Graded filtration projections

PDF p. 10: Section 5 opening; PDF p. 12: Section 6 projection p3

Assume the Axiom of Choice and fix compatible site and universe size bounds for the specified motivic and weight-one constructions. Let $k$ be a field, $X$ a quasi-compact quasi-separated $k$-scheme, and $i\ge 0$. Write $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i}K(X)$ for stage $i$ of the specified motivic filtration of the nonconnective $K$-theory spectrum of perfect complexes. The stage projection $p_i:{\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i}K(X)⟶H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2i,i}(X,\mathbb{Z})$ is induced by the map from that stage to its graded cofiber $\mathbb{Z}(i{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X)[2i]$. Here $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{a,i}(X,\mathbb{Z})$ is degree-$a$ cohomology of the specified weight-$i$ motivic complex. Both the filtration and the complex are restricted from the prime field of $k$.

The image filtration group $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^i{K}_0(X)$ is the image of the stage-to-$K(X)$ homomorphism on ${\pi }_0$; write $q_i:{\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i}K(X)\to F_{\mathrm{m}\mathrm{o}\mathrm{t}}^i{K}_0(X)$ for its canonical corestriction. If $k$ is algebraically closed and $X$ is normal, then for $0\le i\le 2$ the map $q_i$ is an isomorphism. For these indices the same symbol $p_i$ also denotes the transported homomorphism $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^i{K}_0(X)\stackrel{q_i^{-1}}{\to }{\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i}K(X)\stackrel{p_i}{\to }{H}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2i,i}(X,\mathbb{Z})\mathrm{.}$ This convention covers every normal qcqs $k$-scheme, including the normal finite-type case. It requires no finite-type, Noetherian or dimension bound. At stage zero, the equivalence $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{0}K(X)=K(X)$ already identifies the stage and image for every $X$ above. Normal means that every local ring is an integrally closed domain. Finite type means that $X$ has a finite affine cover by spectra of finitely generated $k$-algebras. Choice means that every family of nonempty sets has a choice function; the size bounds fix the small sites and ambient universes of the specified constructions.

For $i=3$ the stage projection has domain ${\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{3}K(X)$. The low-stage identification above does not itself identify this domain with $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^3{K}_0(X)$; image-domain notation at stage three requires a separate justification.

Motivic filtration and image groups

Motivic complexes and cohomology

External actual graded algebra and multiplicative tower spectral sequence

The actual determinant fiber is a natural retract of the rank-zero K-space

1.1 Let $\mathbb{F}$ be the prime field of $k$. Forgetting to $\mathbb{F}$ preserves the underlying qcqs scheme and takes each $k$-morphism to an $\mathbb{F}$-morphism. Under the Choice and size assumptions in [F1], restrict the natural graded equivalence in [F2] along this forgetful functor. By the definitions in [F1], its source and target are exactly the specified tower's graded cofiber and the specified shifted motivic complex on $X$. This applies over every field $k$ without a finite-type or convergence hypothesis. [F1, F2, given]

1.2 Apply [F3] to the given qcqs $X$. Evaluating $d\kappa \simeq \mathrm{i}\mathrm{d}$ at $X$ makes ${\pi }_{1}S(X)\to {\pi }_{1}B(X)$ surjective. The actual fiber sequence therefore has zero boundary into ${\pi }_{0}T(X)$, and its transition ${\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2}K(X)\to {\pi }_0{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{1}K(X)$ is injective. The first-stage augmentation is injective by the rank-fiber assertion in [F3]. Their composite is the second-stage augmentation by [F1], so it too is injective. Stage zero has its augmentation equivalence. Thus every $q_i$ for $0\le i\le 2$ is injective and is surjective onto its image by definition; in particular this applies on the stated normal qcqs schemes. [F1, F3, F4, given]

2.1 The cofiber sequence $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i+1}K(X)\to F_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i}K(X)\to {\mathrm{gr}}^i{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}K(X)$ and step 1.1 give the stage-to-complex map. Applying ${\pi }_0$ identifies its target with $H^0(\mathbb{Z}(i{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X)[2i])=H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2i,i}(X,\mathbb{Z})$. This defines the displayed homomorphism for every $i\ge 0$, naturally under pullback. [F1, F2, F4, step 1.1]

3.1 Each $q_i$ in step 1.2 is an isomorphism of abelian groups. Compose its inverse with the projection in step 2.1 to obtain the displayed image-domain homomorphism. It is the unique map whose composite with $q_i$ is the stage projection, since $q_i$ is surjective. Equivalently, [F4]'s kernel criterion holds because the augmentation is injective. The construction is natural and agrees with the same transport in the finite-type special case. [F1, F4, step 2.1, step 1.2]

4.1 Step 1.2 identifies only the augmentations in stages zero, one and two. It gives no third-stage injectivity, so $p_3$ retains the stage domain of step 2.1 until a separate result supplies image-domain identification or descent. This establishes all stated conventions. [step 2.1, step 1.2, step 3.1] ∎

The supplied article PDF text was read at its conventions on p. 3, Lemma 4.1 and proof on pp. 8–9, Section 5 opening on p. 10, and Section 6 opening on pp. 12–13. The version-specific Elmanto–Morrow reading is recorded in the reference above. The current definitions, graded source input, stage-exactness argument, determinant-fiber proof and its rank-zero and global-Picard suppliers were inspected for these applications. Their underlying motivic, determinant and descent constructions are used through those contracts, not reconstructed here. The exact outside-resolution packet, its low-filtration proposal and the relevant genuine rejected response were consulted as proposals and historical evidence; they confer no acceptance. The separate finite-type low-homotopy lemma remains unchanged.

The supplied article PDF text was read at its conventions on p. 3, Lemma 4.1 and proof on pp. 8–9, Section 5 opening on p. 10, and Section 6 opening on pp. 12–13. The version-specific Elmanto–Morrow reading is recorded in the reference above. The current definitions, graded source input, stage-exactness argument, determinant-fiber proof and its rank-zero and global-Picard suppliers were inspected for these applications. Their underlying motivic, determinant and descent constructions are used through those contracts, not reconstructed here. The exact outside-resolution packet, its low-filtration proposal and the relevant genuine rejected response were consulted as proposals and historical evidence; they confer no acceptance. The separate finite-type low-homotopy lemma remains unchanged.