Algebraic K-theory

Algebraic K-theory studies invariants built from projective modules and their automorphisms over rings. Its lowest groups record the additive projective-module classes and stable invertible matrices, respectively. It differs from Topological K-theory, although vector-bundle clutching and module patching have parallel forms.

A cartesian square of unital rings $A=A_1{\times }_{A_0}{A}_2$ is a Milnor square when at least one map $A_i\to A_0$ is surjective. Finitely generated projective modules over $A$ can then be assembled from projective modules over the two upper rings together with an isomorphism of their reductions over $A_0$.

Evaluation of the cyclic generator at $1$ and at a primitive $p$th root gives the displayed pullback. The maps to the common field reduce integers modulo $p$ and set ${\zeta }_p=1$. If a representative polynomial for the second component has value $c$ at one, adding an integer multiple of ${\Phi }_p$ adjusts that value by a multiple of $p$ without changing the second component. This proves surjectivity onto compatible pairs.

The diagonal maps are extension of scalars; the maps to $A_0$ are differences. The connecting map sends $g$ to the difference of its patched free module and the corresponding free $A$-module. Elementary lifting and common free stabilization prove exactness at every internal term. The displayed fragment does not assert injectivity at its first term or surjectivity at its last.

A compatible triple $(P_1,P_2,\varphi )$ patches to $\{(x_1,x_2):\varphi ({\bar{x}}_1)={\bar{x}}_2\}$. It is finitely generated projective and recovers $P_i$ after extension of scalars. For free modules glued by $g$, the block identity $\mathrm{diag}(g,g^{-1})\in E_{\infty }(A_0)$ lifts through the surjective side and makes the direct sum with inverse gluing free. Complements reduce general projective triples to that free case. Conversely an idempotent over $A$ is exactly a compatible pair of idempotents over its two factors.

Stable invertible matrices, with stabilization $g\mapsto \mathrm{diag}(g,1)$, form the stable general linear group. Its elementary subgroup is its commutator subgroup, and the displayed quotient is its abelianization. Elementary matrices lift along any surjective ring map, even when arbitrary invertible matrices do not lift. Over a field, determinant identifies $K_1$ with its multiplicative unit group.

The union of finite invertible matrices under $g\mapsto \mathrm{diag}(g,1)$. Every element and relation occurs in a finite matrix size; adding identity blocks permits comparisons between different sizes without changing the stable class.

The subgroup generated by elementary transvections in finite matrix sizes, with identity stabilization. A third distinct index expresses each transvection as a commutator; stable block identities show that this subgroup is normal and that its quotient in the stable general linear group is abelian. Thus it is precisely the commutator subgroup.

The Grothendieck group of finitely generated projective right $A$-modules under direct sum. An element is a formal difference $[P]-[Q]$. Equality of two positive classes means $P\oplus T\cong Q\oplus T$ for some projective $T$; adding a complement of $T$ converts this into an isomorphism after a common finite free summand. This stabilization criterion is essential in patching arguments.