Set, class, family?

Set, family, and class; what are they exactly?

A set is a well-defined collection of distinct objects. But not every collection is a set; we will demonstrate an example. In set theory, the comprehension principle states that for a predicate P(x) formed by first-order logic, there exists a set including all objects that satisfy P(x). But an unrestricted form of this principle leads to what’s called Russell’s paradox. We see that letting P(x): x\notin x, where x is a set, and let R=\{x:P(x)\}, the principle asserts that R exists. It can’t be that R\in R, so R\notin R, which implies that P(R) is true and R\in R. But this leads to a contradiction. But in ZFC, which is the canonical axiom system, an Axiom of Separation was imposed to avoid Russell’s paradox. This axiom essentially restricts the comprehension principle, by that using the set construction method, one can not build a set bigger than the previously constructed sets.

Practically, a collection such as R above, which is not a set, is an example of a proper class, while sets are exactly objects that are known as small classes. In category theory, a category whose collection of objects forms a proper class is called a large category. Lastly, what is a family?

A family of sets is a collection of sets (with no restrictions). That is, a collection with repeated sets or a proper class of sets pass as a family of sets.


Coherent Sheaves et al.

I am tired of reading Harder’s chapter on sheaf theory. My goal is to learn coherent sheaves. I got stuck at the section where he gives the conditions a presheaf can be a sheaf. He wants to construct a morphism between two products, \prod_{\alpha\in A}\mathcal{F}(U) and \prod_{(\alpha,\beta)\in A\times A} \mathcal{F}(U_{\alpha}\cap U_{\beta}). He does this using a fact he proved earlier; one can construct such a morphism from a map of the set of indices that goes in the opposite direction, I\rightarrow J in this case, if there is also a morphism that relates \mathcal{F}(U) and \mathcal{F}(U_{\alpha}\cap U_{\beta}). I actually got stuck here, and not on the sheaf definition even though I still don’t understand the double arrows. I understand the earlier construction up to (and including) this commutative diagram: screenshot-from-2016-12-24-20-23-05But I don’t get how the existence of a map \prod_{\i\in I}X_i\rightarrow X_{\tau(j)}\rightarrow Y_j finishes the construction of f_{\tau}. I have a feeling that the inverse arrow from Y_j to the Y_j products finishes the proof, but it is not clear to me how. After this Harder discusses a concrete example of a sheaf–complex manifold etc., which is something I have seen in Neeman’s book. Along the way, actually gives a brief exposition on complex spaces, something that took Neeman a chapter to introduce. I may look into it later, but right now, it is irrelevant to my goal. Finally, before finishing off the chapter by discussing that sheaves with values in category of abelian groups form an abelian category, he talks about sheafification and some functors he calls f_{*}, f^{*}. The following chapter is Cohomology of Sheaves. But I thought the writing was a bit wordy and confusing at times.

I figured, why not actually study Serre’s FAC-Faisceaux Algébriques Cohérents (algebra of coherent beams). My Prof. suggested I read it but then discovered that I can’t read French. Before suggesting the English version, he sent me a list of replacements, I suppose. He was quick to accompany those with the English translation of FAC, which he soon after discovered. Anyway, I have decided to use those books as a back up and actually delve into FAC. I have read that Serre’s writing is particularly concise and clear. This will be my first reading of his work. And the idea doesn’t sound that crazy since my prof actually had recommended it anyway.

On a similar note, I stumbled upon this mathSE question on reading the masters. No surprises; Serre’s GAGA and FAC were mentioned. But apart from those, I was delighted to see that Matt E[merton] gave an answer. I really enjoy his detailed answers. Here are the papers he suggested:

  • Zariski’s paper on Simple points
  • Deligne and Mumford’s paper on Moduli spaces of curves (at least after Hartshorne I, II, and III)
  • Mumford’s book Lectures on Curves on Algebraic Surfaces (“develops a lot of fantastic material and intuitions”)
  • Serre’s GAGA
  • Grothendieck’s paper on Vector bundles on \mathbb{P}^1 (“should be read before…”)
  • Atiyah’s paper on Vector bundles over an elliptic curve
  • Graber, Harris, Starr’s paper Families of rationally connected varieties (proves that “the total space of a family of rationally connected varieties over a rationally connected base is rationally connected.”)

Hodge theory stuff:

  • Clemens and Griffith’s paper The Intermediate Jacobian of the Cubic Threefold (shows that a smooth cubic threefold in \mathbb{P}^4 is not rational)
  • Griffith’s Variations on a theorem of Abel (on Abel-Jacobi theorem in  concrete geometric terms)
  • Deligne’s note Theorie de Hodge I and his paper Theorie de Hodge II (“mark the introduction of Hodge theory into modern alg. geometry as a fundamental tool.”

I may have to learn to read math in French. Definitely worth it, given that EGA, SGA (much of it already translated), Serre’s, and Deligne’s works are in French. Then I don’t also have to luck out on finding an English translation of some paper. Luckily, FAC is available in English.

This turned out to be a long post than I expected. (ノ>ノ). Alas, it will a page I will come back to a lot.