P¹ is simply connected

Posted on 19 April 2019 by Remy

This is a cute proof that I ran into of the simple connectedness of $\mathbb P^1$ . It does not use Riemann–Hurwitz or differentials, and instead relies on a purely geometric argument.

Lemma. Let $k$ be an algebraically closed field. Then $\mathbb P^1_k$ is simply connected.

Proof. Let $f \colon C \to \mathbb P^1$ be a finite étale Galois cover with Galois group $G$ . We have to show that $f$ is an isomorphism. The diagonal $\Delta_{\mathbb P^1} \subseteq \mathbb P^1 \times \mathbb P^1$ is ample, so the same goes for the pullback $D = (f \times f)^* \Delta_{\mathbb P^1}$ to $C \times C$ [Hart, Exc. III.5.7(d)]. In particular, $D$ is connected [Hart, Cor. III.7.9].

But $D \cong C \times_{\mathbb P^1} C$ is isomorphic to $|G|$ copies of $C$ because the action

$\begin{align*} G \times C &\to C \times_{\mathbb P^1} C\\ (g,c) &\mapsto (gc,c) \end{align*}$

is an isomorphism. If $D$ is connected, this forces $|G| = 1$ , so $f$ is an isomorphism. $\qed$

The proof actually shows that if $X$ is a smooth projective variety such that $\Delta_X$ is a set-theoretic complete intersection of ample divisors, then $X$ is simply connected.

Example. For a smooth projective curve $C$ of genus $g \geq 1$ , the diagonal cannot be ample, as $\pi_1(C) \neq 0$ . We already knew this by computing the self-intersection $\Delta_C^2 = 2-2g \leq 0$ , ~~but the argument above is more elementary~~.

References.

[Hart] Hartshorne, Algebraic geometry. GTM 52, Springer, 1977.

Number theory is heavy metal

Posted on 1 April 2019 by Remy

As some of you may be aware, I am a musician as well as a mathematician. I often like to compare my experiences between the two. For example, I found that my approach to the creative process is not dissimilar (in both, I work on the more technical side, with a particular interest in the larger structure), and I face the same problems of excessive perfectionism (never able to finish anything).

This post is about some observations about the material itself, as opposed to my interaction with it.

Universal donor.

A look at the arXiv listing for algebraic geometry shows the breadth of the subject. Ideas from algebraic geometry find their way into many different areas of mathematics; from representation theory and abstract algebra to combinatorics and from number theory to mathematical physics. But when I say algebraic geometry is a universal donor (of ideas, techniques, etc.), I also mean that its applications to other fields far outnumber the applications of other fields to algebraic geometry, and that the field of algebraic geometry is largely self-contained.

Much the same role is played by classical music inside musical composition. Common practise theory is used throughout Western music, whether you’re listening to hip hop, trance, blues, ambient electronic, bluegrass, or hard rock. Conversely, the influence of popular genres music on [contemporary] classical is comparatively little. One could therefore argue that classical music is a universal donor in the field of musical composition.

Universal receptor.

The opposite is the case for number theory. Another vast area, number theory often uses ingenious arguments combining ideas from algebra, combinatorics, analysis, geometry, and many other areas. In practical terms, the amount of material that a number theorist needs to master is immense: whatever solves your particular problem.

So is there any musical genre that plays a similar role? I claim that metal fits the bill. With a vast list of subgenres including thrash metal, black metal, doom metal, progressive metal, death metal, symphonic metal, nu metal, grindcore, hair metal, power metal, and deathcore, metal writing often contains creative combinations of other genres, from classical music to ambient electronic, and from free jazz to hip hop. As Steven Wilson of Porcupine Tree said about his rediscovery of metal:

I said to myself, this is where all the interesting musicians are working! Because for a long time I couldn’t find where all these creative musicians were going… You know in the 70’s they had a lot of creative musicians like Carlos Santana, Jimmy Page, Frank Zappa, Neil Young, I was thinking “where are all these people now?” and I found them, they were working in extreme metal.

