My qualifying exam

I took (and passed!) my oral qualifying exam in May 2022. I write this note to finish processing it, and to remember it. I hope that sharing my experience will be useful to other grad students (specially in arithmetic geometry) preparing for their own quals (''clears throat'' Alexis, Roberto, and Jazz).

Of course, all errors (mathematical and literary) are my fault. 
Please email me with your suggestions and corrections.

The exam

At least for DZB students, the drill is to agree on a syllabus covering the core material of algebraic geometry, and take a ~1.5 hour oral exam with a committee of professors of your choosing. My examiners where David Zureick-Brown (DZB) and Brooke Ullery (BU).

It seems to me that the point of the exam is (1) studying for it, and (2) making sure you have a solid understanding of the big picture.

  • My qual was scheduled originally scheduled for June 19. On one of my meetings with DZB, approximately 2 months before the scheduled date of my qual, I mentioned that I was thinking about pushing the date up a month so that I wouldn't be stressing about studying on a family trip I had in late May. He replied: ''Do it! You can tell me something about all the topics on the syllabus, right?''
  • That story is a classic example of DZB's kind overestimation of the math I know, but the point is that you are not expected to know everything perfectly. For example, it would be highly uniquely if they asked you to prove the Valuative Criterion of Separatedness in full detail. But, you should be able to explain the criterion, its geometric interpretation, and know how to use it!

My syllabus

I wanted my syllabus to contain all the topics of my advisor's qualifying exam syllabus. This coincides with what DZB considers to be the core material of algebraic geometry and algebraic number theory.

My questions

Algebraic Geometry

  1. (DZB) Show that a morphism of sheaves (of sets) is surjective if and only if the induced morphism of stalks is surjective for every point.
    • DZB didn't specify he was asking about sheaves of sets, so I tried to do it for sheaves of abelian groups (where surjective means the cokernel sheaf is zero). He later asked me to do it for sets, so it took me a solid minute to remember what surjective meant from Vakil's book.
  2. (DZB) Can you give me an example of a surjective morphism of sheaves that is not surjective on open subsets.
    • I talked about the sheaf of holomorphic functions on the complex plane, and the exponential morphism of sheaves. (See Example 2.4.10 in FOAG).
  3. (BU) Let \(\, f\colon X \to Y\) be a morphism of schemes. What are the induced maps on stalks?
    • I defined stalks and morphisms of schemes. Then I explained how the universal property of limits yields to the induced maps on stalks.
  4. (BU) How are these maps related to localization?
    • I talked about morphisms between affine schemes, and how the induced map on stalks is the induced map of localization at a prime ideal.
  5. (DZB) This is a trick followup question. Why does the induced morphism of stalks take the maximal ideal to the maximal ideal?
    • This is by definition, since morphisms of schemes are morphisms of locally ringed spaces.
  6. (DZB) Let \(X\) be a scheme satisfying condition (\(*\)). What are Weil divisors on \(X\), and what is the divisor class group?
    • I explained what condition (\(*\)) was and why we wanted each adjective to talk about Weil divisors. I defined prime divisors, Weil divisors, and locally principal divisors. Then I realized I had to talk about the restriction map \(\mathrm{Div}\, X \to \mathrm{Div}\, U\) for every open \(U \subset X\) and principal divisors first, so I defined those. I explained how all of these groups fit in together with a nice diagram (see entry 14.2.7 in FOAG) and defined the class group.
  7. (DZB) What is the class group of projective space?
    • I verbally explained why \(\mathrm{Cl}\, \mathbf{P}^n\) is free of rank one, generated by the class of any hyperplane.
  8. (BU) What is a finite morphism?
  9. (BU) Can you give an example of a morphism of schemes with finite fibers that is not finite?
    • I'm pretty sure this question came up in the scheme theory course tought by Brooke Ullery, and I tought of an example then. So this was a gracious gift from Brooke that I managed to waste! I flipped the question in my head, so I was trying to find a finite map with infinite fibers. This, of course, cannot happen (see Important Exercise 7.3.K in FOAG). I got flustered and my brain stopped working from embarrassment. After admitting defeat, DZB and Brooke kindly mentioned some examples. Here are two:
      • The open embedding \(\mathbf{A}^2_k - \{(0,0)\} \to \mathbf{A}^2\) has finite fibers but is not affine, since the punctured plane is not affine.
      • The map on spectra corresponding to an infinite field extension is affine and has finite fibers, but is not finite!
    • We got sidetracked and DZB seized the chance to mention a cool example of an affine morphism with infinite fibers. Let \(\overline{\mathbf{Q}} \supset \mathbf{Q}\) be an algebraic closure. The morphism \(\mathrm{Spec} \, \overline{\mathbf{Q}}\otimes_{\mathbf{Q}}\overline{\mathbf{Q}} \to \mathrm{Spec} \, \overline{\mathbf{Q}}\), obtained by pulling back the extension map \(\mathrm{Spec} \, \overline{\mathbf{Q}} \to \mathrm{Spec} \, \mathbf{Q}\) over itself, is affine but has infinite fibers! In fact, one can show that \(\mathrm{Spec} \, \overline{\mathbf{Q}}\otimes_{\mathbf{Q}}\overline{\mathbf{Q}} \cong \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\).
  10. (DZB) For the last question, you can choose between: (1) Computing a Cech cohomology example. (2) Explaining what the connecting δ-morphism is.
    • I chose to compute the cohomology of \(\Omega_{\mathbf{P}^1/k}\) since I had that one ready to fire.
  11. (DZB) How could we have known ahead of time what \(H^1\) was?
    • By Serre-duality, \(H^1(\Omega_{\mathbf{P}^1})\) is isomorphic to \(H^0(\mathcal{O}^\vee) \cong H^0(\mathcal{O}) \cong k\).

