The actual graded determinant map on the current K-space

On $\mathrm{Spec}A$ use $\mathcal{W}(A)=C{h}^b(\mathrm{proj}A)$, with quasi-isomorphisms as weak equivalences and degreewise split monomorphisms as cofibrations. Mapping cylinders give factorization; quasi-isomorphisms between bounded projectives have contractible cones, so saturation holds. The completed enhancement comparison identifies its infinity-localization with the supplied $\mathrm{Perf}(A)$. It is idempotent complete. Hence [F1] and [F2] give the actual determinant map on the connective cover of the current K-spectrum and its space.

For every ring map $A\to B$, termwise tensor takes finite projectives to finite projectives and preserves split cofibrations and contractible cones. It is therefore exact on these Waldhausen models, including preservation of weak equivalences. The KM pullback data and [F1] universal construction make the determinant maps natural, with the tensor-composition identifications, at the Picard-one-type level. These are actual maps and natural isomorphisms of the determinant data; a calculation of homotopy groups alone has not supplied them.

Affine schemes cover the big scheme site by open immersions. By sheafification’s universal property and restriction to this covering basis, the actual affine map to the graded-line stack extends to a map from the sheafification of the K-space presheaf to that stack. Compose with the K-space sheafification unit to obtain the claimed map at every qcqs $X$. This construction does not require an equality of global bundle K zero and perfect K zero, or even an assertion that taking a first Postnikov section commutes with global sections. KM pullback composition and the affine exact-model naturality supply the descent morphisms. On a perfect object E its affine restrictions are sent to the alternating determinant with those same quasi-isomorphism transition maps; these glue to exactly KM’s global determinant line by its extended determinant/base-change construction.

The rank of the graded line is Euler rank. Its rank-zero substack is $B{\mathbb{G}}_m$ with its usual symmetry, giving the stated restricted map. An enhanced cofiber sequence is represented locally by a genuine cofibration sequence or a mapping-cylinder replacement of such a sequence; its determinant additivity is the one in [F1]. Taking line isomorphism classes yields $[\det Q]=[\det P]+[\det R]$ for every such cofiber relation, independently of a chosen ordinary triangle diagram. Degree-zero vector bundles and their matrix automorphisms give precisely exterior determinant and matrix determinant. For a global cofiber sequence, use its mapping-cone true triangle and the KM additivity isomorphism, with the quasi-isomorphism identifications of [F1]. These are the same pullback-compatible isomorphisms used by the affine determinant data in steps 1.1–3.1, so the global line relation retains its gluing maps. Mere equality of classes on affine opens is not used to infer equality in the global Picard group.