Not every open immersion is an open immersion

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

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 is typically 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 3 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).

Finiteness is not a local property

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

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

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.

Local schemes

Consider the following definition. It seems to be standard, although I have not found a place where it is actually spelled out in this way.

Definition. A pointed scheme (X,x) is local if x is contained in every nonempty closed subset of X.

Example. If (A,\mathfrak m) is a local ring, then (\Spec A,\mathfrak m) is a local scheme. Indeed, \mathfrak m is contained in every nonempty closed subset V(I) \subseteq X, because every strict ideal I \subsetneq A is contained in \mathfrak m.

We prove that this is actually the only example.

Lemma. Let (X,x) be a local scheme. Then X is affine, and A = \Gamma(X,\mathcal O_X) is a local ring whose maximal ideal corresponds to the point x \in X = \Spec A.

Proof. Let U be an affine open neighbourhood of x. Then the complement V is a closed set not containing x, hence V = \varnothing. Thus, X = U is affine. Let A = \Gamma(X,\mathcal O_X). Let \mathfrak m be a maximal ideal of A; then V(\mathfrak m) = \{\mathfrak m\}. Since this contains x, we must have x = \mathfrak m, i.e. x corresponds to the (necessarily unique) maximal ideal \mathfrak m \subseteq A. \qedsymbol

Classification of compact objects in Top

In my previous post, I showed that compact objects in the category of topological spaces have to be finite. Today we improve this to a full characterisation.

Lemma. Let X be a topological space. Then X is a compact object in \operatorname{\underline{Top}} if and only if X is finite discrete.

This result dates back to Gabriel and Ulmer [GU71, 6.4], as was pointed out to me by Jiří Rosický in reply to my MO question and answer of this account (of which this post is essentially a retelling). Our proof is different from the one given in [GU71], instead using a variant of an argument given in the n-Lab.

Before giving the proof, we construct an auxiliary space against which we will be testing compactness. It is essentially the colimit constructed in the n-Lab, except that we swapped the roles of 0 and 1 (the reason for this will become clear in the proof).

Definition. For all n \in \mathbb N, let X_n be the topological space \mathbb N_{\geq n} \times \{0,1\}, where the nonempty open sets are given by U_{n,m} = \mathbb N_{\geq m} \times \{0\} \cup \mathbb N_{\geq n} \times \{1\} for m \geq n. They form a topology since

    \begin{align*} U_{n,m_1} \cap U_{n,m_2} &= U_{n, \max(m_1,m_2)}, \\ \bigcup_i U_{n,m_i} &= U_{n,\min\{m_i\}}. \end{align*}

Define the map f_n \colon X_n \to X_{n+1} by

    \[(x,\varepsilon) \mapsto \left\{\begin{array}{ll} (x,\varepsilon), & x > n, \\ (n+1,\varepsilon), & x = n. \end{array}\right.\]

This is continuous since f_n^{-1}(U_{n+1,m}) equals U_{n,m} if m > n+1 and U_{n,n} if m = n+1. Let X_\infty be the colimit of this diagram.

Since the elements (x,\varepsilon), (y,\varepsilon) \in X_n map to the same element in X_{\max(x,y)}, we conclude that X_\infty is the two-point space \{0,1\}, where the map X_n \to X_\infty = \{0,1\} is the second coordinate projection. Moreover, the colimit topology on \{0,1\} is the indiscrete topology. Indeed, neither \mathbb N_{\geq n} \times \{0\} \subseteq X_n nor \mathbb N_{\geq n} \times \{1\} \subseteq X_n are open.