Number Theory

DZB asked all the number theory questions.

  1. What is the group structure of \(\mathbf{Q}_p^\times\)?
    • I solved this via limits instead of logarithms. I explained why \(\mathbf{Q}_p\) is isomorphic to \(\mathbf{Z}\oplus\mathbf{Z}_p\), so that it is enough to calculate the structure of the p-adic units. From the isomorphisms \((\mathbf{Z}/2^{n+1}\mathbf{Z})^\times \cong \mathbf{Z}/2\mathbf{Z} \oplus \mathbf{Z}/2^{n-1} \mathbf{Z}\) and \((\mathbf{Z}/p^{n+1}\mathbf{Z})^\times \cong \mathbf{Z}/(p-1)\mathbf{Z} \oplus \mathbf{Z}/p^n \mathbf{Z}\) for \(p > 2\) we get that \(\mathbf{Z}_2^\times \cong \mathbf{Z}/2\mathbf{Z}\oplus \mathbf{Z}_2\) and \(\mathbf{Z}_p^\times \cong \mathbf{Z}/(p-1)\mathbf{Z} \oplus \mathbf{Z}_p\) after taking limits.
  2. Explain the decomposition of primes in Galois extensions of number fields.
    • I wrote down a summary of splitting of primes in Dedekind extensions, mentioned the fundamental identity, and explained how things simplify in the case of galois extensions. I talked about decomposition groups, inertia groups, and Frobeinus, and wrote down the corresponding exact sequence.
  3. How do primes ramify in the decomposition and inertia subfields?
    • I wrote down the corresponding diagram of fixed fields and after a minute of pondering I remembered how this worked.
  4. State the Chebotarev density theorem.
    • I had all the necessary setup on the board already, so this was quick!
  5. Prove that there is always a prime that splits completely.
    • I used characterization of the primes that split as a Chebotarev set and the Chebotarev density theorem to show that there are in fact infinitely many primes that split completely, and that they have natural density \(1/d\) where \(d = [L:K]\) is the degree of the extension.
  6. In the last 8 minutes, explain the main statements of class field theory.
    • I summarized Bjorn Poonen's brief summary of the statements of CFT. I was racing to do this in 8 minutes, so the last minute was a bit messy.
    • DZB pointed out that you can quickly define Adeles over \(\mathbf{Q}\) as the tensor product \(\mathbf{A}_{\mathbf{Q}} = (\mathbf{R}\times \hat{\mathbf{Z}})\otimes \mathbf{Q}\), and then for a number field \(K/\mathbf{Q}\) we have \(\mathbf{A}_K = \mathbf{A}_\mathbf{Q}\otimes_\mathbf{Q} K\).

Some advice

Everybody's approach to learning is different. Mine is to write things down until they make sense, and then working on lots of problems.

  1. Have your syllabus ready one year before, make a study plan, and stick to it (this is the hard part).
  2. Balance theory with practice. Make sure you understand the big picture, but get your hands dirty with calculations along the way.
    • For example, you should be able to explain what fiber products are, the universal property, and how to construct them, but you should also be able to calculate the fibers of the map \(\mathrm{Spec}\, \mathbf{C}[x,y,t]/(xy-t) \to \mathrm{Spec}\, \mathbf{C}[t]\).
  3. Keep permanent records of your solutions.
    • I kept a scan of every problem I solved. 10/10 would do again; can't recommend enough.
  4. Have your friends ask you questions to practice thinking LIVE!!!
    • This was the hardest part for me, since my qual was my first oral exam ever. Thinking on the spot is a skill that can be learned.
    • A good tip for this is to write down the questions on the board; this gives you some extra time to think. If you don't know how to solve a problem right away, you can start writing down the definitions and theorems you suspect might be relevant for the solution.
  5. Create a list of definitions, theorems, and (non-)examples.
    • Chris Keyes created a list of definitions and theorems. I went crazy and TeXed an extended summary of everything. It worked for me, but in retrospect, I think it would have been better to type a minimalistic list of definitions, theorems, and (non-)examples.
    • One of the most common type of questions (and in my opinion the hardest to solve on the spot) are to give examples and counter-examples.
    • Memorization of the important definitions is practical but sometimes detached from reality. Intuition comes from understanding a handful of crucial examples. This is worth spending a lot of time on.
    • You can make flash-cards and use CTRL-f on your document to check your anwers.
  6. Check out Anya Michaelsen's qual materials and advice, from which I emphasize:
    • You will not know everything.
    • Not failing = Passing.
    • Feel free to make simplifying assumptions.

Author: Santiago Arango-Piñeros


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