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

Question

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?

Answer 1

To my knowledge, one can only make this analogy fully consistent with Weibel's homotopy invariant $K$-theory $KH$ and $G$-theory (although the proofs of what I claim below rely heavily on our understanding of classical algebraic $K$-theory). Then, using the canonical map $K(X)\to KH(X)$, the pairing relating $KH$ and $G$ induce the pairing relating $K$ and $G$ which fits in the folkloric description of Poincaré duality relating $K$ and $G$.

Indeed, classical Poincaré duality is a particular instance of Grothendieck-Verdier duality (i.e. Grothendieck duality for ordinary sheaves of abelian groups). Indeed, if $D(X)$ denotes the derived category of sheaves on a (nice locally compact) space $X$, then we have, for any continuous map $f:X\to Y$ a pullback functor $f^*:D(Y)\to D(X)$ which has a right adjoint $f_*: D(X)\to D(Y)$, and there is push-forward functor $f_!:D(X)\to D(Y)$, the right derived functor of the direct image with compact support functor, which has a right adjoint $f^!: D(Y)\to D(X)$. There is natural map $f_!\to f_*$ which is invertible for $f$ proper, and so on.

Now, if $X$ is a space and $a:X\to \{pt\}$ denotes the canonical maps to the point,

• the cohomology of $X$ with coefficients in $\mathbf Z$ is $a_* a^*(\mathbf Z)$
• the cohomology with compact support is $a_! a^*(\mathbf Z)$
• Borel-Moore homology is $a_* a^!(\mathbf Z)$
• Homology is $a_! a^!(\mathbf Z)$

For nice enough spaces (e.g. algebraic varieties), these are perfect complexes of abelian groups (whence dualizable objects in the derived category of abelian groups), and taking the dual in the derived category exchanges $*$ and $!$. In particular, the dual of homology $a_! a^!(\mathbf Z)$ is cohomology $a_* a^*(\mathbf Z)$, while the dual of homology with compact support $a_! a^*(\mathbf Z)$ is Borel-Moore homology $a_* a^!(\mathbf Z)$. Poincaré duality consists in identifying, when $X$ is smooth complex orientable of dimension $d$, $a_! a^* (\mathbf Z)$ and $a_! a^!(\mathbf Z)(-d)[-2d]$ (where $A(-n)=A\otimes H^2(\mathbf P^1(\mathbf C),\mathbf Z)^{\otimes n}$).

Using Morel-Voevodsky's motivic stable homotopy category $SH$, we can extend this to schemes: to simplify, I will restrict to schemes of finite type over a field $k$. Then, given a a commutative motivic ring spectrum $E$ in $SH(k)$, we may define $D(X)$ as the category of $E$-modules in $SH(X)$ (to be precise, of $a^*(E)$-modules in the $(\infty,1)$-category $SH(X)$, where $a:X\to\mathrm{Spec}\, k$ denotes the structural map). And we have most of the features above, replacing $\mathbf Z$ by $E$ (e.g. we have cohomology $a_* a^*(E)$ and so forth). In the case where $E=KGL$ is the object which represents Weibel's $KH$ in $SH$, this gives a context in which Grothendieck's six operations apply.

Here, Poincaré duality identifies $a_! a^* (KGL)$ and $a_! a^!(KGL)(-d)[-2d]$ (for $X$ smooth of dimension $d$). Dually, it corresponds to

$$a_* a^!(KGL)\simeq a_* a^*(KGL)(d)[2d]$$

but Bott-periodicity also says that $KGL\simeq KGL(d)[2d]$.

Furthermore, for possibly singular $X$, one can check that the global sections of $a_* a^!(KGL)$ really give back $G$-theory:

$$\Gamma(\mathrm{Spec} \,k,a_* a^!(KGL))=G(X)$$ (this is essentially a reformulation of $K$-theoretic Poincaré duality as formulated in the question above, of Quillen's localization theorem for $G$-theory and of homotopy invariance for $G$-theory). And we have:

$$\Gamma(\mathrm{Spec} \,k,a_* a^*(KGL))=KH(X)$$

We could define homotopy invariant $K$-theory with compact support: $$KH_c(X):=\Gamma(\mathrm{Spec} \,k,a_! a^*(KGL))$$ and $KH$-homology as $\Gamma(\mathrm{Spec} \,k,a_! a^!(KGL))$.

As for the coniveau spectral sequence (a.k.a the motivic Atiyah-Hirzebruch spectral sequence), applied to $G$-theory, Marc Levine has showed that the $E_2$-term will be motivic cohomology defined through Bloch's cycle complexes (also for $X$ singular): this is why the cohomology defined through Bloch cycle complex should not be called "motivic cohomology" but rather "motivic Borel-Moore homology".

Rationally, $KGL$ is naturally a $H\mathbf Q$-algebra, where $H\mathbf Q$ denotes the $\mathbf Q$-linear motivic Eilenberg-MacLane spectrum. In fact, $KGL$ becomes the free Bott-periodic $H\mathbf Q$-algebra. Furthermore, the category of $H\mathbf Q$-modules in $SH(X)$ is then equivalent to $DM(X,\mathbf Q)$ (the category of motivic sheaves over $X$), and the change of scalars functor from $H\mathbf Q$-modules to $KGL$-modules commutes with the six operations (at least if we restric to compact objects). In particular, one recovers Poincaré duality in $KGL$-modules from the one in motives, but with a "Todd-twist": the classical formulations of Poincaré duality as above involve Thom isomorphisms, which themselves rely on a choice of an orientation. Poincaré duality on homotopy $K$-theory as described above corresponds to the orientation of $KGL$ defined by the multiplicative formal group law, while the one coming from seeing rationalized $KGL$ as a $H\mathbf Q$-algebra corresponds to the additive formal group law. Relating the two through an explicit isomorphism is exactly the purpose of Grothendieck-Riemann-Roch theorems.