Actual first-Chern and determinant maps of the first EM layer

Work in Nisnevich sheaves of spaces on Sm_k. The rank-zero K-space S=Ω∞Fil¹K is connected as a SHEAF, even though S(X) may have many components: at a henselian local stalk the rank map K₀→Z is an isomorphism by [F2]. Its sheaf of fundamental groups is $G_m$ by determinant, since the positive-degree first-stage map is an isomorphism by [F1]. Thus τ≤1S is $B{G}_m$ with its identification fixed by the actual determinant map in [F2]. A map from S to $B{G}_m$ factors uniquely through this pointed one-type.

The map on fundamental-group sheaves induced by κ is the unit section, hence is an isomorphism at local stalks by [F2] and [F3]. Since q₁κ=id by [F3], the fundamental-group map of q₁ is its inverse, namely determinant. Full faithfulness of the pointed one-type classification in [F2] therefore identifies q₁:S→$B{G}_m$ with the actual determinant/Pic map itself, as a map of Nisnevich stacks after the canonical truncation. This retains all descent morphisms and thus determines global Picard classes; the fact that an invertible sheaf is locally trivial has not been used to erase its global class.

Evaluate this equality of actual maps at X and take π₀ and π₁. By [F1] their source groups are reduced K₀ and K₁, and their target groups, under [F3], are Pic(X) and Γ(X,$O_X$)×. By the determinant construction [F2], its π₀ is the class of the determinant graded line after passing to rank zero, which is the usual first Chern class and is det(E) on a vector bundle; its π₁ is the matrix-compatible determinant characterized in [F3]. These are exactly the two formulas, with positive sign fixed in [F3].