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Natural unital bigraded motivic ring with negative zero extension

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

Statement

Let $k$ be any field. On qcqs $k$ -schemes set $C_{b} (X) = Z (b)_{m o t} (X)$ for $b \geq 0$ , and explicitly set $C_{b} (X) = 0$ for $b < 0$ . Define $H^{a, b} (X) = H^{a} (C_{b} (X))$ for every pair of integers. The direct sum $H^{*, *} (X)$ is naturally an associative unital ring, with products of bidegrees $(a, b) + (c, d) = (a + c, b + d)$ , unit in $(0, 0)$ , and $x y = (- 1)^{a c} y x$ . Pullbacks are contravariantly functorial unital ring homomorphisms. This extends the supplied groups without modifying their nonnegative weights.

Facts & Assumptions

[F1]

External EM multiplicative motivic complexes and actual Chern map supplies the natural commutative graded spectrum $G_{b} (X) = C_{b} (X) [2 b]$ , its unit and products; negative pieces are zero.

[F2]

Motivic complexes and cohomology fixes the specified complexes and cohomology groups; Multiplicative filtrations and graded products specifies the associated graded products. A cochain complex $C$ has underlying spectrum with $H^{a} (C) = π_{- a} C$ ; the symmetry on $S^{u} \land S^{v}$ has sign $(- 1)^{u v}$ .

Proof

Given: $k, X$ as stated, all integer bidegrees.

Goal: Construct and justify the ring and pullbacks.

1.1F1F2construct

For $b \geq 0$ identify $H^{a, b} (X) = π_{2 b - a} G_{b} (X)$ by [F1]. Multiplying representatives in homotopy groups with the actual graded product $G_{b} \land G_{d} \to G_{b + d}$ defines the stated bilinear product. If either weight is negative its group is zero; define that product to be zero.

2.1F1F2step 1.1

The unit $S \to G_{0}$ gives $1_{X} \in H^{0, 0} (X)$ . Associativity and the unit identities follow by applying homotopy groups to the coherent diagrams in [F1]. Swapping homogeneous classes contributes $(- 1)^{(2 b - a) (2 d - c)} = (- 1)^{a c}$ ; hence the required graded commutativity follows. Cases with a zero negative-weight factor satisfy these identities as well. Finite-support sums define the direct-sum ring.

3.1F1F2step 1.1step 2.1∎

Each morphism of schemes induces a map of these unital graded algebra objects. Its homotopy-group maps preserve the products, unit, identities and composites. The unique maps on the zero pieces preserve these assertions for negative weights. Restriction from the prime subfield to $k$ is composition of contravariant functors, so all these claims hold for arbitrary $k$ .

Depends on

Used by · 3

Dependency tree · 3 levels

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