of schemes of finite type over a Noetherian base scheme S. Indeed, a morphism in is universally subtrusive if and only if it is universally submersive . In other words, More generally, on the category of all qcqs schemes, neither of the v- nor the h- topologies are finer than the other: and . There are v-covers which are not h-covers (e.g., ) and h-covers which are not v-covers (e.g., where R is a valuation ring of rank 2 and is the non-open, non-closed prime ). However, we could define an h-analogue of the fpqc topology by saying that an hqc-covering is a family such that for each affine open there exists a finite set K, a map and affine opens such that is universally submersive (with no finiteness conditions). Then every v-covering is an hqc-covering. Indeed, any subtrusive morphism is submersive (this is an easy exercise using ). By a theorem of Rydh, for a map of qcqs schemes with Noetherian, is a v-cover if and only if it is an arc-cover (for the statement in this form see ). That is, in the Noetherian setting everything said above for the v-topology is valid for the arc-topology. Notes References Suslin, A., and Voevodsky, V., Relative cycles and Chow sheaves, April 1994, . Algebraic geometry