Related papers: The \'etale local structure of algebraic stacks
The central objective of this article is to provide an elementary proof of the following theorem, of which we are unaware of any trace in the existing literature. If $B$ is a net finite free algebra over a commutative ring $A$, then it is…
We formulate and prove a non-abelian analog of Deligne's Fixed Part theorem on Hodge classes, revisiting previous work of Jost--Zuo, Katzarkov--Pantev and Landesman--Litt. To this aim we study algebraically isomonodromic extensions of local…
The purpose of this paper is to lay the foundations of a theory of invariants in \'etale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and we use the theory we developed to get…
The paper proves that if a reductive group scheme acts properly on a scheme then the geometric quotient exists as an algebraic space. As a consequence we obtain the existence of the moduli spcace of canonically polarized varieties over Spec…
We provide necessary and sufficient conditions for when an algebraic stack admits a good moduli space and prove a semistable reduction theorem for points of algebraic stacks equipped with a $\Theta$-stratification. These results provide a…
We develop the theory of algebraic groups over real closed fields and apply the results to construct a geometric object $\mathcal{B}$ and to prove that $\mathcal{B}$ is an affine $\Lambda$-building. We use a model theoretic transfer…
We determine the structure of all finite-dimensional self-injective algebras over a field whose Auslander-Reiten quiver admits a hereditary stable slice.
In this paper, we study quotients of groupoids and coarse moduli spaces of stacks in a general setting. Geometric quotients are not always categorical, but we present a natural topological condition under which a geometric quotient is…
We prove an analogue for Stokes torsors of Deligne's skeleton conjecture and deduce from it the representability of the functor of relative Stokes torsors by an affine scheme of finite type over C. This provides, in characteristic 0, a…
Given a compact Lie group $G$ acting on a space $X$, the classical Atiyah-Segal completion theorem identifies topological $K$-theory of the homotopy quotient $X/G$ with an explicit completion of $G$-equivariant topological $K$-theory of…
We prove Artin's axioms satisfy a compatibility for composition of 1-morphisms of stacks in groupoids. Consequently, some natural stacks in groupoids are algebraic, including a common generalization of Vistoli's Hilbert stack and the stack…
This note extends some recent results on the derived category of a geometric invariant theory quotient to the setting of derived algebraic geometry. Our main result is a structure theorem for the derived category of a derived local quotient…
We introduce a Grothendieck group of algebraic stacks (with affine stabilisers) analogous to the Grothendieck group of algebraic varieties. We then identify it with a certain localisation of the Grothendieck group of algebraic varieties.…
Given a quotient of a regular noetherian separated algebraic space $X$ over a field by an affine algebraic group $G$ having finite stabilizers (with some mild technical conditions), G. Vezzosi and A. Vistoli defined the geometric part of…
We consider the stack of coherent algebras with proper support, a moduli problem generalizing Alexeev and Knutson's stack of branchvarieties to the case of an Artin stack. The main results are proofs of the existence of Quot and Hom spaces…
In this article, we give a construction of the (un-)stable motivic homotopy category of an algebraic stack in the spirit of Morel-Voevodsky. We prove that this new construction agrees with the stable motivic homotopy category defined by…
Recent results in geometric invariant theory (GIT) for non-reductive linear algebraic group actions allow us to stratify quotient stacks of the form [X/H], where X is a projective scheme and H is a linear algebraic group with internally…
Algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real or quaternionic structure, it is natural to ask for the properties of the groups of real or…
We prove a myriad of results related to the stabilizer in an algebraic group $G$ of a generic vector in a representation $V$ of $G$ over an algebraically closed field $k$. Our results are on the level of group schemes, which carries more…
We prove the generic base change theorem for stacks, and give an exposition on the lisse-analytic topos of complex analytic stacks, proving some comparison theorems between various derived categories of complex analytic stacks. This enables…