Q-construction

Let C be an exact category; i.e., an additive full subcategory of an abelian category that is closed under extension. If there is an exact sequence ${\displaystyle 0\to M'\to M\to M''\to 0}$ in C, then the arrow from M′ is called an admissible mono and the arrow from M is called an admissible epi.

Let QC be the category whose objects are the same as those of C and morphisms from X to Y are isomorphism classes of diagrams ${\displaystyle X\leftarrow Z\to Y}$ such that the first arrow is an admissible epi and the second admissible mono and two diagrams are isomorphic if they differ only at the middle and there is an isomorphism between them. The composition of morphisms is given by pullback.

Define a topological space ${\displaystyle B^{+}C}$ by ${\displaystyle B^{+}C=\Omega BQC}$ where ${\displaystyle \Omega }$ is a loop space functor and ${\displaystyle BQC}$ is the classifying space of the category QC (geometric realization of the nerve). As it turns out, it is uniquely defined up to homotopy equivalence (so the notation is justified.)

Every ring homomorphism ${\displaystyle R\to S}$ induces ${\displaystyle B^{+}P(R)\to B^{+}P(S)}$ and thus ${\displaystyle K_{i}(P(R))=K_{i}(R)\to K_{i}(S)}$ where ${\displaystyle P(R)}$ is the category of finitely generated projective modules over R. One can easily show this map (called transfer) agrees with one defined in Milnor's Introduction to algebraic K-theory.[2] The construction is also compatible with the suspension of a ring (cf. Grayson).

A theorem of Daniel Quillen states that, when C is the category of finitely generated projective modules over a ring R, ${\displaystyle \pi _{i}(B^{+}C)}$ is the i-th K-group of R in the classical sense for ${\displaystyle i=0,1,2}$. The usual proof of the theorem (cf. Weibel 2013) relies on an intermediate homotopy equivalence. If S is a symmetric monoidal category in which every morphism is an isomorphism, one constructs (cf. Grayson) the category ${\displaystyle S^{-1}S}$ that generalizes the Grothendieck group construction of a monoid. Let C be an exact category in which every exact sequence splits, e.g., the category of finitely generated projective modules, and put ${\displaystyle S=\operatorname {iso} C}$, the subcategory of C with the same class of objects but with morphisms that are isomorphisms in C. Then there is a "natural" homotopy equivalence:[3]

${\displaystyle \Omega BQC\simeq B(S^{-1}S)}$.

The equivalence is constructed as follows. Let E be the category whose objects are short exact sequences in C and whose morphisms are isomorphism classes of diagrams between them. Let ${\displaystyle f:E\to QC}$ be the functor that sends a short exact sequence to the third term in the sequence. Note the fiber ${\displaystyle f^{-1}(X)}$, which is a subcategory, consists of exact sequences whose third term is X. This makes E a category fibered over ${\displaystyle QC}$. Writing ${\displaystyle S^{-1}f}$ for ${\displaystyle S^{-1}E\to QC}$, there is an obvious (hence natural) inclusion ${\displaystyle \Omega BQC}$ into the homotopy fiber ${\displaystyle F(BS^{-1}f)}$, which can be shown to be a homotopy equivalence. On the other hand, by Quillen's Theorem B, one can show that ${\displaystyle B(S^{-1}S)}$ is the homotopy pullback of ${\displaystyle BS^{-1}f}$ along ${\displaystyle *\to BQC}$ and thus is homotopy equivalent to the ${\displaystyle F(BS^{-1}f)}$.

We now take C to be the category of finitely generated projective modules over a ring R and shows that ${\displaystyle \pi _{i}B(S^{-1}S)}$ are the ${\displaystyle K_{i}}$ of R in the classical sense for ${\displaystyle i=0,1,2}$. First of all, by definition, ${\displaystyle \pi _{0}B(S^{-1}S)=K_{0}(R)}$. Next, ${\displaystyle GL_{n}(R)=\operatorname {Aut} (R^{n})\to S^{-1}S}$ gives us:

${\displaystyle BGL(R)=\varinjlim BGL_{n}(R)\to B(S^{-1}S)}$.

