Images of support groups in global groups

PDF p. 6: Final two displays of Definition 2.7; PDF p. 7: Equation (2.8), K-zero support-image notation

Assume the Axiom of Choice and the size conventions of the support construction. Fix a field $k$, a quasi-compact quasi-separated $k$-scheme $X$, and a closed subset $Y\subseteq X$ with its reduced induced closed-subscheme structure. Put $U=X\setminus Y$. Let $j\ge 0$ and $i\in \mathbb{Z}$.

Use the support spectra and their support-to-global maps from the preceding support definition. In particular, $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^j{K}_{i,Y}(X)$ is the image of ${\pi }_i{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{j}K(X\mathrm{/}U)$ in ${\pi }_{i}K(X\mathrm{/}U)$, and $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^j{K}_i(X)$ is the image of ${\pi }_i{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{j}K(X)$ in $K_i(X)={\pi }_{i}K(X)$. An image is the subgroup consisting of the values of the indicated homomorphism. Define

$F_{\mathrm{m}\mathrm{o}\mathrm{t}}^j{K}_i(Y;X):=\mathrm{im}(F_{\mathrm{m}\mathrm{o}\mathrm{t}}^j{K}_{i,Y}(X)⟶F_{\mathrm{m}\mathrm{o}\mathrm{t}}^j{K}_i(X)),$

where the arrow is induced by the support-to-global map on $K$-theory. For supported motivic cohomology $H_{\mathrm{m}\mathrm{o}\mathrm{t},Y}^{a,j}(X,\mathbb{Z})$ and global motivic cohomology $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{a,j}(X,\mathbb{Z})$, define

$H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{a,j}(Y;X):=\mathrm{im}(H_{\mathrm{m}\mathrm{o}\mathrm{t},Y}^{a,j}(X,\mathbb{Z})⟶H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{a,j}(X,\mathbb{Z}))$

for every integer $a$; in particular this includes the article's degree $a=2j-i$. The arrow is the support-to-global map. Finally, put

$K_0(Y;X):=\mathrm{im}({\pi }_{0}K(X\mathrm{/}U)⟶K_0(X))\mathrm{.}$

These are image subgroups of the respective global groups; the supported groups are their domains before the final image is taken. No injectivity of a support-to-global map is part of the definition.

Motivic and filtered K theory with support

Motivic filtration and image groups

Motivic complexes and cohomology

1.1 The compatible support and global stage-to-$K$ maps form a commutative square after applying ${\pi }_i$. If an element of $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^j{K}_{i,Y}(X)$ is represented by an element of ${\pi }_i{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{j}K(X\mathrm{/}U)$, its image in $K_i(X)$ is represented by the image of that element in ${\pi }_i{F}_{\mathrm{m}\mathrm{o}\mathrm{t}}^{j}K(X)$. It therefore belongs to $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^j{K}_i(X)$. The map is the restriction of the homomorphism on unfiltered $K$-theory, so it is independent of the chosen representative. [A1, F1, F2, given]

2.1 The image of a group homomorphism is a subgroup: zero is the image of zero, and the difference of two image elements is the image of the difference of their preimages. Apply this to the homomorphism in step 1.1 and to the support-to-global homomorphisms for motivic cohomology and unfiltered $K$-theory. The resulting subgroups are exactly the three displayed definitions. The construction uses no injectivity; it justifies the definitions and their stated targets. [F1, F3, step 1.1] ∎

The supported cohomology supplier uses an explicit $2j$ shift to reconcile the article's displayed ${\pi }_i$ notation with the global cohomological convention. The image in degree $2j-i$ is taken using that supplier's support-to-global map. The construction does not identify $K(X\mathrm{/}U)$ with the $K$-theory of a category of supported perfect complexes.