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Question

Suppose you grasped and enjoyed reading Quillen's "Higher Algebraic K-theory I". Now, if you could go back in time to when you started studying algebraic topology and create a reading list / roadmap with the above paper as a goal, what would this plan look like?

Here's another version of the question: suppose you're the PhD advisor of a student that's recently finished their undergrad degree and in terms of books background has only read and completed most of the exercises in:

  • algebraic geometry: all of Q. Liu's book
  • algebraic topology: first two chapters of W. Massey's intro book
  • differential geometry: first half of J. Lee's smooth manifolds book

Which books and papers (and in what order) should this student master in order to understand (or at least appreciate) most of Quillen 1?

Outlook

The above questions are likely quite vague wrt to "how do I learn modern algebraic k theory?", but hopefully they're somewhat concrete by stating i) the goal [Quillen 1] and ii) the starting maths background.

If it helps, assume a secondary goal is to eventually focus on studying/appreciating arithmetic problems like Parshin's conjecture.

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    $\begingroup$ What do you mean with "has never studied algebraic topology proper"? Has he seen the fundamental group? Homology groups (even deRham)? Some differential topology? $\endgroup$
    – Denis Nardin
    Commented May 23, 2020 at 9:37
  • $\begingroup$ @DenisNardin: great question -- I've attempted to clarify some of the assumed book-level background for the hypothetical new grad student $\endgroup$
    – Quetzalcoatl
    Commented May 23, 2020 at 18:16
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    $\begingroup$ It's not quite what you're asking but Weibel's K-book is quite nice. The first several chapters are from a more historical perspective so you go through elementary matrices, the whitehead group, K_0, K_1, K_2, Milnor K-theory, and then you start going through various constructions of the algebraic K-theory spectrum. $\endgroup$
    – Joe Berner
    Commented May 23, 2020 at 18:41
  • $\begingroup$ On a minimal level, I'd say that reading Kan's On c.s.s. complexes is enough. Of course the real answer is that to learn K-theory it would be a very good idea to learn (the basis of) modern homotopy theory first. $\endgroup$
    – Denis Nardin
    Commented May 24, 2020 at 10:35

1 Answer 1

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For someone who is comfortable with Algebraic Geometry (on the level of Liu's book) but less comfortable with Topology, I'd recommend simply using Srinivas book on Algebraic K-theory.

It's a textbook and it first introduces K-theory "axiomatically", then presents a number of applications in algebraic geometry, and only then starts with the proofs which really need some background in topology. Also, the book pretty much covers a great deal in Quillen I, so if the student works through that book, it could act as a self-contained replacement of (most of) Quillen I.

So, if the student reads Srinivas and Quillen I simultaneously, that should make it waaaay easier.

As an aside: I don't think any Differential Geometry/Differential Toplogy is needed for any of this whatsoever. I think when people use Serre-Swan stuff or Bott periodicity to motivate why one might find Algebraic K-theory cool, they usually do this to address an audience that likes these things already. So for somebody who doesn't know these things in the first place, it's not needed to teach them that first. Rather, try to get them curious by relying on their background. For an Algebraic Geometry student, this might be a question like: How does the localization sequence for Chow groups (or Pic, for somebody on the level of Liu) continue to the left? (leading ultimately to Bloch's higher Chow groups and all that stuff)

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