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 $K_{0}$

Let $R$ be a ring and denote by $\mathord{\mathrm{M}_{\infty}(R)}$ the algebraic direct limit of matrix algebras $\mathord{\mathrm{M}_{n}(R)}$ under the embeddings $\mathord{\mathrm{M}_{n}(R)}\to\mathord{\mathrm{M}_{n+1}(R)}:a\mapsto\left(% \begin{array}[]{cc}a&0\\ 0&0\end{array}\right)$. The zeroth K-group of $R$, $K_{0}(R)$, is the Grothendieck group (abelian group of formal differences) of idempotents in $\mathord{\mathrm{M}_{\infty}(R)}$ up to similarity transformations. Let $p\in\mathord{\mathrm{M}_{m}(R)}$ and $q\in\mathord{\mathrm{M}_{n}(R)}$ 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 $p\oplus q=\mathrm{diag}(p,q)\in\mathord{\mathrm{M}_{m+n}(R)}$. Equivalently, one can work with finitely generated projective modules over $R$.

The functor $K_{1}$

Denote by $\mathrm{GL}_{\infty}(R)$ the direct limit of general linear groups $\mathrm{GL}_{n}(R)$ under the embeddings $\mathrm{GL}_{n}(R)\to\mathrm{GL}_{n+1}(R):g\mapsto\left(\begin{array}[]{cc}g&0% \\ 0&1\end{array}\right)$. Give $\mathrm{GL}_{\infty}(R)$ the direct limit topology, i.e. a subset $U$ of $\mathrm{GL}_{\infty}(R)$ 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 $\mathrm{GL}_{\infty}(R)$, i.e.

 $K_{1}(R)=\mathrm{GL}_{\infty}(R)/[\mathrm{GL}_{\infty}(R),\mathrm{GL}_{\infty}% (R)].$

Note that this is the same as $K_{1}(R)=H_{1}(\mathrm{GL}_{\infty}(R),\mathbb{Z})$, the first group homology group (with integer coefficients).

The functor $K_{2}$

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 $\mathrm{E}_{\infty}(R)$ the direct limit of the $\mathrm{E}_{n}(R)$ using the construction above (note $\mathrm{E}_{\infty}(R)$ is a subgroup of $\mathrm{GL}_{\infty}(R)$). The second K-group of $R$, $K_{2}(R)$, is the second group homology group (with integer coefficients) of $\mathrm{E}_{\infty}(R)$,

 $K_{2}(R)=H_{2}(\mathrm{E}_{\infty}(R),\mathbb{Z}).$

Higher K-functors

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

 $K^{\mathrm{alg}}_{n}(R)=\pi_{n}(B\mathrm{GL}_{\infty}(R)^{+}),$ (1)

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

Rough sketch of suspension:

 $\Sigma R=\Sigma\mathbb{Z}\otimes_{\mathbb{Z}}R$ (2)

where $\Sigma\mathbb{Z}=C\mathbb{Z}/J\mathbb{Z}$. The cone, $C\mathbb{Z}$, is the set of infinite matrices with integral coefficients that 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 many non-trivial coefficients.

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

Algebraic K-theory has a product structure,

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

References

• 1 H. Inassaridze, Algebraic K-theory. Kluwer Academic Publishers, 1994.
• 2 Jean-Louis Loday, Cyclic Homology. Springer-Verlag, 1992.
Title Algebraic K-theory AlgebraicKtheory 2013-03-22 13:31:32 2013-03-22 13:31:32 mhale (572) mhale (572) 10 mhale (572) Topic msc 19-00 msc 18F25 KTheory GrothendieckGroup StableIsomorphism