Flach–Morin's Artin–Verdier duality conjecture

Let $\mathcal X$ be the regular proper arithmetic scheme under consideration, of dimension $d$, and work on its étale site $\mathcal X_{et}$. For any integer $n$, let $\mathbb{Z}(n)$ denote the motivic complex, let $\widehat{H}^i_c$ denote compact-support cohomology with Tate cohomology at all archimedean places, and let $\mathcal{D}(\mathcal{X}_{et})$ be the derived category of étale sheaves. Flach–Morin's Artin–Verdier duality conjecture $\mathbf{AV}(\mathcal{X},n)$. There is a symmetric product map

$$\mathbb{Z}(n)\otimes^L \mathbb{Z}(d-n)\rightarrow \mathbb{Z}(d)$$

in $\mathcal{D}(\mathcal{X}_{et})$ such that the induced pairing

$$\widehat{H}^{i}_c(\mathcal{X}_{et},\mathbb{Z}/m(n))\times H^{2d+1-i}(\mathcal{X}_{et},\mathbb{Z}/m(d-n))\rightarrow \widehat{H}^{2d+1}_c(\mathcal{X}_{et},\mathbb{Z}/m(d))\rightarrow\mathbb{Q}/\mathbb{Z}$$

is a perfect pairing of finite abelian groups for any $i\in\mathbb{Z}$ and any positive integer $m$. This is the torsion-coefficient Artin–Verdier duality input used in the construction of Weil–étale complexes; its resolution status is not specified in the supplied text.