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Products of filtration images

PDF p. 3: Section 2, multiplication

Statement

Let $F$ be a multiplicative filtration on a commutative ring spectrum $E$ , including its multiplicative augmentation as in Multiplicative filtrations and graded products. For integers $a, b, i, j$ , multiplication on $π_{*} E$ sends the product of the image of $π_{i} F^{a} \to π_{i} E$ and the image of $π_{j} F^{b} \to π_{j} E$ into the image of $π_{i + j} F^{a + b} \to π_{i + j} E$ .

Facts & Assumptions

Given: The multiplicatively augmented filtered spectrum $F \to E$ and the four integers.

Goal: Prove the containment of products.

[F1]

Multiplicative filtrations and graded products supplies $F^{a} \land F^{b} \to F^{a + b}$ and its compatibility with multiplication on $E$ .

[F2]

Images of filtration stages supplies the image subgroups. Homotopy groups of spectra admit the pairing induced by smash product, in the convention used in [F1].

Proof

1.1F1F2given

Choose elements $x \in π_{i} F^{a}$ and $y \in π_{j} F^{b}$ lifting the two given image elements. Smashing representatives and applying the stage multiplication gives an element of $π_{i + j} F^{a + b}$ .

2.1F1step 1.1∎

Multiplicative compatibility of the augmentation identifies its image in $π_{i + j} E$ with the product of the two original image elements. Thus that product belongs to the required image subgroup. This proves the statement.

Depends on

Used by · 1

Motivic filtration and image groups

Dependency tree · 3 levels

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