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30

I am surprised that nobody has mentioned (yet) that an essential point of the algebro-geometric notions of (algebraic) stack and algebraic space is to completely shift the burden of construction problems: one gives up on trying to make any kind of actual ringed space at all, and in fact the whole point is to create a kind of "geometry for functors". In ...


26

To dash off a quick answer, Pursuing Stacks is composed of (if memory serves correctly) three themes. The first was homotopy types as higher (non-strict) groupoids. This part was first considered in Grothendieck's letters to Larry Breen from 1975, and is mostly contained in the letter to Quillen which makes up the first part of PS (about 12 pages or so). ...


26

I'll use the definition of stack as a (weak) functor from the category of schemes to that of groupoids (as opposed to the definition as a fibered category over the category of schemes). The prestack associated to the action of $GL_n$ on $\mathbb A_n$ is, by definition, given by $$ X \mapsto \left\{\begin{matrix}\text{Objects: maps $s:X \to \mathbb A^n$}\...


22

At least when the group $G$ is discrete, and when the base is a topological space (as opposed to e.g. a scheme), I would like to advertise the fact that: a $G$-gerbe is the same thing as a fibration whose fiber is $BG$ (the latter is the classifying space of $G$, also known as $K(G,1)$). As an example, by taking $G=\mathbb Z$, we learn that an $S^1$-...


21

For your "special" question, the answer is negative, already when $A$ is an elliptic curve. In fact, a principal $A$-bundle over a smooth projective curve $B$ which is not topologically trivial is never algebraic — see the book by Barth, Hulek, Peters, Van de Ven, ch. V, Proposition 5.3. There are many examples of this situation, for instance Hopf surfaces $(...


19

Here is a simple way to talk about the homotopy type of a stack. Let $\mathfrak{X}$ be a stack and $f: U \to \mathfrak{X}$ a representable surjective submersion (an atlas) from a space $U$ (e.g., the map from the Teichmuller space to the moduli stack.) Now, form the pullback of $f$ along itself: $U\times_{\mathfrak{X}} U$. This comes with two maps to $U$ ...


19

I'm not sure why you're replacing $G$ with $K$. So let me compare $[*/G]$ and $B(G^{an})$. As you say, for the purposes of homotopy theory, $B(G^{an})$ is as good as $BK$. One way to understand a stack $\mathcal{X}$ is via a hypercover, which is a simplicial scheme. The easiest way to get a hypercover is to choose an atlas $U\to \mathcal{X}$ and to ...


19

To answer question 2, the best example I know is $\mathscr{M}_1$, the stack of (proper smooth geom. connected) curves of genus 1. Indeed, Raynaud has contructed an elliptic curve $E\to S$ over a scheme $S$ and an $E$-torsor $X\to S$ which is (an algebraic space but) not a scheme. This implies two things. First, in order to define $\mathscr{M}_1$ we are ...


18

Consider an arbitrary site (or an ∞-site) S. In fact, the constructions below only depend on the underlying topos (or ∞-topos) T of S, and not on S itself. Below “sheaf”, “∞-sheaf”, “stack”, and “∞-stack” are all synonyms for presheaves (of spaces) that satisfy homotopy descent. The nth Weiss topology (n≥0 or n=∞) on T is defined by declaring a family {U_i→...


18

I agree with Tim that calling Pursuing Stacks a "letter to Quillen" is erroneous, especially as Quillen never replied. Grothendieck also wrote: "This is written in English in response to a correspondence in English." At one stage he planned more volumes in French but it seems got diverted from this. I hope the following will be of help, in addition to ...


18

You can get a lot of examples by dimension shifting. Namely, consider any exact sequence of groups $$1\to K\to G \to H\to 1 \; .$$ Fix a $H$-torsor $T$. The stack $\mathcal G_T$ of liftings of the structure group of $T$ to $G$ is clearly a gerbe (the objects are pairs $(T',\alpha)$, where $T'$ is a $G$-torsor and $\alpha : T'\times^G H \simeq T$ is an ...


18

I think the separated condition should be included, at least for purposes of sheaf cohomology. First let's consider the case where Y is a point. if X is a compact Hausdorff space, then sheaf cohomology over X commutes with filtered colimits. This is a nice property which fails if X is non-Hausdorff. For example, take X to be two copies of [0,1] glued ...


17

Probably you want your functor $\varphi$ to restrict to the "identity functor" on the full subcategory of algebraic spaces that are schemes (which doesn't seem to be a purely formal consequence of your hypotheses). And to be "reasonable" you likely want such a hypothetical functor $\varphi$ to carry open immersions to open immersions and etale maps to flat ...


17

A canonical example of a sheaf of sets on a topological space $X$ is the sheaf that sends an open subset $U$ of $X$ to the set of continuous real-valued functions on $U$. The gluing property then says that a continuous function on a union of open subsets $U_i$ of $X$ is the same thing as a collection of continuous functions $f_i: U_i \to \mathbb{R}$ such ...


