# Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?

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?

• 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
• @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