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Extending a nonnegative filtration to integer indices

PDF p. 3: Section 2, filtered spectrum convention

Statement

Let $F$ be a coherent multiplicative decreasing filtration indexed by $N = {0, 1, 2, \dots}$ , with $F^{0} = E$ as commutative ring spectrum and with its maps to $F^{0}$ as multiplicative augmentation. It extends canonically to a multiplicative decreasing filtration indexed by $Z$ , by setting ${\tilde{F}}^{j} = F^{\max (j, 0)}$ . Its stages and products in nonnegative degrees are the original ones, and every stage in nonpositive degree is $E$ with identity transition maps between such stages.

Facts & Assumptions

Given: The specified coherent nonnegative multiplicative tower $F$ .

Goal: Construct and justify the integer-indexed extension.

[F1]

Decreasing spectral filtrations and associated graded specifies transition directions and augmentation; Multiplicative filtrations and graded products specifies the coherent finite-input product data and their compatibility.

Proof

1.1F1givenconstruct

Put $r (j) = \max (j, 0)$ . If $j \geq k$ , then $r (j) \geq r (k)$ , so the original transition $F^{r (j)} \to F^{r (k)}$ defines the extended transition. Monotonicity also gives all composition and identity compatibilities by restriction of the original coherent diagram. For $j \leq 0$ , $r (j) = 0$ , giving the asserted constant stages and identity transitions.

2.1F1givenstep 1.1construct

For any finite list $(j_{1}, \dots, j_{n})$ , the inequality $\sum r (j_{t}) \geq r (\sum j_{t})$ holds, including the empty list. Compose the original $n$ -input product into $F^{\sum r (j_{t})}$ with the transition to $F^{r (\sum j_{t})}$ . Associativity of addition, monotonicity and functoriality of the original transitions ensure that all alternative composites use the same original product followed by the same transition. The original coherent permutation, substitution and unit data therefore give the corresponding coherent data for the extension.

3.1F1step 1.1step 2.1∎

The maps to $F^{0} = E$ provide the compatible multiplicative augmentation. When all indices are nonnegative, $r$ preserves their sum, so the construction recovers the original products and transitions. This proves the extension and its stated properties.

Depends on

Used by · 1

Motivic filtration and image groups

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