Approximation of sheaves on algebraic stacks

Raynaud--Gruson characterized flat and pure morphisms between affine schemes in terms of projective modules. We give a similar characterization for non-affine morphisms. As an application, we show that every quasi-coherent sheaf is the union of its finitely generated quasi-coherent subsheaves on any quasi-compact and quasi-separated algebraic stack.Comment: 17 pages; generalized result to all quasi-compact and quasi-separated algebraic stacks; added some applications; major revision; final versio

A minimal set of generators for the ring of multisymmetric functions

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 TS^d_A(A[x_1,...,x_r]) as an A-algebra. In characteristic zero, i.e. when A is an algebra over the rational numbers, a minimal set of generators has been known since the 19th 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 degree bound previously obtained by Fleischmann.Comment: 25 pages, to be published in Ann. Inst. Fourie

Submersions and effective descent of etale morphisms

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 effective descent for the fibered category of etale 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 hypotheses in many instances.Comment: 42 pages; added sections on descent of locally closed subsets and weakly normal descent; many other changes; final versio

The telescope conjecture for algebraic stacks

Using Balmer--Favi's generalized idempotents, we establish the telescope conjecture for many algebraic stacks. Along the way, we classify the thick tensor ideals of perfect complexes of stacks.Comment: 20 page

Algebraic groups and compact generation of their derived categories of representations

Let $k$ be a field. We characterize the group schemes $G$ over $k$, not necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in terms of their stabilizer groups.Comment: 17 pages; generalized Theorems A and C; final version; some minor corrections and updated citation

Mayer-Vietoris squares in algebraic geometry

We consider various notions of Mayer--Vietoris squares in algebraic geometry. We use these to generalize a number of gluing and pushout results of Moret-Bailly, Ferrand--Raynaud, Joyet and Bhatt. An important intermediate step is Gabber's rigidity theorem for henselian pairs, which our methods give a new proof of.Comment: 26 page