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In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.

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Let $G$ be a (smooth, connected for convenience) algebraic group acting on a variety $X$ (over a reasonable base scheme). The theory of equivariant perverse sheaves is a functor $\operatorname{Perv}_G(...
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$\newcommand{\G}{\mathbb{G}}\newcommand{\A}{\mathbb{A}}\newcommand{\gr}{\operatorname{gr}}$I'm trying to understand the $\Theta$-stratification perspective on the Harder--Narasimhan stratification of $...
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I want to work over the site of smooth manifolds with Grothendieck topology induced by open immersion. I am primarily concerned with the the following question: in what ways can we classify vector ...
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Let $\pi: X' \to X$ be an étale $\Gamma$-cover of curves (over an algebraically closed field) and $G$ a group scheme (e.g. reductive). Of course, $\operatorname{Bun}_G(X)$ can be identified with the ...
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The stratification of $\overline{\mathcal{M}}_{g,n}$ by so called stable graphs is a classical topic about Moduli of curves. A stratum corresponds to a decorated graph $\Gamma$ and in the literature, ...
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Let $M$ denote the coarse moduli space of smooth cubic surfaces. It is a classical result that GIT allows one to naturally construct a compactification $\bar{M}$ of $M$ which is isomorphic to the ...
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The classical Schlessinger conditions for the existence of a formally versal object are formulated for functors $$ F : \mathrm{Art}_k \to \mathrm{Set}, $$ with $F(k) = \{ * \}$. One requires that for ...
mathuser's user avatar
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Let $X$ be a Fréchet manifold. Let $G=\Bbb C^*$ be the multiplicative group of the complex numbers. Let $X_{\text{st}}$ be the smooth stack of associated with $X$: a plot is a smooth map $U\to X$, ...
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Let $X$ be an Artin stack over $\operatorname{Spec}\mathbb Z$. A quasi-coherent sheaf $M\in\operatorname{QC}_X$ is globally flat if $-\otimes M$ is an exact functor $\operatorname{QC}_X\to \...
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First some definitions: Let $\text{Mfd}$ be the category of finite dimensional smooth manifolds. Let $\text{stk}$ be the category of stacks over $\text{Mfd}$. An object of $\text{stk}$ is a smooth ...
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Let $A$ be an ordinary ring, and let $R$ be a connective $\mathbb{E}_{\infty}$-ring. Maps $R\to A$ can be described easily: any such map will factor uniquely through the truncation map $R\to\pi_0 R$. ...
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I am reading Toen's note 'higher stacks — a global overview'. For theory of stacks I read Alpers note 'Stacks and Moduli'. My question is the following: For Artin stacks with some nice properties we ...
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As I'm reading about Gromov-Witten theory in Cox and Katz Mirror Symmetry and Algebraic Geometry, I'm unsure of how exactly integration against the virtual fundamental class exactly works: in their ...
Integral fan's user avatar
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I am reading about stacks and moduli from Alper's notes. I am wondering how to see group scheme action on some quotient stack. Let $X$ be a scheme with an action of a group scheme $G$. Then $[X/G]$ be ...
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Oftentimes when working in the internal logic of a (ringed) topos $X$ (I'm interested in both the $1$-topos and the $\infty$-topos cases), I've found myself wanting to relate the following objects, in ...
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Let $\text{Mfd}^{\llcorner}$ denote the category of smooth manifolds with corners (using whatever reasonable notion of "manifold with corners" you like). (1). It seems to me that notion of a ...
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In Lurie's Spectral algebraic geometry I read the the topos $\mathcal{S}\mathrm{hv}_{\mathrm{Nis}}(X)$ if noetherian, finite krull dimension and quasi compact spectral algebraic space is postnikov ...
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I am new in derived algebraic geometry, mostly self studying. I want to know about theory of derived $n$ stacks, derived category of sheaves over them, their properties like smoothness, flatness, ...
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Let $\mathrm{An}$ be the $\infty$-category of anima (i.e., $\infty$-groupoids). Let $X \colon \mathcal{O}(U)^{\mathrm{op}} \to \mathrm{An}$ be a presheaf on a topological space $U$. Now, consider a ...
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I will assume familiarity with the notions of singular support as defined by Kashiwara and Schapira in ''Sheaves on Manifolds''. Let $M$ be a manifold, one can define a functor : $\mu Sh^{pre} : Op_{T^...
stratified's user avatar
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Let $X$ be a smooth projective curve over $\mathbb C$, $G$ be a connected reductive group over $\mathbb C$. The Hecke stacks is usually defined as the stacks whose $S$-valued points (for any $\mathbb ...
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Let $\mathcal{X}$ be a Deligne–Mumford stack over Sch/k, and let $\mathcal{G} \overset{t}{\underset{s}{\rightrightarrows}} \mathcal{X}$ be a groupoid over $\mathcal{X}$ with $s, t$ étale. Is there a ...
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For a smooth affine algebraic variety $X$ over $\mathbb{C}$, a theorem of Grothendieck gives an equivalence $$H^i_{\text{dR}}(X) \simeq H^i(X(\mathbb{C}),\mathbb{C})$$ between the algebraic de-Rham ...
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I am reading Jarod Alper's book on Stacks and Moduli. My question is regarding ex. 2.4.43. My question is the following: Let $H$ and $G$ be smooth affine group schemes over a scheme $S$. Let $Hom(H, ...
