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Let $X$ be a weighted projective (stacky) line. At least if we work over a field $K$, the Picard group of $X$ is $\mathbb{Z}$. Only powers of one of the two generators have global sections. Call this …

As is well known, there is classification of line bundles on the moduli stack of elliptic curves over a nearly arbitrary base scheme in the paper The Picard group of $M_{1,1}$ by Fulton and Olsson: ev …

The following makes probably sense for any site, but I stick for concreteness to the etale one. Let $f: U \to X$ be an etale morphism. As explained, for example, in Remark 8.16 of Milne's Lecture Not …

Q1: I did not find the definition of a smooth stack in HAG2 (it would be nice if you provide a concrete citation!), but the definition of smoothness I know is: A morphism $X \to Y$ of algebraic stacks …

Just to give a few more details on Daniel's comments: In general, $\mathcal{H}om_{\mathcal{O}_X}(\mathcal{M},\mathcal{N})$ is not the associated sheaf to $Hom_A(M,N)$. Simple example: Take $A = \math …

The answer seems to be positive and actually at least in the context of regular (locally) noetherian Deligne--Mumford stacks. (Actually, Artin stack should also be enough as we can compute the Brauer …

First a disclaimer: I am at best a part-time arithmetic geometer, so please accept my apologies when I am too naive or get something wrong. From time to time I have tried to learn something about Sh …

I do not think that your definition of Katz modular form is exactly correct. A (Katz) weight-$k$ modular form for $\Gamma$ is a section of the line bundle on $\mathcal{Y}(\Gamma)_R$ that evaluates on …

To my knowledge the most general known statement has been proven by Clausen and Mathew in their paper Hyperdescent and étale K-theory as Theorem 1.2. The precise conditions on your commutative ring (o …

Given a scheme $X$ with structure sheaf $\mathcal{O}_X$, we can associate to each $\mathcal{O}_X$-module $\mathcal{F}$ its global sections $\Gamma(\mathcal{F})$, which gets the structure of a $\Gamma( …

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$. Assume $k$ has characteristic $p$ and denote by $A(p)$ the $p$-divisible group of dimension $g$ associated with …

In p-adic properties of modular schemes and modular forms Katz formulates the following base change theorem as Theorem 1.7.1 Let $n\geq 3$ and $\overline{\mathcal{M}}_n$ be the compactified moduli …

Katz defines in Section 2.0 $p$-adic properties of modular schemes and modular forms the Hasse invariant as a mod $p$ modular form $A$ of weight $p-1$. In other words, it is a section of $\omega^{\oti …

Given a commutative $\mathbb{Z}[\frac1n]$-algebra $R$, we can consider the ring of modular forms $M_*(\Gamma_1(n), R)$. If $R$ is a subring of $\mathbb{C}$, these can be defined as those (holomorphic) …

It is well known and more or less proven in Hartshorne's 'Algebraic Geometry' (p. 209) that for every noetherian scheme $X$ and every collection of abelian sheaves $\mathcal{F}_i$ the canonical map $ …