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Let $\mathcal{E}$ be the universal elliptic curve over the moduli stack $\mathcal{M}$ of elliptic curves. As $\mathcal{E}$ is an abelian group scheme over $\mathcal{M}$, we obtain a product-preserving …

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 …

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 $ …

I am looking for examples of Deligne-Mumford stacks whose coarse moduli space is $\mathbb{A}^n$ or at least an open subscheme of $\mathbb{A}^n$ whose complement has codimension $2$. (Thus the whole in …

Recall that for a subgroup $\Gamma \subset SL_2(\mathbb{Z})$ a modular form $f$ of weight $k$ is a holomorphic function from the upper-half plane into the complex numbers such that for any $\begin{p …

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 …

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 …

If you want to learn about stacks, I can recommend 'Fundamental Algebraic Geometry: Grothendieck's FGA Explained'. Vistoli's exposition of the basic theory of stacks is hard to beat, I think. Moreover …

Model categories capture the idea that in many cases you resolve an object by an equivalent object that is better behaved. The standard example is replacing a chain complex by a chain complex of proje …

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 …

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) …

I recently heard a talk about Chow motives and also read Milne's exposition on motives. If I understand it correctly, the naive definition of the Chow ring would be that it simply consists of all alge …

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 …

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 …

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 …