It's well known that for Noetherian separated regular schemes the canonical map $$K(X) \longrightarrow G(X)$$ (Quillen uses $K'$ instead of $G$, though) is a weak equivalence.

This statement is usually called Poincaré duality.

One can also define $K$-theory with compact support (for sufficiently nice schemes $X$) by choosing a compactification $X \hookrightarrow \overline{X}$ and setting $K_c(X)$ as the homotopy kernel of $K (X) \rightarrow K (\overline{X}\setminus X)$. I have no idea on whether such $K_c$ and $K$ enjoy some kind of Poincaré duality.

When I hear something like Poincaré duality I expect some kind of cap product map with some fundamental virtual class $$H^{\bullet} \longrightarrow H_{d -\bullet}^{BM}$$ or, dually, $$H_c^{\bullet} \longrightarrow H_{d - \bullet}$$. Of course, there's a cap product $$K(X) \wedge G(X) \longrightarrow G(X)$$ induced by tensor product which when restricted to tensoring with $\mathscr{O}_X$ gives the Poincaré duality.

However, I'm not satisfied with such analogy. I, hence, ask the following.

1) Is there any sense in which $G$ is a $K$-theory with compact support? Or maybe it's even the opposite: $K$ is a $G$-theory with compact support?

2) If yes, is there any relation between $K_c (X)$ and $G(X)$?

3) If no, is there any kind of duality between $K (X)$ and $K_c (X)$?

4) If I'm actually sounding silly since in ordinary Poincaré duality both sides of the isomorphism are always simultaneously of the same kind (compact or not compact, for instance, $H_{\bullet}^{BM}$ is somehow non compact as $H^{\bullet}$), how can I see the duality as some isomorphism from a cohomology to a homology? In other words, why $K(X)$ should be a cohomology theory and $G(X)$ a homology theory?

5) If one uses some Atiyah-Hirzebruch spectral sequence for $G$-theory, would it be the case that the graded pieces of the $\gamma$ filtration define a motivic cohomology with compact support up to torsion?

6) What about 5 for $K_c$ instead of $G$? What about $G_c$?

7) After applying the Atiyah-Hirzebruch sequence to all the possibilities ($K$, $K_c$, $G$, $G_c$) what sort of Poincaré duality one acquires?

Thanks in advance.

EDIT

I've added new questions in order to correct my lack of attention to concordance of the "kind" (compact or noncompact) of the domain and codomain in the duality.

  • 2
    G-theory is like Borel-Moore homology. The simplest reason is its functoriality: it is covariant for proper maps and contravariant for etale maps, like BM homology. Your $K_c$ is not well-defined (now that we know K-theory satisfies pro-cdh descent, we can define $K_c(X)$, but this involves taking the limit over all nilpotent thickenings of $\bar X-X$). – Marc Hoyois Nov 9 at 15:34
  • @MarcHoyois Thanks for the comment! It's somehow surprising my $K_c$ is ill defined. What exactly will fail? If I recall correctly, Gillet seems to define the $K$-theory of a pair in the analogous way when proving higher GRR, but maybe I'm overseeing something... By the way, do you know some reference where such $K_c (X)$ that you mentioned is studied? – user40276 Nov 9 at 21:04
  • 1
    @user40276 check out section 4.1 in arxiv.org/pdf/1211.1813.pdf – Gasterbiter Nov 10 at 6:25
  • @Gasterbiter Thanks for the comment. That definition still weird to me, though. I guess its the just the best that one can do when something fails to be smooth (be it $\overline{X}$ or $\overline{X}\setminus X$). In any case, I suppose that by dévissage the naive definition is safe enough for $G$. – user40276 yesterday
  • @user40276 yep. if u want to be fancy, $G$ has cdh descent. it's also fine for KH (homotopy K theory) for the same reason – Gasterbiter 22 hours ago

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