Hochschild cohomology and adic completion

Hochschild cohomology is the prominent cohomology theory for associative algebras. In this talk we study relations between the Hochschild cohomology modules of a commutative algebra A, and the a-adic completion operation, for an ideal a in A. We will first recall what is Hochschild (co)-homology and explain its importance, then discuss some basic results about the derived completion and derived torsion functors, and finally apply these results to the noetherian case, and deduce that Hochschild cohomology commutes with adic completion.