I sometimes think of metal as a small microcosmos reflecting the full range of Western (and some non-Western) music, tied together by distorted guitars and fast, technical drumming.

In summary, algebraic geometry is classical music, and number theory is heavy metal.

Perfectionism and the stages of completion

Posted on 23 February 2019 by Remy

This week I finished another in a series of three preprints that in September 2017 I said were ‘almost done’. With a newer result added later, there are still two more papers left in a similar state.

However, the positive news is that I have now finished two things (oh, and there was this thing too), and I am becoming more honest with myself about what it means to finish something. Here are the stages of completion that I currently observe, and my (current/past) responses to them. I hope this reflection may be useful or somewhat amusing to someone.

Mathematical completion

By this I mean the internal conviction that there is probably a correct and complete argument, and you worked out the most important lemmas. This stage often happens in some part in my head, and not everything will have been written down yet (not even the important arguments).

Coincidentally, this is also the stage where my brain jumps to the next problem. Maybe not the best idea. I call this mathematical completion somewhat ironically, because it’s not: there are always more lemmas to prove in better ways, and things that you thought were obvious when they aren’t.

At this stage, I usually have enough down to feel confident giving talks about the matter, as my familiarity with the argument is pretty robust and immune to probing.

Nearly done

This is usually the point where the rough outline is laid down, a structure for the paper has been decided and set up, and the most important arguments are explained. It feels that there are still some small things to fix, but otherwise it shouldn’t take too long. This, like the previous stage, is based on a lie.

This was the stage that my three papers in 2017 were in, two of which have been posted now. The third one needs a more substantial rewrite for stylistic and formatting reasons, but it has long passed the Mathematical completion stage.

Fully written up

This means that there is a document from start to finish, complete with abstract, introduction, and correctly formatted references. Surely it shouldn’t take more than a week to finish, right? Wrong: proofreading. There’s often more missing than you think. Plus, I don’t like excessive updating, so I’d rather get it right the first time around.

There’s an aspect of optimisation too: every result should be written up in the easiest way in the correct generality (this is not always the biggest generality; more about this in a future post). Some results feel too unimportant to warrant a two-page proof, and I try to find a proof that does not take more space than the result deserves. This is hard to combine with the idea of completeness and self-containedness, and it certainly takes time to get it right. Local optimisation does not always align with global optimisation.

What is appropriate depends on the medium: in my dissertation I explain some well-known results in depth if no complete and modern argument is available in the literature. In papers, I try to give the briefest argument using (well-written and modern) accounts from the literature.

I’m done with this shit

Finally, once I (and maybe someone else too) spent enough time proofreading and fixing small gaps, confusions, notational inconsistencies, inaccurate references, and irrelevant distractions, I don’t want to think about it anymore and just dump it on arXiv. This allows other people to look at it, who will immediately find more mistakes or missed opportunities. (This is the best possible outcome; the alternative is that people don’t care at all.)

Errata and addenda

I keep a file with additional changes I want to make at some point in the future. I fix some of them and send it to a journal.

To be continued…

Since none of my papers have appeared in print yet, I cannot comment on the final stage of completion. Let’s hope it happens some day…

The backlog

In the weeks leading to completion, I am strongly inclined only to work on the paper, because it’s so nearly done (right?!). Also, other things don’t seem to matter (mumble something about microeconomics and optimising utility).

After I finish working obsessively on a paper for some weeks, I need some time to catch up on the rest of my life. Maybe as I’m getting more honest with myself, I can also change this habit (although it’s not entirely clear to me that this is desirable!)…

Closing remarks.

This post doesn’t really talk about my excessive¹ perfectionism, about which I may say a thing or two in the future.

¹Meaning it’s hard for me to finish anything at all, or sometimes even to start something.

Rings are abelian

Posted on 14 November 2018 by Remy

In this post, we prove the following well-known lemma:

Lemma. Let $(R,+,\times,0,1)$ satisfy all axioms of a ring, except possibly the commutativity $a + b = b + a$ . Then $(R,+)$ is abelian.

That is, additive commutativity of a ring is implied by the other axioms.

