# Poincaré duality for motivic cohomology

Is Poincaré duality for étale motivic cohomology known for projective regular (not necessarily smooth, just regular) schemes over Dedekind rings?

More precisely, two questions. Let $f: \mathcal{X}\to\text{Spec}(R)$ be a regular projective morphism of pure relative dimension $d$, with $R$ Dedekind, and let $\mathbf{Z}(j)_{\mathcal{X}}$ be the Bloch cycle complex.

(1.1) Is there a perfect pairing:

$$H^i(\mathcal{X}_{\rm et}, \mathbf{Z}(j))\times H^{2d-i}(\mathcal{X}_{\rm et}, \mathbf{Z}(d-j))\to H^{2d}(\mathcal{X}_{\rm et},\mathbf{Z}(d))\simeq \mathbf{Z}\ ?$$

For $f$ smooth projective, same question.

(1.2) What if one replaces $H^{2d-i}(\mathcal{X}_{\rm et},\mathbf{Z}(j))$ with Borel-Moore homology in the appropriate degree and weight?

(2.1) Is there always a multiplication map

$$\mathbf{Z}(i)\otimes^{L}_{\mathbf{Z}}\mathbf{Z}(j)\to\mathbf{Z}(i+j)$$ and is it an isomorphism in $D(\text{Ab})$?

(2.2) Can one relate the two objects in $D(\text{Ab})$: $$\text{Hom}_{D(\text{Ab})}(\mathbf{Z}(i),\mathbf{Z}[p])\ \ \text{and}\ \ \mathbf{Z}(-i)[2d-p]$$ (with a different shift, if necessary)?

Actual scope:

More generally, I'd be interested in any result allowing a description of $$\text{Hom}_{\mathbf{Z}}(H^p(\mathcal{X}_{\rm et},\mathbf{Z}(i)),\mathbf{Z})$$ in terms of the other étale motivic cohomology groups of $\mathcal{X}$ (eg. as a direct sum of some of them), on the condition that such result is valid for $\mathcal{X}$ over a Dedekind base.

I would benefit a lot from a precise reference.

• I belive that there cannot be any result of this sort for a general $X$ (say, over a field). In particular, the group of codimension $0$ cycles is "very small up to torsion", whereas the group of dimension zero cycles if "usually huge". – Mikhail Bondarko Dec 20 '17 at 21:27
• @Mikhail Bondarko I've slightly edited the question to clarify the scope. – user92332 Dec 20 '17 at 21:42
• Regarding 2.1: there is such a multiplication if X is smooth over R, by recent work of Spitzweck (arxiv.org/abs/1207.4078). I believe this remains an open conjecture for more general X (see section 11 in uni-due.de/~bm0032/publ/KthyMotI12.01.pdf). But this map fails to be an isomorphism already over a field. – Marc Hoyois Dec 21 '17 at 1:29
• A significan part of the motivic cohomology groups are "very big"; it follows that their duals are "really huge". – Mikhail Bondarko Dec 23 '17 at 17:56