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Elmanto–Morrow filtration input

PDF p. 3: Section 2, Elmanto–Morrow filtration; PDF p. 4: After Definition 2.2: functorial multiplicative filtration; EM23 Theorem 1.1(1)

Statement

Assume the Axiom of Choice for the selected local perfect-complex $K$ -theory realization. Let $F$ be $Q$ or $F_{p}$ for a prime $p$ . On the opposite of the category of quasi-compact quasi-separated $F$ -schemes, nonconnective algebraic $K$ -theory has a functorial, multiplicative, $N$ -indexed decreasing motivic filtration whose zeroth stage is naturally equivalent to $K$ . In the terminology of coherently multiplicative filtered spectra, the imported multiplicative filtration includes coherent products, unit and transition compatibility; its maps to $K$ are induced through its zeroth stage.

Statement references

Multiplicative filtrations and graded products

External input

This is the existence and zeroth-stage part of the cited Elmanto–Morrow results. It is imported as literature, not proved by the local formal lemmas. The source's later boundedness and convergence assertions require additional hypotheses and are not imported here. The nonconnective $K$ -theory is used through the perfect-complex interface of Nonconnective K theory of perfect complexes, under its stated Choice assumption. This assumption belongs to the selected local realization; it is not an additional geometric hypothesis attributed to Elmanto–Morrow.

Facts & Assumptions

Given: The Axiom of Choice, a prime field $F$ and a qcqs $F$ -scheme.

Goal: Supply the functorial multiplicative filtration and its zeroth-stage equivalence under exactly these hypotheses.

[F1]

The cited Theorem 1.1(1) states the functorial multiplicative filtration on nonconnective $K$ -theory. The initial assertions of Theorems 4.12 and 4.27 state the nonnegative indexing and zeroth-stage identification, respectively in characteristics zero and $p$ .

[F2]

The assumed The Axiom of Choice discharges the explicit Choice premise of Nonconnective K theory of perfect complexes when its common-family enhanced perfect categories and essentially small nonconnective-functor domain are invoked. Definitions 4.9 and 4.23 construct the imported filtration as a pullback of filtered $E_{\infty}$ -algebras, providing the coherent multiplication specified by Multiplicative filtrations and graded products. No independent choice of a filtration is made here.

Proof

1.1F1F2given

Under the inherited Choice premise in [F2], use the selected $K$ -theory realization and apply the characteristic-zero assertion of [F1] when $F = Q$ , and the characteristic- $p$ assertion when $F = F_{p}$ .

2.1F1F2givenstep 1.1∎

The qcqs hypothesis is exactly the common hypothesis of those initial assertions. The stated functoriality and coherent multiplicativity are the imported theorem's data, with the filtered-algebra structure in [F2]; no finite-valuative-dimension conclusion is used.

Depends on

Used by · 5

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External dependencies

Motivic cohomology of equicharacteristic schemes · Version 1 (15 September 2023; internal date 18 September 2023): Theorem 1.1(1) p.3; Definition 4.9 p.21; Definition 4.23 pp.27–28; initial assertions of Theorems 4.12 p.22 and 4.27 p.29; §2.2 pp.11–12

On qcqs schemes over a prime field there is a natural multiplicative nonnegative motivic filtration on nonconnective algebraic K-theory, whose zeroth stage is K.

Hypotheses: The base prime field is Q or F_p for a prime p; schemes are quasi-compact and quasi-separated over that field. No finite-valuative-dimension hypothesis is used for the stated existence and zeroth-stage assertions.

Reading limits: This adjudication read the retrieved v1 PDF text: title/date, Theorem 1.1(1), section 2.2, Definitions 4.9/4.23 and the initial assertions of Theorems 4.12/4.27. These passages establish the indexing, filtered-algebra convention and existence scope used here. The PDF header is dated 15 September 2023 and its internal title date is 18 September 2023. The HTML at the v1 URL displays 24 August 2026; this record uses the dated PDF. No page-image inspection or reconstruction of trace methods, descent foundations or deep proofs is claimed. Later boundedness conclusions are outside this input.

Sources

Motivic cohomology of equicharacteristic schemes · Version 1 (15 September 2023; internal date 18 September 2023): Theorem 1.1(1) p.3; Definition 4.9 p.21; Definition 4.23 pp.27–28; initial assertions of Theorems 4.12 p.22 and 4.27 p.29; §2.2 pp.11–12