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The following is a special case of Example 3 in Laurent Moret-Bailly's answer here. Let $P$ be the set of primes of $\mathbb{Z}$ and let $S \subseteq P$ be an infinite subset, let $X$ be the gluing o …

We say that a field $K$ is $C_{m}$ if it satisfies the following property: for every positive integer $n$ and every sequence of positive integers $(d_{1},\dotsc,d_{r})$ satisfying $d_{1}^{m} + \dotsb …

Let $X$ be a scheme, let $i : Z \to X$ be a closed subscheme, let $Y := \mathrm{Bl}_{Z}(X)$ be the blowup of $X$ at $Z$ with projection $\pi : Y \to X$. Suppose $U \supseteq X \setminus Z$ is an op …

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 $S$ be a connected scheme, let $\pi : \mathbb{P}_{S}^{r} \to S$ be projective $r$-space over $S$, and let $\mathcal{E}$ be a flat and locally finitely presented $\mathcal{O}_{\mathbb{P}_{S}^{r} …

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}^{\ …

Let $S$ be a normal Noetherian scheme, let $U$ be an open subset whose complement has codimension at least $2$, and let $j : U \to S$ be the inclusion. By e.g. SP Tag 0EBJ, the restriction and pushfor …

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 $f : X \to Y$ be an fpqc morphism of schemes, and let $\mathcal{G}$ be an $\mathcal{O}_{Y}$-module (on the small Zariski site) such that $f^{\ast}\mathcal{G}$ is quasi-coherent. Is $\mathcal{G} …

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 $\mathrm{M}$ be a finitely generated submonoid of $\mathbb{Z}^{\oplus d}$ for some $d$, let $A := k[\mathrm{M}]$ be the associated monoid algebra over a field $k$, let $\mathfrak{m} \subset A$ …

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_{ …

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 …

Part (i) of Gabber's "Lemma K" was generalized to quasi-compact quasi-separated schemes by Tabuada, van den Bergh in Theorem 2.3 of Noncommutative motives of Azumaya algebras: Let $X$ be a quasi-comp …

Here are some modern references: [1, Theorem 13.1.15] and [2, Lemma 2.1] [1] Olsson, "Algebraic Spaces and Stacks", Colloquium Publications 62, AMS (2016) [2] Fulton, Olsson, "The Picard group of $\ …