16

The category of maps from a test object $T$ to a quotient stack $[X/G]$ has the following general form. Objects are pairs $(P, f)$, where $P$ is a $G$-torsor over $T$, and $f: P \to X$ is a $G$-equivariant map. Morphisms $(P,f) \to (P',f')$ are torsor isomorphisms $g: P \to P'$ satisfying $f = f' g$. Here, $X$ is the vector representation $\mathbf{O}^n$, ...


16

The two constructions are not quite equivalent. Let me write $\mathbf BG$ for the stack and $B_\bullet G$ for the simplicial scheme to better distinguish between them. There is a third relevant player, $BG$, which is the presheaf of ∞-groupoids on $C$ presented by $B_\bullet G$. The precise relation between these three objects is the following: $\mathbf BG$...


15

This is more a comment than an answer, but its length makes me post it as an answer. I want to react to what I have just read, for the first time, about "Pursuing Stacks" at the nLab, and the words used there as well as in your question. I find it extremely irritating when people only use words such as "ideas" and "conjectures" to describe the content of "...


15

Short answer: yes. As I recall, Teleman constructs such a spectral sequence for fairly general stacks. You can look at his paper The quantization conjecture revisited Annals 2000. But this may fall into the category of "more generality than you need". The case you want can probably done by hand. The point is if $G$ is a finite group acting on a complex ...


14

Tannaka duality makes two complementary statements: first, that an affine algebraic group can be recovered from its category of representations (the "reconstruction problem"), and second, that certain categories are the categories of representations of an affine algebraic group (the "recognition problem"). Lurie's paper generalizes the first statement, but ...


13

Your $F_N$ is the functor people would usually mean when they talk about the functor classifying elliptic curves with full level N structure (though it's a bit nicer if you replace $(Z/nZ )^2$ with $\mu_n \times Z/nZ$, so that the determinant takes values in $\mu_n$ on both sides.) Wikipedia is not wrong. (Wikipedia is surprisingly seldom wrong!) Your F_2 ...


13

Let's work over the complex numbers for simplicity. Let $X$ be a Riemann surface, fix a base point $x \in X$, and let $\Gamma$ be the fundamental group $\pi_{1}(X,x)$. The data of an $n$-dimensional local system on $X$, together with a trivialization at the point $x$, is equivalent to the data of an $n$-dimensional representation of $\Gamma$. This is true ...


12

Suppose that $\mathcal X$ is an algebraic stack with finite inertia (for example, a separated Deligne-Mumford stack); then, by a well-known result of Keel and Mori, there exist a moduli space $\pi \colon \mathcal X \to M$. The stack $\mathcal X$ is called tame when $\mathrm R^i\pi_* F = 0$ for every quasi-coherent sheaf $F$ on $\mathcal X$ and every $i > ...


12

It is certainly not true that $\mathcal X \to \mathcal X^H$ is a good moduli morphism, unless $H$ is linearly reductive, because when you push forward the cohomology of $H$ will come into play. On the other hand $\mathcal X^H \to X$ is a good moduli space, because the pushforward $QCoh(\mathcal X^H) \to QCoh(X)$ can be factored as the pullback $QCoh(\...


12

This would be the "comparison lemma" from SGA 4-1, Éxposé III, Thm. 4.1: if $C$ is a full subcategory of a site $D$, equipped with the induced topology, and if every object of $D$ is covered by objects of $C$, then the restriction functor $Shv(D)\to Shv(C)$ is an equivalence of categories. More generally, for any finite $n$, the $n$-topoi of $n$-stacks on $...


12

Yes, Calabi-Yau manifolds have unobstructed deformations. This is due to Tian and Todorov; there is a nice algebraic proof in a paper by Kawamata, J. Algebraic Geom. 1 (1992), no. 2, 183–190.


12

In modern language, one would say that $D_{qcoh}(-)$ is a sheaf of $(\infty,1)$-categories on the scheme $X$ (so "homotopy stack" = "sheaf of $(\infty,1)$-categories"). If $X$ is affine, or more generally has an ample family of line bundles, the perfect complexes on $X$ are exactly the finitely presented objects (aka compact objects) in $D_{qcoh}(X)$: this ...


12

Often you hear the informal statement that a stack is like a scheme, except with a stabilizer group attached to each point. Your question shows why the intuition from this statement can be misleading. In a sense, you need to also remember the way these groups fit together. But if you try to make that statement precise you'll soon find yourself completely ...


12

Let's start approaching the question from the simplest possible case $Y=*$. What should be the points of $[X/G]$? Recall that the idea here is to generalize the construction of the action groupoid for discrete groups acting on sets to the manifold case. This allows us to remember the stabilizers of points and it is generally a much better behaved notion. ...


11

Here are two applications of stacks to number theory. 1) Section 3 of this paper, which solves the diophantine equation $x^2 + y^3 = z^7$, explains the connection between stacks and generalized Fermat equations. 2) This post explains how stacks fit into the proof of Deuring's formula for the number of supersingular elliptic curves over a finite field.


11

I am not sure if you are only really interested in properly stacky things, but it is perhaps worth pointing out that the result you mentioned from Hartshorne is true in significantly greater generality. For any quasi-compact quasi-separated scheme $X$ (in fact for any spectral space $X$, or for something even slightly weaker) and any filtered system $(\...


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