Related papers: G-torsors over a Dedekind scheme

Girard's Theorem subjects to the area depending interior angles of a spherical triangle. In this paper, we introduce to its analogues for proper de Sitter triangles with non-null edges.

In this paper we prove formal glueing along an arbitrary closed substack $Z$ of an arbitrary Artin stack $X$ (locally of finite type over a field $k$), for the stacks of (almost) perfect complexes , and of $G$-bundles on $X$ (for $G$ a…

We establish a duality between flat affine group schemes and rigid tensor categories equipped with a neutral fiber functor (called Tannakian lattice), both defined over a Dedekind ring. We use this duality and the known Tannakian duality…

Deformations of ordinary varieties of K3 type can be described in terms of displays by recent work of Langer-Zink. We extend this to the general (non-ordinary) case using displays with $G$-structure for a reductive group $G$. As a basis we…

Let $k$ be a field of characteristic $0$ and let $K = k(B)$ be the function field of a geometrically irreducible projective curve $B$ over $k$. Let $A/K$ be a $g$-dimensional abelian variety with $\mathrm{Tr}_{K/k}(A) = 0$. We prove that…

We define a linear structure on Grothendieck's arithmetic fundamental group $\pi_1(X, x)$ of a scheme $X$ defined over a field $k$ of characteristic 0. It allows us to link the existence of sections of the Galois group ${\rm Gal}(\bar k/k)$…

The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. We settle it in the case when $G$ is quasi-split and $R$ is unramified. Some of…

We construct the \'etale motivic Borel-Moore homology of derived Artin stacks. Using a derived version of the intrinsic normal cone, we construct fundamental classes of quasi-smooth derived Artin stacks and demonstrate functoriality, base…

In this article, we give a proof for a geometric presentation theorem for any irreducible scheme $X$ smooth projective over a discrete valuation ring $R$. As a consequence, for any reductive $R$-group scheme $\mathbf{G}$, we prove that any…

We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction…

Let $\mathcal{G}$ be a fusion category acting on a triangulated category $\mathcal{D}$, in the sense that $\mathcal{D}$ is a $\mathcal{G}$-module category. Our motivation example is fusion-weighted species, which is essentially Heng's…

We prove that for any simply connected isotropic reductive group G over a Dedekind domain D, any Zariski-locally trivial principal G-bundle over D is trivial. The corresponding result for quasi-split groups was proved in 1967 by G. Harder.

Let $G$, $G_1$ and $G_2$ be quasi-finite and flat group schemes over a complete discrete valuation ring $R$, $\varphi_1:G\to G_1$ any morphism of $R$-group schemes and $\varphi_2:G\to G_2$ a model map. We construct the pushout $P$ of $G_1$…

Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. We classify the groupoid fibrations over log schemes that arise in this manner in terms of a categorical…

We show that the category of principal ordered face structures is equivalent to the category of multitopes. We show that the category of principal ordered face structures is equivalent to the category of multitopes. On the way we introduce…

For an effective Cartier divisor D on a scheme X we may form an nth root stack. Its derived category is known to have a semiorthogonal decomposition with components given by D and X. We show that this decomposition is 2n-periodic. For n=2…

We prove a K-theoretic excess intersection formula for derived Artin stacks. When restricted to classical schemes, it gives a refinement and new proof of R. Thomason's formula.

We prove the existence theorem for basic elements in the quasi-projective case, extending results of Eisenbud-Evans and Bruns from the affine case. We give several geometric applications. For example, we show that every local complete…

The paper discusses stably trivial torsors for spin and orthogonal groups over smooth affine schemes over infinite perfect fields of characteristic unequal to 2. We give a complete description of all the invariants relevant for the…

This is an introduction to Grothendieck's descent theory, with some stress on the general machinery of fibered categories and stacks.