Proof of Lemma. If X is compact, then my previous post shows that X is finite. Let U \subseteq X be any subset, and let f \colon X \to X_\infty = \{0,1\} be the indicator function \mathbb I_U. It is continuous because X_\infty has the indiscrete topology. Since X is a compact object, f has to factor through some g \colon X \to X_n. Let h \colon X \to X_n \to \N_{\geq n} be the first coordinate projection, i.e.

    \[g(x) = \left\{\begin{array}{ll}(h(x),1), & x \in U, \\ (h(x),0), & x \not\in U. \end{array}\right.\]

Let m \in \N_{\geq n} be a number such that m > h(x) for all x \not\in U; this exists because X is finite. Then g^{-1}(U_{n,m}) = U, which shows that U is open. Since U was arbitrary, we conclude that X is discrete.

Conversely, every finite discrete space X is a compact object. Indeed, any map out of X is continuous, and finite sets are compact in \operatorname{\underline{Set}}. \qedsymbol

[GU71] Gabriel, Peter and Ulmer, Friedrich, Lokal präsentierbare Kategorien. Lecture Notes in Mathematics 221. Springer-Verlag, Berlin-New York, 1971. DOI: 10.1007/BFb0059396.

Compact objects in the category of topological spaces

We compare two competing notions of compactness for topological spaces. Besides the usual notion, there is the following:

Definition. Let \mathscr C be a cocomplete category. Then an object X \in \ob \mathscr C is compact if \Hom(X,-) commutes with filtered colimits.

Exercise. An R-module M is compact if and only if it is finitely generated.

We want to study compact objects in the category of topological spaces. One would hope that this corresponds to compact topological spaces. However, this is very far off:

Lemma. Let X \in \Top be a compact object. Then X is finite.

Proof. Let Y be the set X with the indiscrete topology, i.e. \mathcal T_Y = \{\varnothing, Y\}. It is the union of all its finite subsets, and this gives it the colimit topology because a subset U \subseteq Y is open if and only if its intersection with each finite subset is. Indeed, if U were neither \varnothing nor Y, then there exist y_1, y_2 \in Y with y_1 \in U and y_2 \not\in U. But then U \cap \{y_1,y_2\} is not open, because \{y_1,y_2\} inherits the indiscrete topology from Y.

Therefore, if X is a compact object, then the identity map X \to Y factors through one of these finite subsets, hence X is finite. \qedsymbol

However, the converse is not true. In fact the indiscrete space on a two element set is not a compact object, as is explained here.

Corollary. Let X \in \Top be a compact object. Then X is a compact topological space.

Proof. It is finite by the lemma above. Every finite topological space is compact. \qedsymbol

Originally, this post relied on the universal open covering of my previous post to show that a compact object in \Top is compact; however the above proof shows something much stronger.

A fun example of a representable functor

This post is about representable functors:

Definition. Let F \colon \mathscr C \to \Set be a functor. Then F is representable if it is isomorphic to \Hom(A,-) for some A \in \ob \mathscr C. In this case, we say that A represents F.

Exercise. If such A exists, then it is unique up to unique isomorphism.

Really one should encode the isomorphism \Hom(A,-) \stackrel\sim\to F as well, but this is often dropped from the notation. By the Yoneda lemma, every natural transformation \Hom(A,-) \to F is uniquely determined by the element of F(A) corresponding to the identity of A.

When \Hom(A,-) \to F is a natural isomorphism, the corresponding element a \in F(A) is called the universal object of F. It has the property that for every B \in \mathscr C and any b \in F(B), there exists a unique morphism f \colon A \to B such that f(a) = b.

Example. The forgetful functor \Ab \to \Set is represented by \Z. Indeed, the natural map

    \begin{align*} \Hom(\Z,M) &\to M\\ f &\mapsto f(1) \end{align*}

is an isomorphism. The universal element is 1 \in \Z.

Example. Similarly, the forgetful functor \Ring \to \Set is represented by \Z[x]. The universal element is x.

A fun exercise (for the rest of your life!) is to see whether functors you encounter in your work are representable. See for example this post about some more geometric examples.

The main example for today is the following:

Lemma. The functor \Top\op \to \Set that associates to a topological space (X,\mathcal T_X) its topology \mathcal T_X is representable.

