Units split the K one determinant

For a ring A, P(A) is split exact: a surjection onto a projective module has a section, so its kernel is a direct summand of the finite projective source and is finite projective. This verifies the kernel-of-surjections hypothesis of Gillet–Waldhausen in [F2]. The Waldhausen category C(A) of bounded finite-projective cochain complexes has mapping cylinders, so every map factors as a degreewise split monomorphism followed by a quasi-isomorphism. Its homotopy category after quasi-isomorphisms is the enhanced perfect category’s homotopy category by [F1]; hence a map is a weak equivalence exactly when it becomes an isomorphism (DHKS-saturation). Indeed the cone of a quasi-isomorphism between bounded projectives is bounded acyclic projective and contractible: apply the K-projective mapping property in [F1] to its identity. Thus the comparison hypotheses in [F3] hold.

Compose the actual maps of [F2], the actual Waldhausen/infinity comparison of [F3], j on π₁ from [F3], and the exact affine enhancement equivalence of [F1]. They give an isomorphism $c_A:GL(A{)}_{ab}\stackrel{\sim }{\to }{\pi }_1\mathbb{K}(\mathrm{Perf}(\mathrm{Spec}A))$. Every step carries an automorphism M of Aⁿ to the class of that same degree-zero automorphism: degree-zero inclusion is literal, the S-construction comparisons use its core loop, and associated sheaf sends it to its matrix acting on $O^n$. Thus $c_A([u])=[\mathcal{O}\stackrel{u}{\to }\mathcal{O}]$. This uses π₁ agreement in the positive-degree range of j; no π₀-only comparison is substituted.

This $c_A$ is natural for unital A→B. Extension of scalars takes finite projectives to finite projectives and preserves split monomorphisms. A bounded projective quasi-isomorphism is a homotopy equivalence by step 1.1, so tensor with B preserves it without a flatness hypothesis. Hence it is weakly exact on C(A). All comparisons are induced by the corresponding actual functors in [F1]–[F3], and on matrix generators they send M to its entrywise image. The resulting π₁ naturality square commutes. Define $d_A={\det }_A\circ c_A^{-1}$; [F4] makes these homomorphisms commute with all such ring maps, and $d_A$([u])=u by step 2.1.

Given k∈K₁(X), on each affine open V=Spec(A) take $d_A(k{\mathrm{\mid }}_V)\in {\mathcal{O}}_X(V{)}^{\times }$. On an intersection of two affine opens, restrict further to any affine neighbourhood W inside that intersection. Naturality from step 3.1 identifies both restrictions with $d_{\mathcal{O}(W)}(k{\mathrm{\mid }}_W)$. They therefore agree as sections on the intersection by the sheaf uniqueness axiom. Apply [F5] to glue a unique global unit det_X(k). The same gluing lemma shows det_X is a homomorphism and independent of the affine cover.

A global unit is the automorphism $m_u$ of $O_X$. The natural automorphism map in [F3] defines $s_X$(u)=[$m_u$]. Composition $m_u{m}_v=m_{uv}$ makes $s_X$ a group homomorphism. It is compatible with scheme pullback by [F1] and [F3]. Restriction to an affine V sends its class to the class [u|_V] of step 2.1.

The restriction of det_X($s_X$(u)) to each affine V is u|_V by steps 3.1 and 5.1. Unit sheaf uniqueness in [F5] gives det_X($s_X$(u))=u globally. The right-inverse lemma in [F5] proves surjectivity of determinant and injectivity of $s_X$. For f:Y→X between qcqs schemes, choose an affine V around f(y) and an affine W around y inside f⁻¹(V). Step 3.1 proves determinant naturality on W; sheaf uniqueness then proves it globally. Thus both maps are natural on the stated category.

For an automorphism α of a vector bundle E, choose affine opens where E is free of constant rank. Naturality of the automorphism construction restricts [α] to the matrix class of α there. The top-wedge expansion in [F6] says that its action on det(E) is multiplication by that matrix determinant, exactly the local unit used in step 4.1. These local units glue uniquely; a line-bundle automorphism is canonically multiplication by a unit, because that is true on trivializations and scalar multiplication commutes with transition units. Thus det_X([α]) has the asserted exterior determinant interpretation. The empty scheme has only the unique unit and zero K-group, so the same assertion also includes it. This completes the proof.