User andrew dudzik - MathOverflow

Answer by Andrew Dudzik for Mathematical "urban legends"

2011-05-24T20:10:19Z

When I took analysis from Paul Sally, he claimed that a student once asked him in class, "Professor Sally, why is it called the p-adic norm? If it's a norm, what does it measure?" Without thinking, Paul loudly replied, "Well, it measures the p-ness of a number."

I suspect that he just substituted himself into an existing urban legend, yet I would not be surprised if it were true.

In what sense is the étale topology equivalent to the Euclidean topology?

2011-04-05T03:15:03Z

I have heard it said more than once—on Wikipedia, for example—that the étale topology on the category of, say, smooth varieties over $\mathbb{C}$, is equivalent to the Euclidean topology. I have not seen a good explanation for this statement, however.

If we consider the relatively simple example of $\mathbb{P}^1_\mathbb{C}$, it seems to me that an étale map is just a branched cover by a Riemann surface, together with a Zariski open subset of $\mathbb{P}^1_\mathbb{C}$ that is disjoint from the ramification locus. (If there is a misconception there, small or large, please let me know) The connection to the Euclidean topology on $\mathbb{P}^1_\mathbb{C}$, however, is not obvious to me.

What is the correct formulation of the statement that the two topologies are equivalent, or what is a good way to compare them?

Asymptotics of Power Series With Branch Singularities

2009-11-13T19:39:10Z

I am wondering if there are analytic tools to find asymptotic formulae for the coefficients of a complex power series of a function with branch singularities. For example, it is possible to show using elementary means that, for $q>1$, the coefficients of $\frac{1}{1-z} log(\frac{1}{1-qz})$ are asymptotic to $\frac{1}{1-q^{-1}} \frac{q^n}{n}$, but I would like to know if it is possible to see this from analytic properties of the function itself.

Motivation: There are nice asymptotic formulae for the coefficients of power series of meromorphic functions. As a simple example, the coefficients of an entire function must be $O(\epsilon^n)$ for any $\epsilon$. In general, the sum of the principal parts of the poles of smallest modulus will provide a very good first approximation, and contour integration can be applied to get more delicate bounds, see e.g. Wilf's Generatingfunctionology.

Comment by Andrew Dudzik

2013-05-20T23:11:05Z

Two more useful examples: 1) A descending sequence of discs with empty intersection gives another such skyscraper sheaf (which also comes from a Berkovich point). 2) There are (non-"overconvergent") examples that don't come from Berkovich points, but do come from points in Huber's adic space. For example, let $F(V)=\mathbb{Z}$ if $V$ contains the open unit disc with finitely many open discs of radius $<1$ removed, and $F(V)=0$ otherwise. One way to think of this second example is: the skyscraper sheaf at the origin of the canonical reduction, $\mathbb{A}^1_{\tilde{k}}$ .

Comment by Andrew Dudzik

2012-07-02T02:14:36Z

This is open for $k\geq 7$. The proof for $k=6$ was done by Barajas and Serra using elaborate computer-assisted casework, and many simplifications that rely on the fact that $6+1$ is prime. It is worth noting that when the ratio of two speeds is irrational, the problem is made easier by density arguments, so the essentially hardest case is when all the speeds are integers. Therefore this is a combinatorial number theory question disguised as basic calculus.

Comment by Andrew Dudzik

2012-03-26T18:59:31Z

I think that Hochster uses the older language of preschemes. In modern terms, he proved that the underlying spaces of separated schemes are exactly the open subsets of spectral spaces (his "locally spectral and quasispectral" spaces) but that the underlying spaces of all schemes are more generally the spaces which are locally spectral.

Comment by Andrew Dudzik

2011-10-31T17:36:05Z

I just encountered the paper "Noncommutative Coordinate Rings and Stacks": web.maths.unsw.edu.au/~danielch/paper/stacks.pdf Chan and Ingalls construct a structure sheaf of noncommutative algebras over a stack that meets some restrictive conditions. I'm told that this result has been extended somewhat, but could not find a reference. Their condition seems to be met for $\mathcal{M}_g$, $g\geq 2$. Whether such a relationship exists in general seems to be an open question.

Comment by Andrew Dudzik

2011-10-20T17:02:18Z

Martin Brandenburg [mentioned this][1] as a "common false belief"—the problem is that we might have to take a colimit over an indexing category that is not essentially small. But it should be possible to sheafify presheaves on most reasonable sites. For example, the category of étale maps to a scheme is essentially small (I think), so this problem does not arise. [1]: mathoverflow.net/questions/23478/…

Comment by Andrew Dudzik

2011-10-13T16:38:58Z

I agree that it's too bad that there isn't a better website for basic early-graduate-level questions, but there are books, courses, TAs, and other students to draw on. This graduate student avoids posting any question unless he thinks it will be of some interest to a nonzero number of research mathematicians. By the way, William Fulton has made his lovely little book on algebraic curves available for free online: math.lsa.umich.edu/~wfulton/CurveBook.pdf

Comment by Andrew Dudzik

2011-10-13T16:21:40Z

Another direct approach: since $\mathbb{C}[x] \cong \mathbb{R}[x] \otimes_\mathbb{R} \mathbb{C}$, by rearranging tensor products we can see that the right-hand side is $(\mathbb{C}\otimes_\mathbb{R} \mathbb{C})[x,y]$.

Comment by Andrew Dudzik

2011-10-06T16:51:03Z

It's formatted like a silent film of a Noh play, and based on an obscure film by Yukio Mishima. These facts alone make it challenging for a modern Western viewer to issue sensible criticism. Interesting to note that Mishima's wife didn't want the original film released—likely not because of how it treated women, but because of how it treated harakiri.

Comment by Andrew Dudzik

2011-09-14T04:26:20Z

Don't forget that sometimes that automorphism is trivial...

Comment by Andrew Dudzik

2011-05-30T14:09:25Z

But $(\underline{f})$ may not be contained in $k[\underline{f}]$.

Comment by Andrew Dudzik

2009-11-15T00:44:01Z

This is pretty clear, so I'll mark it as an accepted answer shortly, but for now I'm holding out to see if someone can demonstrate how to work my example.