Questions tagged [algebraic-stacks]
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293 questions
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Reconstructing a scheme from its quotient stack
Let $X/S$ be a scheme over a base $S$, with a group action by a nice group $G/S$ (typically $G/S$ affine smooth).
Can we reconstruct $X$ from its quotient stack $[X/G]$?
It seems that we can expect $X$...
- 645
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0
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80
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Relation between Chow groups and K theory
I am reading about Chow groups and algebraic K-theory of schemes. I get to know that for smooth schemes the re is a strongly convergent spectral sequence
$$E_2^{p,q} = CH^{-q}(X,-p-q) \implies K_{-p-q}...
- 613
5
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1
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367
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Check that a Sheaf is Invertible Etale Locally
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 ...
- 5,998
3
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Concrete reasons to study derived categories of quasi-coherent sheaves on algebraic stacks?
Title says it all. There seems to be a lot of work done on derived categories of quasi-coherent sheaves on stacks, and yet I couldn't find many "external" applications. I'm mainly wondering ...
- 143
2
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103
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Representability of stack of finite maps between curves
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 ...
- 223
2
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0
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245
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Does automorphism of classifying stack come from automorphism of group?
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 ...
- 253
5
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144
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Why do monoidal functors between categories of quasicoherent sheaves commute with external tensor products
I'm reading Lurie's paper on Tannaka duality for geometric stacks. Very roughly, my question is, why do monoidal functors, from which we try to build geometric morphisms, commute with certain algebra ...
- 1,374
5
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1
answer
327
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Comparison between pushforward-pullback and quasi-coherent pushforward-pullback
In the following, an algebraic stack means a stack over the big fppf site of a scheme, admitting a smooth, representable, surjective morphism from a scheme (no separation hypotheses), and let $\mathsf{...
- 1,349
3
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0
answers
134
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The étale fundamental group of an DM stack acting on a locally constant sheaf
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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114
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Compactifications of product of universal elliptic curves
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 ...
- 12.1k
5
votes
2
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419
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Enough injectives in the category of quasi-coherent sheaves on a stack
For a scheme $X$, I have a reference - https://stacks.math.columbia.edu/tag/077P - that says there are enough injectives in the category $\text{QCoh}(X)$. I am looking for a reference that says the ...
- 988
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305
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Gluing together the moduli stacks of elliptic curves over Z[1/2] and Z[1/3]?
I have a long-running desire to understand what is the "global" moduli stack of elliptic curves, as a stack over $\mathrm{Spec}(\mathbb{Z})$.
Recently I was pointed to Katz and Mazur's book, ...
- 35.4k
5
votes
2
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241
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References for orbifold curves
I am looking for a good reference (if there is any) for the theory of orbifold curves from the perspective of stacks. By an orbifold curve I mean something like a $1$-dimensional irreducible Deligne-...
- 341
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1
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628
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Canonical comparison between $\infty$ and ordinary derived categories
This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
- 1,349
2
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156
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Classifying stack for finite flat group scheme
Let $G$ be a finite flat non-smooth group scheme over an algebraically closed field $k$, for example, $G$ can be $\operatorname{Spec}(\overline{\mathbb{F}}_p[t]/(t^p))$. Then the classifying stack $\...
- 455
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1
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260
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Is a finite morphism of Deligne-Mumford stacks proper?
The situation that I am in is the following. Let $\mathcal{X}$ be a smooth Deligne-Mumford stack over a field $k$. Let $X$ be a $k$-scheme together with a morphism $\pi;\mathcal{X}\rightarrow X$ (you ...
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139
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What are the categories of IND and PRO schemes?
below is a mathexchange question with no answers so I drop it here.
I have some difficulties to figure out what the category of IND-schemes and PRO-schemes are, in particualer the relations with ...
- 1,270
5
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0
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169
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Description of pull-back of coherent sheaves under a smooth morphism of Artin stacks
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 $\mathscr{X}$ ...
- 737
3
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149
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Relationship between $\infty$-categories of ind-coherent sheaves on the base and total space
My question is vaguely as follows, let $G\to E\to X$ be a principal $\infty$-bundle for some group object $G$ in a category of ($\infty$-)stacks. Can we recover the category $\operatorname{IndCoh}(E)$ ...
