arXiv subjects
Publications and source records attributed to Mohammed Moutand.
Let $(X,\bar x)$ be a pointed connected noetherian scheme. In this note, we give characterizations for the vanishing of the second étale homotopy group $π^{\rm ét}_2(X,\bar x)$ in terms of splitting profinite-étale covers of $X$, and by means of universal covering spaces of the Artin-Mazur-Friedlander étale homotopy type $Et(X)$. In particular, this provides certain classes of schemes for which the Brauer map is surjective.
Let $X$ be a smooth variety over a field $K$ with function field $K(X)$. Using the interpretation of the torsion part of the étale cohomology group $H_{\text{ét}}^2(K(X), \mathbb{G}_m)$ in terms of Milnor-Quillen algebraic $K$-group $K_2(K(X))$, we prove that under mild conditions on the norm maps along unramified extensions of $K(X)$ over $X$, there exist cohomological Brauer classes in $H_{\text{ét}}^2(X, \mathbb{G}_m)$ that are representable by Azumaya algebras on $X$. Theses conditions are almost satisfied in the case of number fields, providing then, a partial answer on a question of Grothendieck.
Extending a result of Schröer on a Grothendieck question in the context of complex analytic spaces, we prove that the surjectivity of the Brauer map $δ: Br(X) \rightarrow H_{\rm ét}^2(X,\mathbb{G}_{m, X})_{\rm tor}$ for algebraic schemes depends on their étale homotopy type. We use properties of algebraic $K(π, 1)$ spaces to apply this to some classes of proper and smooth algebraic schemes. In particular we recover a result of Hoobler and Berkovich for abelian varieties. Further, we give an additional condition for the surjectivity of $δ$ which involves pro-universal covers. All proposed conditions turn out to be equivalent for smooth quasi-projective varieties.