Let $M$ be a compact oriented manifold of dimension $n$. It is well-known that there is a perfect intersection pairing $$H_k(M;\mathbb{Z})_{torsion\,\,free}\otimes H_{n-k}(M;\mathbb{Z})_{torsion\,\,free}\rightarrow \mathbb{Z},$$ but there is also a perfect linking pairing $$H_k(M;\mathbb{Z})_{torsion}\otimes H_{n-k-1}(M;\mathbb{Z})_{torsion}\rightarrow \mathbb{Q}/\mathbb{Z}.$$ Is there an algebraic geometry analogue? For instance, etale cohomology is first defined over $\mathbb{Z}_\ell$ then tensored with $\mathbb{Q}_\ell$. It seems plausible that the "right definition" of $\ell$-adic etale cohomology would have $\ell$-torsion and that there would be a linking pairing on the torsion part. The same question applies for crystalline cohomology, or for any Weil cohomology theory. Or is this a feature unique to singular cohomology?
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$\begingroup$ This should be fine, and as you suggest, at least for smooth proper varieties over an algebraically closed field of characteristic $\neq \ell$. You'll have to be careful about Tate twists in your switch from homology to cohomology, though. $\endgroup$– David HansenCommented Oct 25, 2016 at 0:09
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4$\begingroup$ It seems to me that the existence of these two pairings is a formal consequence of universal coefficients formula and Poincare duality for cohomology with integral coefficients. The latter holds for $l$-adic($l\neq p$) and crystalline cohomology of smooth proper varieties. $\endgroup$– SashaPCommented Oct 25, 2016 at 7:05
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