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.

**EDIT2**

Given the comments below by Marc Hoyois and Gasterbiter, $K_c (X)$ should be defined as the homotopy colimit over $r$ of $K (\overline{X}, r (\overline{X}\setminus X))$, where the prefix $r$ denotes the infinitesimal thickening of order $r$ (following the notation of https://arxiv.org/abs/1211.1813).

Also, as noted below, $G$ should behave as a Borel-Moore homology. The analogy, therefore, is that

$$K(X) \longrightarrow G(X)$$ is the analogous of the first duality expressed above (cohomology-BM homology), whereas $$K_c(X) \longrightarrow G_c(X)$$ should correspond to the second duality (the compact version), where $G_c (X) := G (\overline{X}, \overline{X}\setminus X)$ (Btw, how do I state these dualities using the six functor formalism instead of using this "underline $c$"?).

Therefore, only the last questions remain. I will restate them here.

1) If one applies he Atiyah-Hirzebruch spectral sequence to $K$, $K_c$, $G$ and $G_c$, then what will be the graded pieces of the $\gamma$-filtration up to torsion (take $X$ as general as possible)? Or even better, in the level of spectra, what kind of decomposition one acquires?

For instance, in the case of smooth $X$, $K(X) \wedge \mathbb{S}_{\mathbb{Q}} \cong \bigvee_i H \mathbb{Q} \wedge (\mathbb{P}^1)^{\wedge i}$ (I have no idea what happens when $X$ is not smooth, though).

2) Does one recover some kind of Poincaré duality from the graded pieces mentioned in 1?