#### DMCA

## Tame stacks in positive characteristic

Citations: | 56 - 14 self |

### Citations

789 | Étale cohomology - Milne - 1980 |

502 | The irreducibility of the space of curves of given genus - Deligne, Mumford - 1969 |

294 | Complexe cotangent et déformations - Illusie - 1971 |

282 |
Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research
- Mumford
- 1970
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Citation Context ...ivial normal subgroup schemes. In particular, the Frobenius kernel G1 of G is a normal subgroup scheme of G, which does not coincide with the identity, unless G is trivial: so we have that G = G1. In =-=[21]-=-, p. 139 one says that G has height 1. Connected group schemes of height 1 are classified by their p-Lie algebras (see, e.g., [21], p. 139). Lemma 2.15 (Jacobson [15], Chapter 5, Exercise 14, p. 196).... |

240 |
Moret-Bailly L., Champs Algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge
- Laumon
- 2000
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Citation Context ...n+1 G′ n+1 . Note that any such morphism ρn+1 is automatically an isomorphism since Gn+1 and G ′ n+1 are flat over Sn+1. Let LBGs ∈ Dcoh(OBGs) denote the cotangent complex of BGs over s as defined in =-=[19]-=- (and corrected in [22]). Lemma 2.18. We have LBGs ∈ D [0,1] coh (OBGs) Proof. Consider the map p : s → BGs corresponding to the trivial torsor. The map p is faithfully flat so it suffices to show tha... |

205 | Criteres de platitude et de projectivite, - Raynaud, Gruson - 1971 |

200 |
Finite groups
- Gorenstein
- 1980
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Citation Context ...s constant, so this homomorphism is trivial. Equivalently, H is central in G. Let A be a commutative k-algebra. The groups H(A) and G(A)/H(A) are commutative, hence, by “calculus of commutators” (see =-=[11]-=-, Section 6, in particular Lemma 6.1), we have a bilinear map G(A) × G(A) −→ H(A) (x, y) ↦→ [x, y] This is functorial in A, therefore the commutator gives a bilinear map G × G → H, and since H is cent... |

187 | Quotients by groupoids
- Keele, Mori
- 1997
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Citation Context ...tia). If T → S is a morphism, and ξ is an object of M(T ), then the group scheme Aut T (ξ) → T is the pullback of I along the morphism T → M corresponding to ξ. Under this hypothesis, it follows from =-=[16]-=- that there exists a moduli space ρ: M → M; the morphism ρ is proper. Definition 3.1. The stack M is tame if the functor ρ∗ : QCoh M → QCoh M is exact. When G → S is a finite flat group scheme, then t... |

181 | Compactifying the space of stable maps.
- Abramovich, Vistoli
- 2002
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Citation Context ...on of tame stack, and prove the key local structure theorem (Theorem 3.2). In a sequel to this paper, we develop the theory of twisted curves and twisted stable maps with a tame target, in analogy to =-=[1]-=-. This is for us the main motivation for the introduction of tame stacks. In Appendix A we discuss rigidification of stacks. Discussion of rigidification has appeared in several places in the literatu... |

173 | Cohomologie non abélienne - Giraud - 1965 |

123 | Compactifying the Picard Scheme, - Altman - 1980 |

120 | Grothendieck topologies, fibered categories and descent theory, in: Fundamental algebraic geometry, - Vistoli - 2005 |

97 | Twisted bundles and admissible covers,
- Abramovich, Corti, et al.
- 2003
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Citation Context ...s of Propositions 3.7 and 3.6 and of Theorem 3.2. ♠ Appendix A. Rigidification In this section we discuss the notion of rigidification, where a subgroup G of inertia is “removed”. This was studied in =-=[2, 27, 5]-=- when G is in the center of inertia, the general case briefly mentioned in [17]. In what follows, when we refer to a sheaf on a scheme T this will be a sheaf on the fppf site of T . Let S be a scheme,... |

93 | Elements de Geometrie Algebrique, Inst. Hautes Etudes Sci. - Grothendieck - 1961 |

