Inaccessibility of the flat topology

Abstract

We prove that singleton covers in the flat topology on affine schemes are not closed under $\kappa$-cofiltered limits for any regular cardinal $\kappa$. Therefore, for every accessible flat sheaf there exists a strictly finer topology for which it is still a sheaf. The flat topology thus contrasts with other big topologies, such as the arc and pure topologies.

arXiv comment2 pages