# 17 definitions of algebraic K-theory of a ring. Which should I take?

There are multiple definitions of algebraic K-theory, but I have trouble differentiating between them. Could someone help me out? Let $$R$$ be a commutative ring. I would like to define $$K_n(R)$$, and for simplicity $$n \geq 0$$ is OK for me now.

• The plus construction gives us $$B\operatorname{GL}(R)^+$$, whose homotopy groups give me $$K_n(R)$$ for all $$n \geq 0$$.
• The Q-construction takes as input an exact category $$\mathcal{C}$$. At this point we can pick the finitely generated $$R$$-modules, or the projective finitely generated $$R$$-modules, or the finitely generated free $$R$$-modules, or the coherent $$R$$-modules. This gives us four K-theory groups $$K_n(R)$$ for $$n \geq 0$$.
• All the exact categories give Waldhausen categories, to which we can apply the S-construction, yielding four more K-groups.
• Segal's construction involving $$\Gamma$$-spaces takes in a symmetric monoidal category $$\mathcal{C}$$. We can take any of the above categories, along with two choices of monoidal structure, namely $$\otimes$$ or $$\oplus$$, yielding another eight K-theory groups $$K_n(R)$$ for $$n \geq 0$$.

Ideally, I'd like to see these definitions compared to modern ones. I've been told that one should think of algebraic K-theory as an $$\infty$$-group completion procedure, but I have yet to find a readable reference on this.

• Little remark: The $+$-construction does not give $K_0$ which is of course quite crucial. That is also one reason that it is unnatural/ad hoc and which (probably) led Quillen to work on the $Q$-construction.
– Con
Commented Apr 7, 2020 at 9:34
• Which definition you want can depend on what you want to do with it. Are you working towards a particular result or area of research, or are you just curious about all the ways these things interact? Commented Apr 7, 2020 at 9:51
• @KReiser I just want an overview on which is which, because it's all kinda confusing to me. That said, I am mostly after the plus comparison and how it relates to two $\infty$-categorical construction. Basically I have a stable symmetric monoidal $\infty$-category whose K-theory I want to consider, and I want to see if it can be computed by a variation of Quillen's plus construction. But then it struck me that I do not really know what any of these things mean...
– Jim
Commented Apr 7, 2020 at 10:02
• The best way to say it is that all these constructions are deeply interrelated and equivalent. Maybe I will write down an answer later. I will just note that the $\Gamma$ construction per-se doesn't produce $K$-theory. It is a delooping constuction. What this means is that if you throw in a group like monoid, it will output an sequence of deloopings for this space, so in otherwords it outputs some form of connective spectra from an infinite loop space. Commented Apr 24, 2020 at 15:24
• If what you're after is the $\infty$-categorical formulation, let me suggest to watch Thomas Nikolaus' talks at the Newton institute on exactly this topic : newton.ac.uk/seminar/20180702113012301 (there are 4 lectures) Commented Jan 10, 2021 at 17:06

Let me make a few comments relating the various classical constructions you mentioned.

1. The $$BGL(R)^+$$ construction produces an infinite loop space. It's constructed by taking $$BGL(R)=\varinjlim BGL_n(R)$$ and then attaching $$2$$-cells to kill the commutator subgroup of $$\pi_1$$ and attaching $$3$$-cells to preserve the original homology. It might seem ad-hoc but there was good reason to come up with this construction. You can read more about the motivation in Segal's contribution to the AMS tribute to Quillen. The original source is Quillen's seminal paper.

2. The Quillen $$Q$$-construction is far more general and applies to a notion of exact category (i.e. an additive category with a class of exact sequences satisfying some properties), for example coherent sheaves or vector bundles over schemes. Essentially you take a an additive category $$\mathcal{C}$$, give it an exact structure, make a $$Q$$ consturction out of it i.e. a category $$Q\mathcal{C}$$ and then take its classifying space $$BQ\mathcal{C}.$$ The $$K$$-groups are the homotopy groups of the loop space $$\Omega BQ\mathcal{C}.$$ In fact, $$\Omega BQ\mathcal{C}$$ is an infinite loop space.

3. In the $$Q$$-construction for a scheme you get two groups related to the category of coherent sheaves and the category of vector bundles. However there are theorems in the $$Q$$ construction of $$K$$-theory which allows you relate these two groups in nice cases. For example if you are over a smooth projective variety, then every coherent sheaf has a finite resolution by locally free ones, and by Quillen's Resolution theorem, the $$K$$-theories are the same. You can translate this to the language of commutative algebra if you want.

4. In the case of a ring $$R$$ and $$P(R)$$ the category of finitely generated proejctive modules, both constructions the $$Q$$ construction and the plus construction are equivalent i.e. $$K_{0}(R)\times BGL(R)^+\simeq \Omega BQP(R).$$
This is the work of Grayson, which he attributes to Quillen.

5. Then you have the Waldhausen $$S$$-construction, which is the most general of all which you mentioned. This is a functorial way to construct connective spectra as opposed to constructing just the infinite loop space. More precisely, for a Waldhausen Category $$\mathcal{C}$$, the Waldhausen $$S$$-construction gives you not only an infinite loop space $$\Omega |wS_\bullet\mathcal{C}|$$ but also infinite deloopings $$|wS_{\bullet}S_{\bullet}\mathcal{C}|,|w S_{\bullet}^3\mathcal{C}|\ldots,$$ and where $$|wS_{\bullet}^{n}\mathcal{C}|\simeq \Omega|wS_{\bullet}^{n+1}\mathcal{C}|.$$

6. Every exact category is (canonically) a Waldhausen categoy. In this case the $$Q$$-construction and the Waldhausen construction are related via a simplicial subdivision method called Segal's subdivision. This is proved in Waldhausen's original paper. Functionally this means that there is an equivalence $$\Omega BQ\mathcal{C}\simeq \Omega |wS_\bullet\mathcal{C}|.$$

7. Segal's $$\Gamma$$-machinery is an infinite loop space machine. It takes in an infinite loop space and outputs connective spectra. So for example you can feed it the $$Q$$-construction of Quillen and you will get connective spectra out. Note that the $$Q$$-construction already knows about the additive structure and in particular about $$\oplus$$ (the finite bi-poduct). If you also have a symmetric monoidal structure given by $$\otimes$$, over which $$\oplus$$ distributes , then your spectra inherits a coherent multiplication $$KC\wedge KC\to KC$$, where $$KC$$ is the $$K$$-theory spectrum of $$C$$ (in other words it is an $$E_\infty$$-ring spectrum).

8. Finally, there are several other infinite loop space machines such as May's delooping via the bar construction, but these are all equivalent. This is a theorem of May and Thomason.

Edit: A sketch of the $$\infty$$-categorical construction of the K-theory spectrum of a commutative ring (more generally a symmetric monoidal $$\infty$$-category) is given in the appendix of the Bhatt-Scholze paper on the Witt vector affine Grassmanian.