On continuous and differential Hochschild cohomology (Q1283354)

For \(A\), the algebra of smooth complex functions over a manifold \(M\), and \(V^*\) the space of skew multivector fields over \(M\), the author constructs canonical isomorphisms between Hochschild cohomology spaces \(H^*_{\text{cont}}(A,A)=H^*_{\text{diff}}(A,A)=V^*\). So, continuous and differential deformation theories of \(A\) coincide. The author, after some classical facts about Hochschild cohomology of commutative algebras, constructs the Koszul resolution of \(A\), embedded in the bar resolution, when \(M\) is an open subset \(U\) of \(\mathbb{R}^n\) and proves the topological part and the differential case of the main theorem. After developing the techniques of restriction and extension of cochains from one manifold to another the general case of the main result is proved.