Extension Groups of Tautological Sheaves on Hilbert Schemes of Points on Surfaces

In this thesis cohomological invariants of tensor products of tautological objects in the derived category of Hilbert schemes of points on surfaces are studied. The main tool is the Bridgeland-King-Reid-Haiman equivalence between the derived category of the Hilbert scheme and the equivariant derived category of the cartesian power of the surface. The work of Scala on this topic is further developed leading to a new description of the image of tensor products of tautological bundles under the BKRH equivalence. This description leads to formulas for the Euler characteristics of triple tensor products of tautological objects for arbitrary n and for arbitrary tensor products in the case n=2. Furthermore a formula for the extension groups between tautological objects is proven and the Yoneda product is described.