First Chern class factors through determinant

PDF p. 8: Proposition3.2, perfect-complex first class and determinant

Assume the Axiom of Choice and the fixed compatible site-size and universe bounds required by the supplied perfect K-theory, derived-section and weight-one constructions. Let $k$ be any field and $X$ any qcqs $k$-scheme. For the specified perfect-complex spectral $K_0(X)$, the first motivic Chern class is the actual rank-reduced first-layer map constructed by rank reduction, and it factors naturally as $K_0(X)\stackrel{{\det }_X}{\to }\mathrm{Pic}(X)\stackrel{{\gamma }_X\sim }{\to }{H}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2,1}(X,\mathbb{Z})\mathrm{.}$ The first arrow is the extended perfect-complex determinant and the second is the actual EM weight-one orientation. For a line-filtered bundle its determinant is the tensor product of the line factors and the first classes add. On actual and virtual vector-bundle classes this agrees with their supplied projective-relation and total-series first coefficients. No global equality of vector-bundle K zero with perfect K zero is assumed.

Canonical first motivic class on every perfect K zero class