(Here, ${\displaystyle BGL(R)}$ is either the classifying space of the category ${\displaystyle GL(R)}$ or the Eilenberg–MacLane space of the type ${\displaystyle K(GL(R),1)}$, amounting to the same thing.) The image actually lies in the identity component of ${\displaystyle B(S^{-1}S)}$ and so we get:

${\displaystyle f:BGL(R)\to B(S^{-1}S)^{0}.}$

Let ${\displaystyle S_{n}}$ be the full subcategory of S consisting of modules isomorphic to ${\displaystyle R^{n}}$ (thus, ${\displaystyle BS_{n}}$ is the connected component containing ${\displaystyle R^{n}}$). Let ${\displaystyle e\in \pi _{0}(BS)}$ be the component containing R. Then, by a theorem of Quillen,

${\displaystyle H_{p}(B(S^{-1}S)^{0})\subset H_{p}(B(S^{-1}S))=H_{p}(BS)[\pi _{0}(BS)^{-1}]=H_{p}(BS)[e^{-1}].}$

Thus, a class on the left is of the form ${\displaystyle xe^{-n}}$. But ${\displaystyle x\mapsto xe^{m}}$ is induced by the action of ${\displaystyle R^{m}\in S}$. Hence,

${\displaystyle H_{p}(B(S^{-1}S)^{0})=\varinjlim H_{p}(BS_{n})=\varinjlim H_{p}(BGL_{n}(R))=H_{p}(BGL(R)),\quad p\geq 0.}$

Since ${\displaystyle B(S^{-1}S)^{0}}$ is an H-group,

${\displaystyle \pi _{1}(B(S^{-1}S)^{0})=\pi _{1}(B(S^{-1}S)^{0})^{\text{ab}}=H_{1}(B(S^{-1}S)^{0})=H_{1}(BGL(R))=H_{1}(GL(R))=GL(R)^{\text{ab}}=K_{1}(R).}$

It remains to see ${\displaystyle \pi _{2}}$ is ${\displaystyle K_{2}}$. Writing ${\displaystyle Ff}$ for the homotopy fiber, we have the long exact sequence:

${\displaystyle \pi _{2}(BGL(R))=0\to \pi _{2}(B(S^{-1}S)^{0})\to \pi _{1}(Ff)\to \pi _{1}(BGL(R))=GL(R)\to K_{1}(R).}$

From homotopy theory, we know the second term is central; i.e., ${\displaystyle \pi _{1}(Ff)\to E(R)}$ is a central extension. It then follows from the next lemma that ${\displaystyle \pi _{1}(Ff)}$ is the universal central extension (i.e., ${\displaystyle \pi _{1}(Ff)}$ is the Steinberg group of R and the kernel is ${\displaystyle K_{2}(R)}$.)

Proof: The homotopy type of ${\displaystyle Ff}$ does not change if we replace f by the pullback ${\displaystyle {\widetilde {f}}}$ along the universal covering of Y ${\displaystyle \to Y}$. Thus, we can replace the hypothesis by one that Y is simply connected and ${\displaystyle H_{p}(X,\mathbb {Z} )\simeq H_{p}(Y,\mathbb {Z} ),p\geq 0}$. Now, the Serre spectral sequences for ${\displaystyle Ff\to X\to Y}$ and ${\displaystyle *\to Y\to Y}$ say:

${\displaystyle {}^{2}E_{pq}=H_{p}(Y,H_{q}(Ff,\mathbb {Z} ))\Rightarrow H_{p+q}(X,\mathbb {Z} ),}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle {}^2 E'_{pq} = H_p(Y, H_q(*, \Z )) \Rightarrow H_{p+q}(Y, \Z ).}

By the comparison theorem for spectral sequences, it follows that ${\displaystyle {}^{2}E_{0q}={}^{2}E'_{0q}}$; i.e., ${\displaystyle Ff}$ is acyclic. (Coincidentally, by reversing argument, one can say this implies ${\displaystyle H_{p}(X,\mathbb {Z} )\simeq H_{p}(Y,\mathbb {Z} );}$ thus, the hypothesis of the lemma.) Next, the spectral sequence for the covering ${\displaystyle {\widetilde {Ff}}\to Ff}$ with group ${\displaystyle G=\pi _{1}(Ff)}$ says:

${\displaystyle {}^{2}E_{pq}=H_{p}(G,H_{q}({\widetilde {Ff}},\mathbb {Z} ))\Rightarrow H_{p+q}(Ff,\mathbb {Z} )=H_{p+q}(*,\mathbb {Z} ).}$

An inspection of this spectral sequence gives the desired result.

• Daniel Grayson, Higher algebraic K-theory II [after Daniel Quillen], 1976
• Srinivas, V. (2008), Algebraic K-theory, Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.), Boston, MA: Birkhäuser, ISBN 978-0-8176-4736-0, Zbl 1125.19300
• Weibel, Charles (2013), The K-book: An introduction to algebraic K-theory