Algebraic K-Theory

K₀(R) classifies projective modules; K₁(R) captures invertible matrices mod elementary row operations

Ring R

K₀ — Projective Modules

Click bundles to take direct sums:

K₁ — Units & GL

Grothendieck construction:
K₀(R) = group completion of (Proj-R, ⊕)
Elements: [P] - [Q] for f.g. proj. modules

Bass–Heller–Swan:
K₁(R[x,x⁻¹]) ≅ K₁(R) ⊕ K₀(R) ⊕ NK₁

Milnor K₂: universal central extension of E(R)