Fabien Morel, LMU, Munich and IAS

Abstract: In this talk I will discuss some computations related to the study A^1-connected smooth projective schemes X over a fixed field k (mostly perfect). For a smooth projective k-scheme, A^1-connected means A^1-chain connected, and rational smooth projective k-schemes are A^1-connected on nice field (char 0 for instance) as we observed some time ago with A. Asok. After briefly introducing the cellular A^1-homology sheaves (with Anand Sawant) I will explain why these are good for, for instance by understanding the top dimensional one. I will explain some of the basic facts and difficulties involved concerning strictly A^1-invariant sheaves, and give some examples and computations of those. In particular I will explain the case of smooth projective rational surfaces, where Poincare' duality holds (over a perfect field).