Bigraded cohomology and the projective bundle contract

PDF p. 8: Projective bundle construction of Chern classes

Assume the Axiom of Choice. Fix a category $\mathcal{C}$ of schemes and scheme morphisms that is closed under the following operations: for every $X$ in $\mathcal{C}$, every finite locally free ${\mathcal{O}}_X$-module $E$ of constant positive rank, and every morphism $f:Y\to X$ in $\mathcal{C}$, the projective bundle $\pi :{\mathbb{P}}_X(E)\to X$, its base change ${\mathbb{P}}_Y(f^{\ast }E)\to Y$, and the induced base-change morphism are in $\mathcal{C}$. A finite locally free module of rank $r$ is an ${\mathcal{O}}_X$-module locally isomorphic to ${\mathcal{O}}_X^r$; a line bundle is one of rank one.

An additively oriented bigraded theory with the projective bundle contract on $\mathcal{C}$ consists of the following data and axioms.

1. For each $X$ and each $(a,b)\in {\mathbb{Z}}^2$, an abelian group $H^{a,b}(X)$. Their direct sum $H^{\ast ,\ast }(X)$ is an associative unital ring, with unit $1_X\in H^{0,0}(X)$ and bilinear multiplication sending $H^{a,b}(X)\times H^{c,d}(X)$ to $H^{a+c,b+d}(X)$. It is graded-commutative in the first degree: $xy=(-1{)}^{ac}yx$ for homogeneous $x$ and $y$ of first degrees $a$ and $c$.

2. For every morphism $f:Y\to X$ in $\mathcal{C}$, a degree-preserving unital ring homomorphism $f^{\ast }:H^{\ast ,\ast }(X)\to H^{\ast ,\ast }(Y)$. Pullback is contravariantly functorial: $({\mathrm{id}}_X{)}^{\ast }=\mathrm{id}$ and $(f\circ g{)}^{\ast }=g^{\ast }{f}^{\ast }$.

3. A natural group homomorphism $c_1:\mathrm{Pic}(X)\to H^{2,1}(X)$ for every $X$. Here $\mathrm{Pic}(X)$ is the group of isomorphism classes of line bundles under tensor product, and the target uses addition. Naturality means $f^{\ast }{c}_1(L)=c_1(f^{\ast }L)$. Thus $c_1(L\otimes M)=c_1(L)+c_1(M)$ and $c_1({\mathcal{O}}_X)=0$.

4. For every rank-$r>0$ bundle $E$ on $X$, use the quotient convention ${\mathbb{P}}_X(E)={\mathrm{Proj}}_X\mathrm{Sym}(E)$, with tautological quotient line bundle ${\mathcal{O}}_{{\mathbb{P}}_X(E)}(1)$. Put $\xi =c_1({\mathcal{O}}_{{\mathbb{P}}_X(E)}(1))$. For every $(a,b)$, the homomorphism ${⨁}_{m=0}^{r-1}{H}^{a-2m,b-m}(X)⟶H^{a,b}({\mathbb{P}}_X(E)),(x_m{)}_m⟼{\sum }_{m=0}^{r-1}{\pi }^{\ast }(x_m){\xi }^m$ is an isomorphism. This is the projective bundle contract. The rank-zero case is not included in this isomorphism.

The contract is an abstract set of data and axioms. It does not itself assert that motivic cohomology satisfies them; that assertion requires the separate motivic projective-bundle input.

Projective bundle in the quotient convention

Picard group of a scheme

The Axiom of Choice

The published projective-bundle definition supplies the quotient construction and its tautological quotient under the stated choice assumption. The published Picard definition supplies the group of line-bundle classes, so the domain of $c_1$ and its additivity condition are well-defined. The pullback and ring axioms make each displayed summand a homomorphism of abelian groups; its bidegree is $(a,b)$ because $\xi$ has degree $(2,1)$. The sum is finite, and requiring the resulting map to be an isomorphism is a coherent condition on the specified data. The conditions are consistent: taking every $H^{a,b}(X)$ to be the zero group, with its zero ring multiplication and unit $0$, every pullback and every $c_1$ to be the unique map, makes both sides of the contract zero. This example establishes only consistency of the abstract contract, not its realization by motivic cohomology.