Given a ring, you may want also to have a dualizing complex for it. When you have a coalgebra, it has been "dualized" already, being the dual thing to an algebra. So the coalgebra is a dualizing complex over itself; while having a dualizing complex for a ring makes it more like a coalgebra.
Given a coalgebra, though, you may want to "dedualize" it back, which means having a dedualizing complex for it. When you have a ring, it is already a dedualizing complex over itself. Having a dedualizing complex for a coalgebra makes it more like a ring.
The conventional derived category of modules over a ring is tautologically equivalent to itself, but constructing an equivalence between the coderived and contraderived categories of modules over a (Noetherian) ring requires a dualizing complex. This is the Iyengar-Krause covariant version of the Serre-Grothendieck duality.
The coderived category of comodules and the contraderived category of contramodules over a coalgebra are always equivalent to each other. This is called "the co-contra correspondence". But constructing an equivalence between the conventional derived categories of comodules and contramodules over a (co-Noetherian) coalgebra requires a dedualizing complex. This is called the Matlis-Greenlees-May duality.
To obtain a dualizing complex over the ring of functions on an affine variety over a field, one considers the structure sheaf of the spectrum of the field and pulls it back to the variety by the Hartshorne-Deligne extraordinary inverse image functor p^!. To obtain a dedualizing complex over the coalgebra dual to the ring of functions on the formal completion of a variety over a field at a closed point, one considers the structure sheaf of this variety and pulls it back to the formal completion of its closed point by the right derived functor of set-theoretically supported sections Ri^!.
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