he graded centre of the derived category of a ring or space is aturally a graded commutative ring and target of a universal characteristic" homomorphism from the Hochschild cohomology ing, represented by the Atiyah-Chern character . ne may thus view Hochschild cohomology as an approximation o the centre of the derived category and Grothendieck residues llow to bound the kernel of that map, which is generally neither njective nor surjective. We will survey the very few examples, here the situation is better understood. In terms of noncommutative geometry, the situation can be described hrough the Fourier-Mukai transform afforded by the diagonal embedding of a space and it gives rise to the semiregularity map for algebraic cycles.