For integral homology groups there is the notion of linking form (<http://www.map.mpim-bonn.mpg.de/Linking_form>)
$$
Tor(H\_{l}(X,\mathbb{Z}))\times Tor(H\_{n-l-1}(X,\mathbb{Z}))\rightarrow \mathbb{Q}/\mathbb{Z}
$$
for the torsion part of the homology groups. This can be defined by using the Bockstein map associated with the sequence $\mathbb{Z}\rightarrow \mathbb{Q}\rightarrow \mathbb{Q}/\mathbb{Z}$.


Maybe this is a very trivial generalization, but is there an analogous notion for homology groups with coefficients in an arbitrary abelian group $A$? I am looking for something like
$$
Tor(H\_{l}(X,A))\times Tor(H\_{n-l-1}(X,A^{\vee}))\rightarrow \mathbb{Q}/\mathbb{Z}
$$
where $A^{\vee}$ is the dual group. If yes, what is the Bockstein map one should use?



Linking form for homology with general coefficients
Expanding on Ryan's comment, one way to get the torsion linking form you mention (at least for compact oriented manifolds) is through the sequence of isomorphisms $$Tor(H\_l(X;\mathbb{Z}))\cong Tor(H^{n-l}(X;\mathbb{Z})) \cong Ext(H\_{n-l-1}(X);\mathbb{Z})\cong Ext(Tor(H\_{n-l-1}(X));\mathbb{Z})\cong Hom(Tor(H\_{n-l-1}(X));\mathbb{Q}/\mathbb{Z}) .$$
The first isomorphism is Poincar'e duality, and the second is the universal coefficient theorem (using that the Ext summand corresponds to the torsion subgroup). The third is a basic property of Ext (write the homology group as a sum of a torsion group and a free group - this uses that the homology is finitely generated from the compactness assumption). The final can be seen to come from the short exact sequence you mention: take the Hom/Ext exact sequence and note that $Ext(-,\mathbb{Q})=0$ always and $Hom(A,\mathbb{Z})=0$ when $A$ is torsion.


So all this can be generalized in circumstances in which these properties all hold. For example, if $R$ is a Dedekind ring and $X$ is a compact $R$-orientable manifold, I believe everything should go through to give a pairing $Tor(H\_l(X;R))\otimes Tor(H\_{n-l-1}(X;R))\to Q(R)/R$, where $Q(R)$ is the field of quotients of $R$.


Offhand, I'm not sure about a version in which the coefficients are modules over $R$ (in your original question, modules over $\mathbb{Z}$). One would have to think through in what cases versions of these isomorphisms still hold.


\*Note: You don't mention manifolds explicitly, but the site you link to for your basic definitions does assume the context of closed, oriented manifolds.


\*\*It does take a little work to show that this definition of the linking form is equivalent to the one using the Bockstein.