KAK's user avatar
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Let $G$ and $E$ be 0-truncated group objects in the infinity category of stacks on a Grothendieck site. Suppose $E$ is commutative. Then it turns out that the classifying stack $BE$ of $E$-torsors ...
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I am new to algebraic geometry. I am reading about equivariant Chow groups from the paper "equivariant intersection theory" by Edidin and Graham. I was trying to calculate some basic ...
KAK's user avatar
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I am new to algebraic spaces and stacks. My question is the following: Let $X$ be a scheme and $G$ be a group scheme action on $X$. Let $[X/G]$ be the quotient stack. Then when the natural map $\pi: ...
KAK's user avatar
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I am new to algebraic space s and stacks. My question is the following: Let $X$ be a scheme and $G$ be a group scheme acting on $X$. We have the natural map $\pi : X \to X/G$. When $\pi$ will be a ...
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I am new to stacks and algebraic spaces. I have the following question: Let $X$ be a scheme and $G$ be a group scheme action on $X$. Then $X/G$ exists as an algebraic space. Let $\pi: X \to X/G$ be ...
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We want to define what is a morphism of bundles over an algebraic stack. If $X$ is an algebraic stack and $V_n$ is the stack of rank $n$ vector bundles, a vector bundle on $X$ will be a morphism of ...
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This is an interesting observation of mine when exploring moduli of elliptic curves. Let's fix a base field $k$ and assume that we have two elliptic curves $E$, $E'$ which are isomorphic over $\bar k$...
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This question is related to a previous question of mine, which has so far gone unanswered. For a fixed site $\mathcal C$, the fibred categories over $\mathcal C$ form a (strict) $2$-category, see here....
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A question about following statement from Martin Olsson's book on Stacks. In the proof of Proposition 13.2.9. (p 269) is claimed that certain sheaf $K$ on a nodal curve $C$ is invertible it suffice to ...
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I am started learning about stacks. I am reading Alper's notes Stacks and Moduli. I am getting difficulty to verify the following is a cartesian square. Can someone give some hint how to start? For ...
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Let $X$ be a variety over $k$ and $G$ be a finite abelian group. Then we know that $H_{fppf}^{2}(X,G)$ is in bijective correspondence with isomorphism classes of $G$-banded gerbes. Now we consider a ...
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I am interested in the following moduli problem: The moduli functor $\mathcal{F}$ has $T$-points: a nodal $n$-pointed curve $C/T$ of genus $g$. a nodal $b$-pointed curve $D/T$ of genus $h$. a finite ...
Matthias's user avatar
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Let $k$ be a field and $G$ be a group. For simplicity, let us assume $\operatorname{char}k=0$ and $G$ is a finite abelian group. We do not assume $k$ is algebraically closed. If there is an ...
Mike's user avatar
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Let $\overline{M}_{0,n}$ be the fine moduli space of stable n-pointed curves of genus $g=0$. Let $[(D_{0},p_1,...,p_n)] \in \overline{M}_{0,n}$. Suppose that each component of $D_0$ contains at least ...
Manoel's user avatar
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Let $W$ be a connected and separated Deligne-Mumford stack of finite type over an algebraically closed field $k$, and $w\in W$ a geometric point. Consider the étale fundamental group $\pi_1^{ét}(W,w)...
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The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories". Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the ...
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I have found similar questions littered throughout this site and math.SE (for example [1], [2], [3],…), but I feel like like most of them usually just say that non-trivial automorphisms prevent the ...
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If $G/k$ is an algebraic group, a necessary condition for $BG$ to be separated over $k$ is that $G$ is proper over $k$. Assuming that $G$ is smooth and connected we are reduced to the case of an ...
Aitor Iribar Lopez's user avatar
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Let $C$ be a site and $p : F \mapsto C$ $p' : F' \mapsto C$ two categories fibred in groupoids over $C$ which are stacks (see, e.g., the definition here https://stacks.math.columbia.edu/tag/0268). For ...
Analyse300's user avatar
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I'd like to premise that while I know the definition of (differentiable) stack, I'm not really into the language of schemes so my understanding of what is a moduli stack is pretty concrete and ...
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Let $G$ be an affine group scheme over $\mathbb{C}$. I am interested in understanding the differences between different notions of sheaves on the stack $pt/G = BG$. For any algebraic stack $X$ one can ...
arczn's user avatar
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I'm currently working on a structure result for a certain spectral moduli problem, and I've been running into the problem of having to define what, precisely, is meant by the term "stratification&...
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This questions is about the distinction between: Lie groupoids: we require source and target maps to be submersions. This implies that the domain of the composition map, $G_1 \;{}_s\!\times_t G_1$, ...
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Let $X_k$ be a smooth quasi-projective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to characteristic $0$, and let $(X_K)^{\text{an}}$ denote the rigid analytic space associated ...
kindasorta's user avatar
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I am new to these formalisms, so pardon me if the question is basic. Let $\mathscr{X}$ be an Artin stack (you can take it to be Deligne-Mumford stack if it helps). By a coherent sheaf on $\mathscr{X}$ ...
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If $G$ is a nice enough group acting on a nice enough space $X$, then the relative de Rham complex $$\Omega^\bullet_{X/(X/G)}\ \simeq\ \mathcal{O}_X\otimes\text{Sym}\,\mathfrak{g}^*[-1]$$ is given by (...
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