Search Results

Let $S$ be a scheme, let $T$ be an $S$-scheme, and let $M$ be a set. Let $M_{S}$ be the disjoint union of $M$ copies of $S$, considered as an $S$-scheme. (Notation from [SGA 3, Exp. I, 1.8].) Then $S$ …

For any scheme $X$, let $\operatorname{Br}X$ denote the (Azumaya) Brauer group of $X$, namely the Morita equivalence classes of Azumaya $\mathcal{O}_{X}$-algebras. Is the functor $$\operatorname{B …

Let $\mathscr{M}_{1,1,\mathbb{Z}}$ denote the moduli stack of elliptic curves. Does there exist a scheme $X$ and a finite group $G$ acting on $X$ such that $\mathscr{M}_{1,1,\mathbb{Z}}$ is isomor …

Let $A$ be a finitely generated $\mathbb{Z}$-algebra. Is $\operatorname{Pic}(A)$ finitely generated (as an abelian group)? Thoughts: We may assume that $A$ is reduced since $\operatorname{Pic}( …

Let $(A,\mathfrak{m})$ be a local ring, and let $A^{\mathrm{sh}}$ be the strict henselization of $A$ at $\mathfrak{m}$. Let me denote $A^{\mathrm{sh},\mathrm{fin}}$ for the filtered colimit of fini …

Let $X$ be a quasi-compact scheme. Say that $X$ has property $\mathbf{P}_{n}$ if $X$ admits an open cover $X = \bigcup_{i=1}^{n} U_{i}$ such that each $U_{i}$ is affine and each pairwise intersection …

Let $k$ be an algebraically closed field. Let $X$ be an integral $k$-scheme, separated and of finite type over $k$. Let $d := \dim X$, let $T := (\mathbb{G}_{m,k})^{d}$ be the $d$-dimensional torus, a …

Consider the commutative diagram \begin{array}{ccc} \operatorname{Br}k & \xrightarrow{f_1} & \operatorname{Br} \mathbb{P}_{k}^{n} \\ \scriptsize{f_2}\ \downarrow & \swarrow \scriptsize{f_3}& \downarr …

Yes, the point is that $\mathbb{P}_{k}^{1} \times \mathbb{P}_{k}^{1}$ is regular of dimension at most 2. Extend $E$ to a coherent sheaf $E'$ on $\mathbb{P}_{k}^{1} \times \mathbb{P}_{k}^{1}$, then tak …

(For $i=0$, the map $H_{\mathrm{et}}^{0}(\operatorname{Spec} A,\mathbb{G}_{m}) \to H_{\mathrm{et}}^{0}(\operatorname{Spec} A[t],\mathbb{G}_{m})$ is an isomorphism if and only if $A$ is reduced.) For $ …

Here are some thoughts in the case of gluing a DVR along an automorphism of its fraction field: Setup: Let $A$ be a DVR with uniformizer $\pi$ and fraction field $K$, and let $\varphi : K \to K$ be a …

Let $A_{1},A_{2}$ be discrete valuation rings whose fraction fields are isomorphic. Let $A_{i}^{\mathrm{sh}}$ be the strict henselization of $A_{i}$, and let $K_{i}$ be the fraction field of $A_{i}^{\ …

This is a question of DeMeyer (see the last paragraph of [1]): What's an example of a local ring $A$ with residue field $k$ such that the restriction map on Brauer groups $\varphi : \operatorname{ …

Here are some positive results and counterexamples for etale cohomology. Definition: Let $X$ be a scheme. Say that $X$ has property "$AF_{n}$" if for every collection $x_{1},\dotsc,x_{n} \in X$ of $n$ …

Let $A$ be a ring. Is the sequence \begin{align} \textstyle A \to \prod_{\mathfrak{p}} A_{\mathfrak{p}} \rightrightarrows \prod_{\mathfrak{p}_{1},\mathfrak{p}_{2}} A_{\mathfrak{p}_{1}} \otimes_{A} A_{ …