Algebraic and Arithmetic Geometry Seminar

On the canonical, fpqc, and finite topologies: the state of the art

by Luisa Fiorot (Università di Padova)

Google Meet

Google Meet


This talk is based on joint work with Yves André dedicated to the study of the behaviour of ?finite coverings of (affine) schemes with regard to two Grothendieck topologies: the fpqc topology and the canonical topology (i.e., the ?finest topology for which all representable presheaves are sheaves).

After a short introduction to Grothendieck topologies, we shall focus on the category of affine schemes and present our reconstruction of Olivier's proof that the canonical topology coincides with the effective descent topology: covering maps are given by universally injective ring maps. Grothendieck proved that the fpqc topology is coarser than the canonical topology. Actually, it is strictly coarser: we propose new examples of finite coverings which separate the canonical, fpqc and fppf topologies. The key result is that finite coverings of regular schemes are coverings for the canonical topology, and even for the fpqc topology (but not necessarily for the fppf topology).

"Splinters" are those affine Noetherian schemes for which every finite covering is a covering for the canonical topology. Time permitting, we present in geometric terms the state of the art about them.