For any smooth surface S we will introduce an associative algebra acting on the homology of (suitable) moduli spaces (or stacks) of coherent sheaves on S, by elementary modifications at points of S. This algebra includes the algebra of Nakajima operators but also the algebra of multiplication by tautological classes. We will give a presentation of that algebra as well as an application to the proof of the P=W conjecture relating the perverse filtration on the cohomology of moduli spaces of stable Higgs bundles on curves and the weight filtration on the cohomology of character varieties. This is a combination of joint work with T. Hausel, A. Mellit, A. Minets and E. Vasserot.