Motivic filtration and image groups

PDF p. 2: Introductory filtration; PDF p. 3: Section 2 opening; PDF p. 8: Section 4 opening

Assume the Axiom of Choice and fix a field $k$, algebraically closed under the article convention unless otherwise specified. Write ${\mathrm{S}\mathrm{c}\mathrm{h}}_k$ for the category of quasi-compact quasi-separated $k$-schemes and $k$-morphisms, as in Article scheme conventions. For $X$ in this category, let $K(X)$ be the nonconnective $K$-theory spectrum of perfect complexes, and let $K_i(X)={\pi }_{i}K(X)$ for every integer $i$, as in Nonconnective K theory of perfect complexes.

The motivic filtration $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{\bullet }K$ is the Elmanto–Morrow functorial multiplicative nonnegative filtration supplied by Elmanto–Morrow filtration input, restricted from the prime field of $k$ to $k$-schemes. Extend it to integer indices by the construction of Extending a nonnegative filtration to integer indices. Thus it is a functor

$F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{\bullet }K:{\mathrm{S}\mathrm{c}\mathrm{h}}_k^{\mathrm{o}\mathrm{p}}⟶\mathrm{CAlg}(\mathrm{FilSp}),$

where $\mathrm{FilSp}$ is the category of integer-indexed decreasing spectra with coherent transitions and $\mathrm{CAlg}$ denotes its coherently commutative, associative and unital algebra objects, as specified in Multiplicative filtrations and graded products. Its zeroth stage is naturally equivalent to $K(X)$; we use that equivalence to write $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{0}K(X)=K(X)$. Its multiplication includes maps

$F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{a}K(X)\wedge F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{b}K(X)⟶F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{a+b}K(X)$

for all integers $a,b$, coherently compatible with transitions, pullback and the multiplication on $K(X)$.

For $i\in \mathbb{Z}$ and $j\ge 0$, define the image filtration group

$F_{\mathrm{m}\mathrm{o}\mathrm{t}}^j{K}_i(X)=\mathrm{im}({\pi }_i{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{j}K(X)⟶K_i(X))\mathrm{.}$

The arrow is induced by the transition to stage zero and its identification with $K(X)$. Its image is a subgroup, and the induced map onto that subgroup is surjective. The same formula also makes sense for negative $j$ under the stated extension, in which case it gives all of $K_i(X)$.

1.1 Invoke the perfect-complex $K$-theory interface in [F2] under the assumed Choice premise. The structure morphism of each $k$-scheme composes with $\mathrm{Spec}k\to \mathrm{Spec}\mathbb{F}$ for its prime field $\mathbb{F}$. Its underlying scheme remains qcqs, and every $k$-morphism is an $\mathbb{F}$-morphism. Restrict the functor in [F2] along this forgetful functor; no finite-type, regularity or finite-dimensional assumption is needed. [F1, F2, given]

2.1 Apply the extension in [F3] to the restricted nonnegative tower. Its stage zero is naturally equivalent to $K$, and its coherent products give the displayed maps. The extension and all maps are functorial because they are built from the original transitions and products. The equality at stage zero is notation for the supplied natural equivalence. [F2, F3, step 1.1]

3.1 Apply the image lemma in [F4] to each stage-to-$K$ map. The subgroup and surjectivity statements follow. Nesting, pullback compatibility and product containment follow from the remaining lemmas in [F4]. If $j<0$, the extended stage and its map to $K$ are identified with $K$ and its identity, so the image is the whole group. This justifies the definition and its stated properties. [F3, F4, step 2.1] ∎

The external construction is not reconstructed here. No completeness, boundedness, associated-graded identification or injectivity of a stage-to-$K$ map is asserted by this definition.