Proof. By distributivity, we have $2(a+b) = 2a + 2b$ , so multiplication by $2$ is a homomorphism. By our previous post, this implies $R$ is abelian. $\qedsymbol$

Criteria for groups to be abelian

Posted on 14 November 2018 by Remy

This is a review of some elementary criteria for a group to be abelian.

Lemma. Let $G$ be a group. Then the following are equivalent:

$G$ is abelian,

The map $G \to G$ given by $g \mapsto g^2$ is a group homomorphism;

The map $G \to G$ given by $g \mapsto g^{-1}$ is a group homomorphism;

The diagonal $G \subseteq G \times G$ is normal.

Proof. We prove that each criterion is equivalent to (1).

For (2), note that $(gh)^2 = ghgh$ , which equals $gghh$ if and only if $gh = hg$ .

For (3), note that $(gh)^{-1} = h^{-1}g^{-1}$ , which equals $g^{-1}h^{-1}$ if and only if $gh = hg$ .

For (4), clearly $\Delta_G \colon G \hookrightarrow G \times G$ is normal if $G$ is abelian. Conversely, note that $(e,h)(g,g)(e,h^{-1}) = (g,hgh^{-1})$ , which is in the diagonal if and only if $gh = hg$ . $\qedsymbol$

Not every open immersion is an open immersion

Posted on 13 September 2018 by Remy

An immersion (or locally closed immersion) of schemes is a morphism $f \colon X \to Y$ that can be factored as $X \to U \to Y$ , where $X \to U$ is a closed immersion and $U \to Y$ is an open immersion. If it is moreover an open morphism, it need not be an open immersion:

Example. Let $X$ be a nonreduced scheme, and let $X_{\text{red}} \to X$ be the reduction. This is a closed immersion, whose underlying set is the entire space. Thus, it is a homeomorphism, hence an open morphism. It is not an open immersion, for that would force it to be an isomorphism. $\qedsymbol$

Remark. However, every closed immersion is a closed immersion; see Tag 01IQ.

Timeline of postdoc applications

Posted on 18 April 2018 by Remy

This is an attempt to give a rough outline of the timeline of the last year of a PhD in mathematics, especially with the eye on applying for US postdocs through MathJobs. I will say a few words about European postdocs as well.

This is the type of information I wish I had available before I started the process, so I’m hoping it may be of help to some people. I take full responsibility for all inaccuracies; but if you do find that this information is incorrect or incomplete, let me know in the comments below!

Preparing your material.

Of course the correct time to start preparing your material depends on a lot of factors, including the timing of your first application deadline. I personally think it can’t hurt to start thinking about letter writers and get started on your research and teaching statements at the beginning of the summer, but I also know people who only really get started after the summer. At the very least, talk to your advisor early to make a plan for the hiring season (some advisors will provide more guidance than others).

An important thing to keep in mind is that everything takes time. You want to give your letter writers at least one months notice, if not two. But when you ask them, it’s good to have at least some draft of your research statement ready (although it might be ok to supply the final version a bit later, especially if you ask your references early). Some faculty only need one week to write a letter; others will take a bit more time with it (e.g. if they have a lot of commitments, or because they are more careful or meticulous with everything they do). Allow for extra time if you want a letter in the summer, as the professor you’re asking might be travelling or out of office.

If you’re going for the very early UK deadlines (see Deadlines below), this means that you should probably get started on your research statement by early July the latest, and preferably contact letter writers by that time, if not earlier.

A word on MathJobs.

Most US applications are facilitated by the MathJobs. This website run by the AMS contains a list of all academic jobs in mathematics (from postdoc to full-time faculty, at both research universities and more teaching-oriented colleges) of most US universities and institutes, as well as a few in other countries. You can also apply to a few non-academic jobs through MathJobs.

If you’re interested in positions outside the US, be aware that the procedures and requirements are often very different, which can be difficult to navigate. Few European postdoc positions are advertised on MathJobs, and the ones that are will probably receive significantly more applications than the ones that aren’t.

