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Using the flatification by blow-up result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of descent for the fibered category of étale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination of noetherian...

The purpose of this article is to give, for any (commutative) ring $A$, an explicit minimal set of generators for the ring of multisymmetric functions ${\mathrm{T}S}_A^d\left(A[x_1,\cdots ,x_r]\right)={\left(A{[x_1,\cdots ,x_r]}^{{\otimes }_{A}d}\right)}^{{𝔖}_d}$ as an $A$-algebra. In characteristic zero, i.e. when $A$ is a $\mathbb{Q}$-algebra, a minimal set of generators has been known since the 19^th century. A rather small generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound on the generators, improving the...