Explicit versions of the local duality theorem in $\mathbb{C}^n$

We consider versions of the local duality theorem in $\mathbb{C}^n$. We show that there exist canonical pairings in these versions of the duality theorem which can be expressed explicitly in terms of residues of Grothendieck, or in terms of residue currents of Coleff-Herrera and Andersson-Wulcan, and we give several different proofs of non-degeneracy of the pairings. One of the proofs of non-degeneracy uses the theory of linkage, and conversely, we can use the non-degeneracy to obtain results about linkage for modules. We also discuss a variant of such pairings based on residues considered by Passare, Lejeune-Jalabert and Lundqvist.