You can make an account when you start preparing your material, but not all jobs will be advertised yet by the beginning of September. Therefore, it’s also ok to wait until your first deadline gets a bit closer. It’s probably smart to have an account at least about a month before your first deadline, so that you have ample time to familiarise yourself with the website’s functionality, as well as to leave your references some time to upload their letters once you requested this.

You only have to submit most of your material once (e.g. one research statement, one teaching statement, one CV, one publication list, etc), with the exception of the cover letters that should be made individually for each application. Similarly, your references will only need to upload a letter once, and you can then choose which ones you want to include with each application.

Once your files are in the system, applying to a university is just a few clicks plus the preparation of a cover letter. If you’ve done a few, eventually this will take no more than 15-20 minutes for most universities (unless you want to prepare special materials for some places). In particular, once your material is ready and your references have uploaded their letters, you can apply to all universities you’re interested in.

You should apply to universities as early as possible, and not wait for the deadline to get close. This way, they can start reviewing your file, and be prepared when the official process starts. You can always update your material later; all positions you applied for will then see the new version only.

Deadlines.

There are a lot of different choices you can make when you’re applying, and for most people not everything below will be relevant. Here are some key deadlines:

Late August: deadlines for some Junior Research Fellow positions in the UK (especially Oxbridge colleges and some universities in London). These are positions for 3-4 years, often with no mandatory teaching requirement (you get paid for additional teaching). This is the earliest deadline that I’m aware of.
Mid October: NSF deadline. This only applies to US students, but the NSF’s schedule dictates the overall timeline and some other aspects of the application process (see Offers below).
September-early November: first university deadlines. There will be a few universities with early deadlines; e.g. Stony Brook used to be a bit earlier than the others. Specific deadlines for some of these universities can vary greatly from year to year, so I can’t make more precise statements.
Mid November-early December: most top university deadlines. The vast majority of strong US universities, as well as a lot of other universities, have application deadlines between 15 November and 1 December. More universities follow gradually, but it seems to taper off after January 1.
There will be more deadlines until as late as May, possibly even later. Many European institutes have later deadlines than their North-American counterparts, although it seems to become more common for European institutions to try to compete with American institutions for the top candidates.

There are also some special fellowships that you can only be nominated for. The most famous one is obviously the Clay Fellowship, but there are many others, like the Miller Fellowship. Some people ask their advisor to nominate them; it is not clear to me whether this is socially accepted, nor whether this is what the system was designed for. I did not do this myself, but it definitely happens. If you want to go down this route, you should familiarise yourself with the relevant deadlines.

Offers.

The information below is largely based on the following answer on StackExchange. I strongly encourage you to read that post as well, because it contains more specific advice on what to do if certain scenarios do or do not happen.

Everything is coordinated by the AMS common deadline (late January) that is agreed by most US universities. This in turn is linked to the NSF announcement date: usually universities will not force you to accept an offer before a week or so after the NSF’s are announced.

Late December-early January: first offers. It seems to be the case that only the top 10 or so US universities (if even that) give out offers before January 7.
Mid January: a lot of universities will give their first offers in this period, and few top people will be making choices. The NSF’s are typically announced mid-late January as well, and everything is based on this.
Late January: first decisions. Most first-round offers will be accepted or turned down now. This means that there will be a lot of activity, because universities want to try to get their top candidate before they accept an offer somewhere else. If you didn’t have an offer before, your chances are now increasing dramatically, because top applicants will be turning down offers if they have multiple.
Early February-mid April: a lot more offers. There are still a lot of offers given out after the AMS common deadline. A big difference, however, is that these later offers typically give you less time to reply (sometimes only a few days). You might be able to stretch this a bit if you have other offers pending, but everything is moving quite fast now.
15 April: everything finalised. Just like with PhD applications, most universities will finalise their postdoc hiring by tax day (but usually much earlier).

There are often possibilities of deferring one offer for at most one year (although the NSF does not have this option), but once you accept an offer it’s considered bad form to back out again. Thus, even if you get a better offer three days after you accepted (e.g. because of a deadline), you should still go to the place whose offer you accepted. (There might be options to defer the other offer and quit your first position after a year; I don’t know what the acceptable practises for this are.)

