Alphabeta Math
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Stable determinant commutes with ring maps

PDF p. 9: Surjectivity of the determinant on K1

Statement

For a unital homomorphism f:A→B of commutative unital rings, entrywise application of f induces GL(A)ab→GL(B)ab. The stable determinant homomorphisms commute with this map and with f:A×→B×.

Facts & Assumptions

Given: A unital homomorphism f:A→B of commutative unital rings.

Goal: Prove the stated naturality square.

[F1]

Stable determinant is induced by the permutation formula and factors through abelianization. Determinant on the stable general linear group

Proof

technique · Apply the ring homomorphism to the determinant formula
1.1givenalgebra

If MN=NM=I, then f(M)f(N)=f(N)f(M)=I, so entrywise application gives compatible homomorphisms GLn(A)→GLn(B), hence a homomorphism of stable groups and of their abelianizations. A unit and its inverse likewise map to a unit and its inverse.

2.1F1step 1.1algebra∎

Applying f to the finite permutation formula gives f(det⁡M)=det⁡(f(M)). The equality is compatible with stabilization and passes to the abelianized groups by [F1]. This is the desired square.

Depends on

Used by · 1

Dependency tree · 1 level

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