Dualizing and support shifts on the complex line

A positively complex-oriented open disk $U\subset \mathbb{C}$, with closed-disk compactification $D=\bar{U}$, has compact-support cochain complex with a single cohomology generator in degree $2$ over $k$: the closed disk with its boundary is the finite relative cellular model $(D,\mathrm{\partial }D)$ whose quotient is generated by the oriented $2$-cell, which computes compact supports on $U$ by extension by zero to $D$. Hence the dual local model has $k$ concentrated in degree $-2$, and the orientation normalization of [F1] at $d=1$ gives ${\omega }_{\mathbb{C}}=k_{\mathbb{C}}[2]$. This uses the same positive complex orientation generator as the diagonal Thom calculations.

Closed-support cohomology at the origin. By [F2] the object $i^{!}{k}_{\mathbb{C}}$ is $R{\mathrm{\Gamma }}_{\{0\}}(\mathbb{C},k_{\mathbb{C}})$, the derived functor of $i^{-1}{\mathrm{\Gamma }}_{\{0\}}$, and the localization triangle of [F2] computes it as the relative cohomology of the pair $(U,U\setminus \{0\})$ for a small open disk $U$ around the origin; radial retraction of the punctured disk to the circle gives the relative cellular model with a single generator in degree $2$, so $i^{!}{k}_{\mathbb{C}}=k[-2]$. Applying the shift $[2]$ of [F4] gives $i^{!}{\omega }_{\mathbb{C}}=i^{!}(k_{\mathbb{C}}[2])=k[-2][2]=k$, consistently with dualizing transitivity of [F1]. The duality route produces the same answer: $D_{\mathbb{C}}{k}_{\mathbb{C}}=k_{\mathbb{C}}[2]$, $i^{\ast }(k_{\mathbb{C}}[2])=k[2]$ and $D_{\{0\}}(k[2])=k[-2]$, with the same positive orientation generator used in both computations. The displayed computation involves only the finite relative cellular model of a disk and is choice-free; AC is present only because the consumed Verdier supplier assumes it.