Visas.

Please note that the following should not be considered legal advise, and you should consult your current or future institution, or an immigration lawyer, with any visa questions.

If you are not a US national, there are a few options for postdoc visas (depending on the hosting institution). Most people will apply to one of the following:

F-1 OPT (Optional Practical Training): if you have been an F-1 student in the US during your PhD, you may apply for Optional Practical Training for your postdoc. This is for 12 months plus a 24 month STEM extension, but you have to deduct any time you held a pre-completion OPT.
J-1 Exchange visitor: this is the typical visa you would get if you did not have an F-1 prior to starting your postdoc. You could also apply for this if you did have an F-1, for instance if you want to bring a spouse or dependent on a J-2.
H-1B specialty occupation: these are harder to get, and most universities will not offer these for postdocs. These are for three years (extendable to six), after which you have to apply for a Green Card if you want to stay.

If you are planning to stay in the US on your F-1 post-completion OPT, you should be aware of the rather complicated timeline. The following four statements turn out to be difficult to combine:

You cannot apply for post-completion OPT more than 90 days before the end of your programme.
It takes at least 3-4 months to process your application.
After applying for OPT, you cannot enter the US unless you have your Employment Authorisation Document (EAD) in hand. It is strongly recommended you stay in the US until it arrives.
The US Citizenship and Immigration Services do not send EADs to foreign addresses.

It is probably best to have a visa strategy ready by late February, so that you are ready to start the process by early March (assuming your programme ends late May or early June).

There are a lot of people applying around the same time, so the waiting time will increase dramatically if you wait even a few weeks. You should apply as early as you can.

Finiteness is not a local property

Posted on 27 January 2018 by Remy

In this post, we consider the following question:

Question. Let $A$ be a Noetherian ring, and $M$ and $A$ -module. If $M_\mathfrak p$ is a finite $A_\mathfrak p$ -module for all primes $\mathfrak p \subseteq A$ , is $M$ finite?

That is, is finiteness a local property?

For the statement where local means the property is true on a cover by Zariski opens, see Tag 01XZ. Some properties (e.g. flatness) can also be checked at the level of local rings; however, we show that this is not true for finiteness.

Example 1. Let $A = \mathbb Z$ , and let $M = \bigoplus_{p \text{ prime}} \mathbb Z/p\mathbb Z$ . Then $M_{(p)} = \mathbb Z/p\mathbb Z$ , because localisation commutes with direct sums and $(\mathbb Z/q\mathbb Z)_{(p)} = 0$ if $q \neq p$ is prime. Thus, $M_{(p)}$ is finitely generated for all primes $p$ . Finally, $M_{(0)} = 0$ , because $M$ is torsion. But $M$ is obviously not finitely generated.

Example 2. Again, let $A = \mathbb Z$ , and let $M \subseteq \mathbb Q$ be the subgroup of fractions $\frac{a}{b}$ with $\gcd(a,b) = 1$ such that $b$ is squarefree. This is a subgroup because $\frac{a}{b} + \frac{c}{d}$ can be written with denominator $\lcm(b,d)$ , and that number is squarefree if $b$ and $d$ are. Clearly $M$ is not finitely generated, because the denominators can be arbitrarily large. But $M_{(0)} = \mathbb Q$ , which is finitely generated over $\mathbb Q$ . If $p$ is a prime, then $M_{(p)} \subseteq \mathbb Z_{(p)}$ is the submodule $\frac{1}{p}\mathbb Z_{(p)}$ , which is finitely generated over $\mathbb Z_{(p)}$ .

Another way to write $M$ is $\sum_{p \text{ prime}} \frac{1}{p}\mathbb Z \subseteq \mathbb Q$ .

Remark. The second example shows that over a PID, the property that $M$ is free of rank $1$ can not be checked at the stalks. Of course it can be if $M$ is finitely generated, for then $M$ is finite projective [Tag 00NX] of rank $1$ , hence free since $A$ is a PID.

