Determinant descends on the actual perfect K zero

Assign to a perfect object $P$ the Picard class $d(P)$ of the underlying line of its global graded determinant in [F1]. Derived-isomorphism invariance makes $d(P)$ depend only on its object equivalence class, and $d(0)=0$. Forgetting the grading is done only on isomorphism classes, after using the graded determinant data.

For any complex representing $P$, the identity mapping cone is acyclic, and the termwise split short-exact complex diagram $0\to P\to \mathrm{Cone}(1_P)\to P[1]\to 0$ is a true triangle of perfect complexes. Its determinant additivity gives $d(P)+d(P[1])=d(\mathrm{Cone}(1_P))=d(0)=0$. This proves the inverse-line identity without assuming triangle functoriality in the ordinary derived category.

Given an enhanced cofiber sequence $P\to Q\to R$, pass to its distinguished triangle in the supplied sheaf-derived category. Represent its first morphism by a roof and replace its source by the quasi-isomorphic complex $P^{\mathrm{\prime }}$ in that roof, obtaining an actual complex map $a:P^{\mathrm{\prime }}\to Q$ whose cone represents $R$. All three objects are perfect by [F2], although these representing complexes need not be globally bounded complexes of bundles. The globally defined termwise split short-exact complex diagram $0\to Q\to \mathrm{Cone}(a)\to P^{\mathrm{\prime }}[1]\to 0$ is a KM true triangle. Thus [F1] and step 2.1 give $d(R)=d(Q)+d(P^{\mathrm{\prime }}[1])=d(Q)-d(P)$, so every actual cofiber relation is killed. Only existence of these isomorphisms is needed for the equality of Picard classes.

Extend $P\mapsto d(P)$ to the free abelian object-class group. By step 3.1 it descends through all its cofiber relations, and the object-class isomorphism of [F2] gives the homomorphism on the specified spectral $K_0$. For a morphism $f:Y\to X$ of qcqs schemes, [F1] identifies the global determinant of $L{f}^{\ast }P$ with the pullback determinant of $P$; the naturality of the object-class map in [F2] therefore makes these homomorphisms natural in $X$. Exactness of the supplied derived pullback, rather than flatness of $f$, is what is used.

On any open with a bounded finite locally free representative, [F1] gives the alternating tensor formula in the Statement. In degree zero it is the exterior determinant of [F3], and step 2.1 gives the shift formula. These are formulas for restrictions of an already global line: the line and its descent isomorphisms come from [F1]. They do not assert that an arbitrary perfect class is globally a bundle difference or admits a global bounded bundle representative. This proves the full qcqs statement.