The complex tautological Euler class restricts to the projective-fiber generator

Put $N=n-1$. The pullback $j_b^{\ast }{\gamma }_E$ has fiber precisely the line represented by each point of $P(E_b)$. The chosen complex-linear identification therefore identifies it with the usual tautological line over ${\mathbb{C}\mathbb{P}}^N$. The identification preserves the frames $(v,iv)$; changing a complex line frame by $a+ib\mathrm{\ne }0$ has positive real determinant $a^2+b^2$. Both bases are in Thom scope by [F1], also applied to the trivial rank-$n$ bundle over a point. Thus [F3] gives $j_b^{\ast }x=x_{\mathrm{t}\mathrm{a}\mathrm{u}\mathrm{t}}$ with exactly the stated orientation.

The integral cellular cochain complex of ${\mathbb{C}\mathbb{P}}^N$ has one copy of $\mathbb{Z}$ in each even degree $0,2,\dots ,2N$ and zero in every other degree. Every differential is zero. Hence its integral cohomology is $\mathbb{Z}$ in those degrees and zero elsewhere, including all degrees above $2N$. Its degree-zero unit is a generator.

For $N\ge 1$, the sphere bundle of its tautological complex line is $S^{2N+1}$: the homeomorphism sends $(\mathrm{\ell },v)$ with $v\in \mathrm{\ell }$ of unit length to $v$, and its inverse sends $v$ to $(\mathbb{C}v,v)$. Both maps are continuous in the quotient and bundle charts of [F1]. For $2\le k\le 2N$, both flanking groups in [F2] vanish by [F5], since $1\le k-1<k<2N+1$. Thus multiplication by $x_{\mathrm{t}\mathrm{a}\mathrm{u}\mathrm{t}}$ is an isomorphism $H^{k-2}\to H^k$ in this range. Starting at the unit, its powers generate all the even groups in step 1.2. This proves the integral basis assertion and degree-two generation; vanishing above the top degree comes from step 1.2, not from an induction through the exceptional top sphere group.

With ${\mathbb{F}}_p$ coefficients the same cellular cochain complex has one copy of ${\mathbb{F}}_p$ in each even degree and zero differentials. The natural coefficient map reduces each integral cell coordinate modulo $p$. Each power in step 2.1 is an integral generator, hence has coordinate $+1$ or $-1$ and reduces to a basis vector over ${\mathbb{F}}_p$. Together with step 1.1 this proves the reduction assertion.

When $n=1$, step 1.2 with $N=0$ gives $H^2=0$ and the single basis element $1$, integrally and after reduction. No use of step 2.1 is needed. An empty base contributes no fibers. Rank zero is excluded; the separate convention in [F1] gives empty projectivization with no $x$. The determinant argument in step 1.1 proves independence of each complex-linear identification, so no varying sign is introduced across components. AC is inherited from the stated suppliers; choosing a frame for one fixed fiber introduces no further choice assumption.