4 problems

Wittenberg's descent conjecture. Let $X$ be a smooth $k$-variety and let $G$ be a linear algebraic $k$-group. Let $f:Y\to X$ be a $G$-torsor with $Y$ rationally connected. Assume t…

Let $A,A_1,A_2,A_{12}$ be noetherian $\mathbb{E}_1$-Banach rings in the indicated simplicial ind-Banach categories. Suppose that their homotopy rings fit into a short strictly exac…

Atlas descent conjecture. The following conditions on $F$ are equivalent: (1) $F$ satisfies descent along atlases; (2) $F$ satisfies hyperdescent. In particular, the full subcatego…

Let $(X,\mathcal{A})$ be a compact ringed space with soft structure sheaf $\mathcal{A}$. Let $U_{\bullet}$ be a hypercover of $X$, and let $\mathfrak{L}(X)$ denote the dg-category…