- 1,374
2
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254
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Finite generation of stack cohomology
Let $X$ be an Artin stack of finite type. Does it follows that its (say, $\ell$-adic or de Rham) cohomology $\text{H}^*(X)$ is a finitely generated algebra?
For instance, $\text{H}^*(\text{B}\mathbf{G}...
- 5,701
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181
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Dualizing sheaf for classifying stack and duality
For an algebraic group $G$ there should be an equivalence $\operatorname{Rep}(G) \simeq \operatorname{IndCoh}(BG)$. I'm trying to understand what the dualizing sheaf (or complex) of $BG$ is. Here's ...
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2
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1
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390
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Hypercover and hyper descent
I am trying to understand the descent condition using hypercovers. The condition says that a hyper cover of a scheme $X$ is a simplicial set $Y_{\bullet}$ that satisfies the condition $Y_n\rightarrow ...
- 23
1
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0
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180
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Moduli stack of l-adic sheaves?
Let us work over a field $k$. Then for any smooth affine group scheme $G$ over $k$, we can consider the stack quotient $BG := [\text{pt} / G]$ which classifies étale $G$-torsors.
Let $\ell$ be a prime ...
- 426
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1
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403
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Grothendieck purity for Brauer groups of stacks
Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...
- 388
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263
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An algebraic stack is an algebraic space if and only if it has the trivial stabilizer group
Let $G\to S$ be a smooth affine group scheme over a scheme. Let $U$ be a scheme over $S$ with an action of $G$. Let $[U/G]$ be the quotient stack.
In Alper's note: Stacks and Moduli, there is a result ...
- 11
1
vote
1
answer
235
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Derived McKay correspondence between a weighted projective plane and a Hirzebruch surface
Let $k$ be an algebraically closed field of $\text{ch}(k) =0$.
Let $\mathbb{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a stack.
Let $\mathbf{P}(1,1,2)$ be the weighted ...
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255
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Confusion in identification of quasicoherent sheaves on BG and G -representations
I asked this question on MSE a few days ago, but didn't get a response and also managed to confuse a senior colleague with it since then. This is probably a stupid question, so please bear with me.
...
- 1,071
1
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0
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84
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Algebraizable image of a morphism of Galois cohomology stacks
Assume I have a surjective morphism of algebraic group schemes over $\mathbb{Q}_p$, $\mathcal{G}\longrightarrow S$, equipped with a section, and assume both of these group schemes are equipped with an ...
- 2,907
2
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0
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296
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Grothendieck duality for root stacks
Let $k$ be an algebraically closed field of characteristic $0$.
Let $X$ be a projective scheme over $k$, let $D = \sum_{i=1}^d$ be a simple normal crossing divisor.
Let ${\bf a} = (a_1,\cdots,a_d)$ be ...
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1
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279
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On sheaf quotient
Let $X$ be a scheme and $G$ be a group acting on $X$. Suppose the action is not free. Consider the quotient sheaf $X/G.$ Can we directly prove that the sheaf quotient is not an algebraic space?
- 494
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$D^b_\text{Coh}(X)$ for a smooth, proper Deligne-Mumford stack
Let $X$ be a smooth, proper DM stack over a field $k$. I see in Hall-Rydyh's paper "Perfect Complexes on Algebraic Stacks" (https://arxiv.org/abs/1405.1887) a discussion of compact ...
- 511
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0
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147
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On the stack of bundles
Let $K$ denote the function field $\mathbb C((t))$ and let $X$ be a smooth projective curve of genus $g\geq 2$ over $K$. Let $r\geq 2$ be some positive integer. Let $B$ denote the moduli of vector ...
- 494
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0
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277
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How to define Cartier divisor and Weil divisor on algebraic stack?
How to define Cartier divisor and Weil divisor on algebraic stack? Do they correspond to line bundles on stack like the case of schemes? In case of a Deligne-Mumford stack, can we have a simpler ...
- 523
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1
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127
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Covering a stack by an open substack that contains all points of finite type
Let $X = \text{Spec}A$ be an affine scheme. It is well known that if $U$ is an open subset which contains $\text{SpecMax}A$, then $U\supseteq X$. The previous statement generalizes to arbitrary ...
