The field comparison preserves refined smooth Gysin and the point class

The Thom class of a line bundle is determined by the pointed first-Chern class on its projective completion. The sum/product construction and projective-bundle detection determine the rank-$d$ Thom class from these line classes. Thus an orientation-preserving multiplicative coefficient map preserves the actual Thom class, as well as the Thom isomorphism $a\mapsto {\pi }^{\ast }a\mathrm{t}\mathrm{h}(N)$. Equivalently use the top-Chern definition on the projective completion and its unique refined lift. This uses the actual top class, not Navarro's malformed displayed Thom expansion.

Homotopy purity identifies $V\mathrm{/}(V\setminus Z)$ with $\mathrm{T}\mathrm{h}(N)$ in the motivic category. This is a map of motivic spaces before applying any coefficients, so the coefficient comparison commutes with it. Define refined Gysin by composing the Thom isomorphism from step 1.1 with this purity identification. The resulting square commutes; composing with the coefficient-independent forget-support arrow gives the ordinary Gysin square. The deformation construction in the cited RR source agrees with this construction on smooth regular pairs. For a nonseparated smooth pair, perform this construction on affine opens of the ambient scheme; their closed intersections with $Z$ are affine smooth pairs. Naturality on their open overlaps and Nisnevich descent glue the same purity and Thom maps. Thus the application of the primary purity theorem uses only separated finite-type pairs; descent handles the stated nonseparated scope.

For a smooth rational point of codimension three, $N$ is a rank-three vector space. Its Thom class is the product of the three normalized Cartier zero-section classes, hence is the positive point class. Units are preserved, so $i_{!}(1)$ has precisely the same normalization in the two theories. In the supported cycle model these three regular Cartier parameters cut out the reduced point with intersection multiplicity $+1$; the divisor bonding normalization is the $+1$ boundary in Spitzweck Lemma 6.6. In the article application over an algebraically closed field this is exactly the ${\delta }_x$ of [F2], on its stated pure-threefold neighbourhood. This also agrees with the three-coordinate-hyperplane test in ${\mathbb{P}}^3$. No cancellation of a forget-support map or division is used.