Derived Hochschild functors over commutative adic algebras

Let $\k$ be a commutative ring, and let $(A,\mfrak{a})$ be an adic ring which is a $\k$-algebra. We study complete and torsion versions of the derived Hochschild homology and cohomology functors of $A$ over $\k$. To do this, we first establish weak proregularity of certain ideals in flat base changes of noetherian rings. Next, we develop a theory of DG-affine formal schemes, extending the Greenlees-May duality and the MGM equivalence to this setting. Finally, we define complete and torsion derived Hochschild homology and cohomology functors in this setting, and show that if $\k$ is noetherian and $(A,\mfrak{a})$ is essentially of finite type (in the adic sense) over $\k$, then there are formulas to compute them that stay inside the noetherian category. In the classical case, where $\k$ is a field, we deduce that topological Hochschild cohomology and discrete Hochschild cohomology are isomorphic.