To say I am a novice in $K$-theory is to overstate my experience with the field. I've been reading the various wiki articles so as to have some preparation before jumping in, and I couldn't answer the following question to myself:

I understand that $K$-theory had started with the Grothendieck-Riemann-Roch in mind, and that the only thing that was needed for that purpose from $K$-theory was just to define the Grothendieck Group ($K_0$). Once the idea of the Grothendieck group was established, this was generalized to topological spaces, as well as for other kinds of modules. Then comes the step I don't understand -- it seems that people were then trying to find the higher $K$-groups that would make $K$-theory into a cohomological theory. Milnor came up with Milnor $K$-theory, which I understand from wiki is different from later notions of higher $K$-theory. But why would one leap from the concept of the Grothendieck group to thinking that this construction is the $0^{th}$ step in a cohomological theory? What was the context/motivation for that?

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    $\begingroup$ I am not an expert on the history of $K$-theory, but I am pretty sure that the most important motivation for higher algebraic $K$-theory was the existence of higher topological $K$-theory. Higher topological $K$-theory was constructed by Atiyah and Hirzebruch (using the Bott periodicity theorem). $\endgroup$ – Johannes Ebert Jun 12 '11 at 19:04
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    $\begingroup$ I think Milnor's book (Intro to Algebraic K-theory) gives a pretty good indication of why people expected a "higher" K-theory (even if some of the reasons presented in the book later turned out to be mistaken, e.g. the Mayer-Vietoris sequence doesn't extend past K_2 in general). But the book makes it clear that the three functors K_0, K_1, and K_2 (which all have geometric meanings/applications) are linked in ways that resemble a cohomology theory. $\endgroup$ – Dan Ramras Jun 12 '11 at 19:46

The idea of considering higher K-groups comes from topology, and is due to Atiyah, Bott, and Hirzebruch. Atiyah and Hirzebruch defined topological K theory and observed that Bott periodicity says that $K(X)$ is more or less the same as $K(S^2X)$. This suggested to them defining a generalized cohomology theory of period 2 by using all the groups $K(S^nX)$ (this was the first example of a generalized cohomology theory). Once one realizes that topological $K^0$ can be extended to topological $K^n$, it does not take much imagination to suggest that algebraic $K^0$ also has an extension to algebraic $K^n$. (Of course, finding this extension was much harder than guessing it existed.)


After having defined K0, a natural things to do is to study its functoriality properties. You do that, and you notice some exact sequences... that you happen to be able to extend a bit by using the functors K1, and then later K2.

It is then natural (specially if you know what cohomology is) to try to find long exact sequences that extend the above sequences... A lot of people tried to do that.

Quillen's brilliant idea was to define algebraic K-theory as the homotopy groups of an appropriately constructed space. In that way, the long exact sequences came as natural consequences of known long exact sequences in topology.
Slogan: Homotopy theory is the mother of all long exact sequences.

Aside: I also recommend reading Thomason's work on algebraic K-theory of schemes: it's beautifully written!

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    $\begingroup$ I might be wrong, but I think it is unlikely that Quillen was the first to realize that the algebraic K-groups should be the homotopy of some space. Swan and Karoubi (and quite possibly others) had the same idea. The problem, of course, was to find the right space. $\endgroup$ – Steven Landsburg Jun 12 '11 at 23:59
  • $\begingroup$ I chose the other answer because it was more directed at the question, but this was also a very helpful answer. Do you have a precise name for Thomasson's work, so I can look it up? $\endgroup$ – James D. Taylor Jun 13 '11 at 12:39
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    $\begingroup$ Thomason, R. W.; Trobaugh, Thomas: Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III, 247–435, $\endgroup$ – André Henriques Jun 13 '11 at 16:34
  • $\begingroup$ In Milnor's book on the functor $K_2$, he motivates the story in the way André mentions: a pushout diagram of rings gives rise to a Mayer-Vietoris sequence for $K_0$ and $K_1$. Milnor was motivated by this observation when giving a definition of $K_2$. Regarding uniqueness, note that $K_0(R)$ is the universal receptacle for invariants of f.g. projective $R$-modules additive on exact sequences. Likewise, $K_1(R)$ can be viewed as the universal gadget for invariants of f.g. projective $R$-modules equipped with automorphisms. There's a related description for $K_2$, but I'm out of space. $\endgroup$ – John Klein Nov 29 '11 at 22:43
  • $\begingroup$ @JohnKlein That's sad! Could you please expand on a universal characterization of $K_2$? It is very interesting! $\endgroup$ – მამუკა ჯიბლაძე Oct 17 '16 at 9:42

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