- 2,907
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0
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76
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Function vanishing on the image of a morphism of algebraic stacks
Let $f: X\longrightarrow Y$ be a morphism of algebraic stacks, and assume I know the Krull dimension of the ring of global functions on $Y$ to be $n$. Assume that the stack $X$ has "stacky" ...
- 2,907
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0
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223
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etale fundamental group of global quotient algebraic stacks
I am reading about fundamental group of algebraic stacks, and in my opinion, the class of quotient stacks is an important one in algebraic stacks. However, I am not clear about how to compute ...
- 455
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0
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144
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Cartesian square in the category of Algebraic stacks
Suppose we have a commutative diagram of Artin stacks
$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\...
- 494
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165
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Topological property of an algebraic stack and its presentation
I started to learn algebraic stacks this January. I found there are several properties of algebraic stacks which are defined in terms of their underlying topological spaces, for example, connectedness,...
- 455
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156
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Moduli of morphisms between varieties
Let $\mathcal{M}, \mathcal{N}$ be two "well-behaved" (i.e. representable by an algebraic stack parametrising flat, proper, surjective, finitely presented morphisms with geometric fibres ...
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204
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Closed embedding into weighted projective stack/space
Let $ Z $ be a proper or even, projective scheme. I have two related questions. (everything is over complex numbers)
(1) What is a criteria for a map $ \phi: Z \rightarrow \left[ \mathbb{A}^{n+1} - (0)...
- 936
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0
answers
109
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Defining log prestacks (and their structures)
It's possible to define log schemes, and Olsson's thesis defines log stacks.
Is there a definition of log prestack in the literature? Is it easy to extend Gaitsgory-Rozenblyum type results (...
- 5,701
8
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2
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376
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Residual gerbes and coarse moduli space of a DM stack
Let $X$ be a nice DM stack (Noetherian, separated), and $X_0$ its coarse
moduli space which exist by the Keel-Mori theorem.
(I like the exposition in D. Rydh. “Existence and properties of geometric ...
- 645
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0
answers
238
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Local structure of smooth morphisms of stacks
Let $\varphi:X \to Y$ be a smooth morphism of schemes.
There is a well-known local structure theorem: Zariski locally $\varphi$ is given by the composition of an etale morphism and an affine space (...
- 21.1k
2
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214
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Explicit computation of inertia stacks
I am learning algebraic stacks myself with some reference recommended by my friends. I know from here Section 6.1 that an explicit description of the fibre product of stacks $\mathfrak{X}\times_\...
- 455
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1
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370
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How the automorphism group of an elliptic curve acts at the localization of the stack $\mathcal{M}_{1, 1, k}$ at the corresponding point
I am studying the enlightening article "The Picard Group of $\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof.
Setting
Let $\mathcal{M}_{1, 1, k}$ ...
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2
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304
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Smallest atlas for algebraic stack
Let $X$ be an algebraic stack of finite type over a field.
Is there an intrinsic way to calculate the minimum of the dimensions of all atlases of $X$?
By intrinsic here I mean using constructions such ...
- 447
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171
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Is $\operatorname{Rep}(G,\operatorname{SL}_2)$ representable by an algebraic space?
Let $G$ be a finite group. Consider the category of rigid analytic spaces over $\operatorname{Spf}\mathbb{Q}_p$, and let $\operatorname{Rep}(G, \operatorname{SL}_2)$ be the fibred category above it, ...
- 2,907
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0
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155
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Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?
I need the reference to a detailed proof the following fact.
Let $g\geq 2$ be an integer. Let $\mathcal M^{ss}_g: Sch\rightarrow Groupoids$ be the prestack which sends a scheme $T$ to the groupoid of ...
- 494
1
vote
0
answers
213
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Quotient stack is an algebraic space when $G$ is finite and acts freely
I have been following Jarod Alper's lecture series on YouTube on Stacks https://youtube.com/playlist?list=PLhFI5R_xInjdhtWuhgYlA8NZGXO-unnl4
From what I understand -
If a smooth affine group scheme $...
1
vote
0
answers
306
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Decomposition of vector bundles on the inertia stack of a DM stack
Let $X$ be a tame DM stack over $\mathbb{C}.$ Let $IX$ denote the inertia stack of $X.$ Let $K(IX)$ denote the Grothendieck group of vector bundles on $IX.$ By the discussion on page 20 of Toen's ...
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