51905/how-to-picture-mathbbc-p/51912#51912 Comment by Phillip Williams Phillip Williams 2011-01-13T16:14:16Z 2011-01-13T16:14:16Z I like this picture; thanks! I'll have to think about it more... http://mathoverflow.net/questions/51905/how-to-picture-mathbbc-p/51960#51960 Comment by Phillip Williams Phillip Williams 2011-01-13T15:47:31Z 2011-01-13T15:47:31Z I thought someone might mention Berkovich space! I like the pictures in your book... http://mathoverflow.net/questions/51905/how-to-picture-mathbbc-p/51919#51919 Comment by Phillip Williams Phillip Williams 2011-01-13T15:40:21Z 2011-01-13T15:40:21Z @Pete: In regards to 4, there are many things in mathematics that give me the impression of being &quot;important&quot; though I could not tell you why anyone thinks so, as far as my own understanding of anything goes. I'm trying to repair gaps in my understanding of algebraic number theory; part of what I'm seeking is some idea as to why something I want to run from seems to be showing up all over the place. So it's helpful for me to know how the experts treat it; your comments, and those of others, do seem to be providing this for me, so thank you! http://mathoverflow.net/questions/41836/elementary-proof-of-nakayamas-lemma/41842#41842 Comment by Phillip Williams Phillip Williams 2010-10-12T00:00:40Z 2010-10-12T00:00:40Z Thanks, this is helpful.