Proof. Consider the topological space Y = \{0,1\} with topology \{\varnothing, \{1\},\{0,1\}\}. Then there is a natural map

    \begin{align*} \Hom(X,Y) &\to \mathcal T_X\\ f &\mapsto f^{-1}(\{1\}). \end{align*}

Conversely, given an open set U, we can associate the characteristic function \mathbb I_U. This gives an inverse of the map above. \qedsymbol

The space Y we constructed is called the Sierpiński space. The universal open set is \{1\}.

Remark. The space Y^I represents the data of open sets U_i for i \in I: for any continuous map f \colon X \to Y^I, we have U_i = f^{-1}(Y_i), where Y_i = \pi_i^{-1}(\{1\}) \subseteq Y^I. If Z_i denotes the complementary open, then the U_i form a cover of X if and only if \bigcap_{i \in I} Z_i = \varnothing. This corresponds to the statement that f lands in Y^I\setminus\{(0,0,\ldots)\}.

Thus, the open cover Y^I\setminus\{0\} = \bigcup_{i \in I} Y_i is the universal open cover, i.e. for every open covering X = \bigcup U_i there exists a unique continuous map f \colon X \to Y^I\setminus\{0\} such that U_i = f^{-1}(Y_i).

A weird Dedekind domain

Here is a random question that I was wondering about at some point (just out of curiosity):

Question. Does there exist a Dedekind domain R that has infinitely many points whose residue field has characteristic 0 and finitely many points (at least one) whose residue field has positive characteristic?

I don’t fully recall why this question came up, but it had something to do with a similar property that was satisfied by an object involved in the definition of the Fargues–Fontaine curve. However, we don’t need such deep theory to discuss this elementary commutative algebra question.

Lemma. Such a Dedekind domain exists.

Proof. We will construct R as a localisation of \mathbb Z[x]. Recall that prime ideals of \mathbb Z[x] come in four types:

  • The generic point (0), of height 0;
  • Height 1 primes (p) for every prime p \in \mathbb Z;
  • Height 1 primes (f) for every irreducible polynomial f \in \mathbb Z[x];
  • Height 2 closed points (p,f) for p \in \mathbb Z a prime and f \in \mathbb Z[x] a polynomial whose reduction \bar f \in \mathbb F_p[x] is irreducible.

We first localise at (p,x); then the only primes we have left are the ones contained in (p,x). This is the generic point, the height 1 prime (p), the height 1 primes (f) where f \in \mathbb Z[x] is an irreducible polynomial whose constant coefficient is divisible by p (e.g. (x), (x-p), \ldots), and exactly one height 2 prime (p,x).

Next, we invert x; denoting the resulting ring by R. This gets rid of all prime ideals containing x, which are (p,x) and (x). In particular, there are no more height 2 primes, so R is 1-dimensional. It is a normal Noetherian domain because it a localisation of a normal Noetherian domain. Therefore, R is a Dedekind domain.

The primes of R are (0) with residue field \mathbb Q(x); the prime (p) with residue field \mathbb F_p(x); and the prime ideals (f) with f \neq x a polynomial whose constant coefficient is divisible by p, whose residue field is a finite extension of \mathbb Q. \qedsymbol

Remark. The ring R we constructed is essentially of finite type over \mathbb Z (a localisation of a finite type \mathbb Z-algebra). There are no examples of finite type over \mathbb Z, because by Chevalley’s theorem the image of \operatorname{Spec} R \to \operatorname{Spec} \mathbb Z would be constructible. However, no set of the form \{(0)\} \cup S for S \subseteq \operatorname{Spec} \mathbb Z finite is constructible. (Alternatively, the weak Nullstellensatz implies that every closed point of a finite type \mathbb Z-algebra has residue characteristic p > 0.)

I was a little surprised that we can make examples when we drop the finite type assumption. I don’t know if this type of ring has ever been used for anything.