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I am looking for a guidance in $K$-theory. My master thesis was in the field of Algebraic K-theory and its relation and interaction with the field of Algebraic Topology. I mainly had concentrated on the study of the third K-group of an infinite field. I studied the Anderie Suslin paper, which was titled as the "$K_3$ of a field, and the Bloch group". Entering to this field required my hard work on algebraic topology. By now I am a Ph.D. student, and I do not have my master supervisor here. I always see $K$-theory as a giant which has many things in his heart. I am much interested in it, but I do not how should I walk into this realm. I may have a question from the experts: I know some preliminaries, and I want to fulfill my dreams about $K$-theory, what is the main plan? What kind of requirements I need?

I have studied several books in this branch, Hatcher for first, Sirinivas, Atiyah, and many lectures and some papers. I have good background in Homology and Algebraic geometry too.

Your comments and suggestions would be greatly appreciated.

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  • $\begingroup$ For stable homotopy theory, I would also recommend the book ncatlab.org/nlab/show/Introduction+to+Stable+Homotopy+Theory . I also think that Quillen's "Higher Algebraic K-theory I" is a must read classic in K-theory. $\endgroup$ – user40276 Jun 3 '18 at 3:17
  • $\begingroup$ @user40276 Oh absolutely! And I could add also Quillen's On the cohomology and K-theory of the general linear group over a finite field. I was just trying to limit the list to a manageable starting point in my answer, there are so many great papers in the subject... $\endgroup$ – Denis Nardin Jun 3 '18 at 6:35
  • $\begingroup$ @DenisNardin Indeed, there are a lot of classics which probably would include most of Thomason's and Quillen's papers. By the way, my comment was meant to be an expression of only my opinion and not a critic to your answer. $\endgroup$ – user40276 Jun 4 '18 at 0:31
  • $\begingroup$ @user40276 I didn't take it as a criticism, I just wanted to agree that there's a big world to be explored. I'd love if other people would write their own answers with their favorite introductory papers. I keep stumbling in gems I had overlooked :) $\endgroup$ – Denis Nardin Jun 4 '18 at 7:09
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I think that doing algebraic K-theory properly certainly requires a good background on stable homotopy theory, that is to say the homotopy theory of spectra. Unfortunately there are not many textbooks in the subject. Let me mention two of them:

  • Stable homotopy and generalized cohomology by J. Frank Adams is an old classic. Its treatment of some topics is far from modern though, and in particular the development of localizations is flawed and should be complemented by reading Bousfield's original papers. He also doesn't talk about commutative (i.e. $E_∞$) ring spectra, which you're going to need.

  • Symmetric spectra by S. Schwede. A much more modern approach, covering a variety of topics you're going to need. Just don't get too hung on the model categorical subtleties of the model he chose (I'm thinking mainly semistability here), 'cause they won't come up in practice.

  • Categories and cohomology theories by G. Segal. This is a short paper, but if you want to learn about group completion and its relation to spectra, reading this is probably the quickest thing you can do. Despite its age it is surprising modern in its approach. Moreover it is a pleasure to read.

When you have a sufficiently good background in homotopy theory that the words spectra and group completion don't make you scream in terror, it's time to start with actual algebraic K-theory. Here are some useful starting points

  • The K-book by Charles Weibel has a lot of classical material and it is a useful bridge from the low dimensional, hand-defined groups to the more modern algebraic K-theory spectrum. It is a bit long though, and I'd treat it more as a reference than a book to be read from top to bottom.

  • Higher Algebraic K-theory of Schemes and of Derived Categories by Thomason and Trobaugh. One of the fundamental papers on algebraic K-theory. It also has a decent introduction to Waldhausen's S-construction and it is worth reading in full. Highly highly recommended.

  • On the Lichtenbaum-Quillen Conjectures from a Stable Homotopy-Theoretic Viewpoint by Stephen A. Mitchell. This is a general survey of algebraic K-theory. It is extremely useful and will acquaint you with most classical theorems that you might find used in more specialized papers. Absolutely on a to-read list.

  • Algebraic K-theory and traces by I. Madsen. Filling a gap in the previous survey, this talks about trace methods, the best way we have to actually compute the K-theory groups.

  • Algebraic K-theory of spaces Waldhausen's original paper on his approach to algebraic K-theory. It's worth taking a look at some of the proofs, and the motivation is explained clearly and naturally.

I also think some familiarity with ∞-category theory might be useful from a technical standpoint, but you shouldn't enter into a full dive into the foundations. This is something best coordinated with your advisor, who will know how much of it is actually useful for you. As a rule of thumb, if you're spending more than a week on it (at first, of course), you're doing too much.

Feel free to jump into the homotopy theory chatroom to hang around and ask questions if you feel like it.

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