K_1, the Determinant, and Elementary Matrices

Stabilize the general linear group

K_0 was built from modules; K_1 is built from their automorphisms — that is, from invertible matrices. Form GL(n, R) and embed GL(n, R) ↪ GL(n+1, R) by sending A to the block matrix [A, 0; 0, 1]. The union of all these is the stable general linear group GL(R) = colim GL(n, R). Stabilizing throws away the accident of a particular size and keeps only what survives in all large matrices — the same instinct as passing from free to stably free.

Inside GL(n, R) sit the elementary matrices e_ij(r) = I + r·E_ij, the identity with a single off-diagonal entry r in position (i, j), i ≠ j. These are exactly the matrices that perform a single row operation. The subgroup they generate, stably, is E(R) ⊆ GL(R). The definition is then disarmingly short: K_1(R) = GL(R) / E(R).

The Whitehead lemma

Why is K_1 = GL/E a sensible thing — in particular, why is it abelian? Because E(R) turns out to be exactly the commutator subgroup of GL(R). This is the Whitehead lemma, the technical heart of K_1, and it rests on a clean matrix identity you can check by hand.

Two facts give E(R) = [GL(R), GL(R)].

(1) Elementary matrices are commutators (n >= 3).
    For distinct i, j, k the Steinberg relation says
        [ e_ik(r), e_kj(1) ] = e_ij(r).
    So each generator of E is a commutator => E <= [GL, GL].

(2) Every commutator is elementary, stably. For g in GL(n, R),
    the 2n x 2n block matrix
        [ g, 0 ; 0, g^{-1} ]
    lies in E(R), because
        [ g, 0 ; 0, g^{-1} ]
          = [ I, g ; 0, I ] [ I, 0 ; -g^{-1}, I ] [ I, g ; 0, I ] [ 0, -I ; I, 0 ],
    and each factor is a product of elementary matrices.
    Apply this to a commutator [a,b]: the block-diagonal
        [ [a,b], 0 ; 0, I ] = [ [a,b], 0 ; 0, [a,b]^{-1} ] * [ I, 0 ; 0, [a,b] ]
    is elementary => [GL, GL] <= E.

Hence E(R) = [GL(R), GL(R)], so
        K_1(R) = GL(R) / E(R) = GL(R)^{ab}      (abelianization).

Whitehead lemma: E(R) is the commutator subgroup of GL(R), so K_1 is the abelianization of the stable GL.

Recovering the determinant

Now the payoff. The determinant det: GL(n, R) → R^× is a homomorphism, it is unchanged by row operations, so it kills E(R), and it stabilizes (det of [A, 0; 0, 1] equals det A). Therefore det descends to a map K_1(R) → R^×. Over a field, this map is an isomorphism: K_1(field k) ≅ k^×. So the determinant was K_1 all along — the universal invariant of an invertible matrix that survives stabilization.

For more general rings, det K_1(R) → R^× is still surjective (the 1×1 matrix [u] has determinant u), and its kernel SK_1(R) is the genuinely interesting, hard part. For a Euclidean domain SK_1 vanishes and K_1 ≅ R^×; for the ring of integers of a number field it also vanishes, a theorem of Bass–Milnor–Serre. The Whitehead group of a group ring, K_1(Z[G]) modulo trivial units, is what topologists use to classify when a homotopy equivalence is a genuine deformation.