Motivic projective bundle formula

PDF p. 2: Motivic projective bundle property; PDF p. 8: Orientation for the splitting principle

Assume AC. On the article category of qcqs $k$-schemes, the supplied motivic cohomology, with $H^{a,b}(X)=0$ explicitly for $b<0$, is an additively oriented bigraded theory satisfying the projective bundle contract. This holds over any field $k$ (hence in particular the article’s algebraically closed base), without smoothness, reducedness, Noetherianity or dimension hypotheses. Its ring is the actual EM graded cup-product ring and its orientation is EM Definition5.2. For a bundle $E$ of constant rank $r>0$, the quotient convention ${\mathbb{P}}_X(E)={\mathrm{Proj}}_X\mathrm{Sym}(E)$ with $\xi =c_1(\mathcal{O}(1))$ gives, for every $(a,b)\in {\mathbb{Z}}^2$, ${⨁}_{m=0}^{r-1}{H}^{a-2m,b-m}(X)\stackrel{{\sum }_m{\pi }^{\ast }(-){\xi }^m}{\to }{H}^{a,b}({\mathbb{P}}_X(E)),$ a natural isomorphism. No rank-zero projective-bundle assertion is required.

Bigraded cohomology and the projective bundle contract