Higher pushforwards along finite morphisms

Posted on 12 January 2018 by Remy

This post is about one of my favourite answers I have given on MathOverflow, although it seems to have gone by mostly unnoticed. In the post, Qixiao asks (essentially) the following:

Question. If $f \colon X \to Y$ is a finite morphism of schemes, is the pushforward $f_* \colon \Sh(X) \to \Sh(Y)$ exact?

Note that this is true on the subcategory of quasicoherent sheaves because affine morphisms have no quasicoherent higher pushforwards. Also, in the étale topology the pushforward along a finite morphism is exact on the category of all abelian sheaves; see e.g. Tag 03QP.

However, we show that the answer to the question above is negative.

Example. Let $Y$ be the spectrum of a DVR $(R,\mathfrak m)$ , let $R \to S$ be a finite extension of domains such that $S$ has exactly two primes $\mathfrak p, \mathfrak q$ above $\mathfrak m$ , and let $X = \Spec S$ . For example, $R = \Z_{(5)}$ and $S = \Z_{(5)}[i]$ , or $R = k[x]_{(x)}$ and $S = k[x]_{(x)}[\sqrt{x+1}]$ if you prefer a more geometric example.

By my previous post, the global sections functor $\Gamma \colon \Sh(Y) \to \Ab$ is exact. If the same were true for $f_* \colon \Sh(X) \to \Sh(Y)$ , then the global sections functor on $X$ would be exact as well. Thus, it suffices to prove that this is not the case, i.e. to produce a surjection $\mathscr F \to \mathscr G$ of sheaves on $X$ such that the map on global sections is not surjective.

The topological space of $X$ consists of closed points $x,y$ and a generic point $\eta$ . Let $U = \{\eta\}$ and $Z = U^{\operatorname{c}} = \{x,y\}$ ; then $U$ is open and $Z$ is closed. Hence, for any sheaf $\mathscr F$ on $X$ , we have a short exact sequence (see e.g. Tag 02UT)

$0 \to j_! (\mathscr F|_U) \to \mathscr F \to i_* (\mathscr F|_Z) \to 0,$

where $j \colon U \to X$ and $i \colon Z \to X$ are the inclusions. Let $\mathscr F$ be the constant sheaf $\Z$ ; then the same goes for $\mathscr F|_U$ and $\mathscr F|_Z$ . Then the map

$H^0(X,\mathscr F) \to H^0(X,i_*(\mathscr F|_Z)) = H^0(Z,\mathscr F|_Z)$

is given by the diagonal map $\Z \to \Z \oplus \Z$ , since $X$ is connected by $Z$ has two connected components. This is visibly not surjective. $\qedsymbol$

Cohomology of a local scheme

Posted on 6 January 2018 by Remy

This is a continuation of my previous post on local schemes. Here is a ridiculous lemma.

Lemma. Let $(X,x)$ be a local scheme, and let $\mathscr F$ be any abelian sheaf on $X$ . Then $H^i(X,\mathscr F) = 0$ for all $i > 0$ .

Proof. It suffices to show that the global sections functor $\Gamma \colon \Sh(X) \to \Ab$ is exact. Let $\mathscr F \to \mathscr G$ be a surjection of abelian sheaves on $X$ , and let $s \in \mathscr G(X)$ be a global section. Then $s$ can be lifted to a section of $\mathscr F$ in an open neighbourhood $U$ of $x$ . But the only open neighbourhood of $x$ is $X$ . Thus, $s$ can be lifted to a section of $\mathscr F(X)$ . $\qedsymbol$

What’s going on is that the functors $\mathscr F \mapsto \Gamma(X,\mathscr F)$ and $\mathscr F \mapsto \mathscr F_x$ are naturally isomorphic, due to the absence of open neighbourhoods of $x$ .

Remark. It seems believable that there are suitable site-theoretic versions of this lemma as well. For example, a strictly Henselian local ring has no higher cohomology in the étale topology. The argument is essentially the same: every open neighbourhood of the closed point has a section; see e.g. the proof of Tag 03QO.

Lovely little lemmas

Or maybe ‘amusing observations’

Author Archives: Remy