52 research outputs found
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 be a field. We characterize the group schemes over , not
necessarily affine, such that 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
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