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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 $n$ …

From Section 3 of the paper, we see that the moduli stack of elliptic curves (over a ring $A$) has a quotient stack presentation $$ \mathcal{M}_{1,1,A} \simeq [U/G] $$ where $U = \operatorname{Spec} A …

(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 $ …

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_{ …

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 completeness, the spectral sequence (2.3.14.1) mentioned is \begin{align*} E_{2}^{s,t} = \check{H}^{s}(\mathscr{X},\underline{\mathscr{H}}^{t}(F)) \implies H^{s+t}(C/X,F) \end{align*} By minimalit …

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 …

Let $S,S'$ be schemes, let $\pi : S' \to S$ be a morphism which is faithfully flat and locally of finite presentation, set $S'' := S' \times_{S} S'$ and $S''' := S' \times_{S} S' \times_{S} S'$ with p …

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{ …

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 …

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 …

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 …

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 …

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 $A_{1},A_{2}$ 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}^{\ …