Algebraic K-theory

Algebraic K-theory is a series of functors on the category of rings.Broadly speaking, it classifies ring invariants, i.e. ring properties that are Morita invariant.

The functor $K_{\mathrm{0}}$

Let $R$ be a ring and denote by ${\mathrm{M}}_{\mathrm{\infty }}(R)$ the algebraic direct limit of matrix algebras ${\mathrm{M}}_n(R)$ under the embeddings.The zeroth K-group of $R$, $K_0(R)$, is the Grothendieck group (abelian group of formal differences) of idempotents in ${\mathrm{M}}_{\mathrm{\infty }}(R)$ up to similarity transformations.Let $p\in {\mathrm{M}}_m(R)$ and $q\in {\mathrm{M}}_n(R)$ be two idempotents.The sum of their equivalence classes $[p]$ and $[q]$ is the equivalence class of their direct sum:$[p]+[q]=[p\oplus q]$ where $p\oplus q=\mathrm{diag}(p,q)\in {\mathrm{M}}_{m+n}(R)$.Equivalently, one can work with finitely generated projective modules over $R$.

The functor $K_{\mathrm{1}}$

Denote by ${\mathrm{GL}}_{\mathrm{\infty }}(R)$ the direct limit of general linear groups ${\mathrm{GL}}_n(R)$ under the embeddings.Give ${\mathrm{GL}}_{\mathrm{\infty }}(R)$ the direct limit topology, i.e. a subset $U$ of ${\mathrm{GL}}_{\mathrm{\infty }}(R)$ is open if and only if$U\cap {\mathrm{GL}}_n(R)$ is an open subset of ${\mathrm{GL}}_n(R)$, for all $n$.The first K-group of $R$, $K_1(R)$, is the abelianisation of ${\mathrm{GL}}_{\mathrm{\infty }}(R)$, i.e.

$$K_1(R)={\mathrm{GL}}_{\mathrm{\infty }}(R)/[{\mathrm{GL}}_{\mathrm{\infty }}(R),{\mathrm{GL}}_{\mathrm{\infty }}(R)].$$

Note that this is the same as $K_1(R)=H_1({\mathrm{GL}}_{\mathrm{\infty }}(R),\mathbb{Z})$,the first group homology group (with integer coefficients).

The functor $K_{\mathrm{2}}$

Let ${\mathrm{E}}_n(R)$ be the elementary subgroup of ${\mathrm{GL}}_n(R)$.That is, the group generated by the elementary $n\times n$ matrices $e_{ij}(r)$, $r\in R$,where $e_{ij}(r)$ is the matrix with ones on the diagonals, the value $r$ in row $i$, column $j$and zeros elsewhere.Denote by ${\mathrm{E}}_{\mathrm{\infty }}(R)$ the direct limit of the ${\mathrm{E}}_n(R)$ using the construction above (note ${\mathrm{E}}_{\mathrm{\infty }}(R)$ is a subgroup of ${\mathrm{GL}}_{\mathrm{\infty }}(R)$).The second K-group of $R$, $K_2(R)$, is the second group homology group (with integer coefficients) of ${\mathrm{E}}_{\mathrm{\infty }}(R)$,

$$K_2(R)=H_2({\mathrm{E}}_{\mathrm{\infty }}(R),\mathbb{Z}).$$

Higher K-functors

Higher K-groups are defined using the Quillen plus construction,

$$K_n^{\mathrm{alg}}(R)={\pi }_n(B{\mathrm{GL}}_{\mathrm{\infty }}{(R)}^{+}),$$(1)

where $B{\mathrm{GL}}_{\mathrm{\infty }}(R)$ is the classifying space of ${\mathrm{GL}}_{\mathrm{\infty }}(R)$.

Rough sketch of suspension:

$$\mathrm{\Sigma }R=\mathrm{\Sigma }\mathbb{Z}{\otimes }_{\mathbb{Z}}R$$(2)

where $\mathrm{\Sigma }\mathbb{Z}=C\mathbb{Z}/J\mathbb{Z}$.The cone, $C\mathbb{Z}$, is the set of infinite matrices with integral coefficientsthat have a finite number of non-trivial elements on each row and column.The ideal $J\mathbb{Z}$ consists of those matrices that have only finitely manynon-trivial coefficients.

$$K_i(R)\cong K_{i+1}(\mathrm{\Sigma }R)$$(3)

Algebraic K-theory has a product structure,

$$K_i(R)\otimes K_j(S)\to K_{i+j}(R\otimes S).$$(4)