84 |
Versal deformations and algebraic
- Artin
- 1974
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Citation Context ...on of an action of G on a quasi-coherent sheaf. For example, if α: F → F ⊗OS π∗OG is a coaction, then F G is the kernel of α − ι, where ι: F → F ⊗OS π∗OG is the trivial coaction, given by s ↦→ s ⊗ 1. =-=(4)-=- Suppose φ : H → G is surjective, with kernel a flat group scheme K. For F ∈ QCoh H (S) we have φ∗F = F K with the induced action of G. On the other hand if F ∈ QCoh G (S) then the adjunction morphism... |

50 |
Catégories Tannakiennes
- Saavedra
- 1972
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Citation Context ...morphism F G → F ′G is also surjective. Since S is noetherian, every quasi-coherent sheaf with an action of G is a direct limit of coherent subsheaves with an action of G (see, e.g. Lemma 2.1 [30] or =-=[28]-=-). By replacing F ′ with an arbitrary coherent subsheaf and F with its inverse image in F , we may assume that F ′ is coherent. Let {Fi} the inductive system of coherent G-equivariant subsheaves of F ... |

50 |
Algebraic K-theory of group scheme actions, Algebraic topology and algebraic K-theory, Annals of Math Studies 113
- Thomason
- 1987
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Citation Context ...induced morphism F G → F ′G is also surjective. Since S is noetherian, every quasi-coherent sheaf with an action of G is a direct limit of coherent subsheaves with an action of G (see, e.g. Lemma 2.1 =-=[30]-=- or [28]). By replacing F ′ with an arbitrary coherent subsheaf and F with its inverse image in F , we may assume that F ′ is coherent. Let {Fi} the inductive system of coherent G-equivariant subsheav... |

49 | Group actions on stacks and applications.
- Romagny
- 2005
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Citation Context ...s of Propositions 3.7 and 3.6 and of Theorem 3.2. ♠ Appendix A. Rigidification In this section we discuss the notion of rigidification, where a subgroup G of inertia is “removed”. This was studied in =-=[2, 27, 5]-=- when G is in the center of inertia, the general case briefly mentioned in [17]. In what follows, when we refer to a sheaf on a scheme T this will be a sheaf on the fppf site of T . Let S be a scheme,... |

28 | On coverings of Deligne-Mumford stacks and surjectivity of the Brauer map
- Kresch, Vistoli
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Citation Context ... with trivial generic stabilizers. Therefore every regular Deligne–Mumford is an étale gerbe over a regular Deligne–Mumford stack with trivial generic stabilizers. This fact was used, for example, in =-=[18]-=-, without adequate justification. Proof. Let us define a fibered category X over S in the following fashion. The objects of X are the objects of X. Let f : T → T ′ be a morphism of S-schemes, ξ and ξ ... |

19 |
Lie algebras. Republication of the 1962 original
- Jacobson
- 1979
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Citation Context ...ivial: so we have that G = G1. In [21], p. 139 one says that G has height 1. Connected group schemes of height 1 are classified by their p-Lie algebras (see, e.g., [21], p. 139). Lemma 2.15 (Jacobson =-=[15]-=-, Chapter 5, Exercise 14, p. 196). Let G be a non-abelian group scheme of height 1. Then G contains αp, and hence is not linearly reductive. Proof. Considering the p-lie algebra g of G, we need to fin... |

16 | et al. Schémas en groupes - Demazure, Grothendieck |

14 | Uniformization of Deligne-Mumford curves
- Behrend, Noohi
(Show Context)
Citation Context ...s of Propositions 3.7 and 3.6 and of Theorem 3.2. ♠ Appendix A. Rigidification In this section we discuss the notion of rigidification, where a subgroup G of inertia is “removed”. This was studied in =-=[2, 27, 5]-=- when G is in the center of inertia, the general case briefly mentioned in [17]. In what follows, when we refer to a sheaf on a scheme T this will be a sheaf on the fppf site of T . Let S be a scheme,... |

9 | Théorie des Intersections et Théorème de - Berthelot, Grothendieck, et al. - 1971 |

6 | Keel-Mori theorem via stacks, preprint available at http://math.stanford.edu/~conrad/papers/coarsespace.pdf - Conrad |

6 | On the geometry of Deligne-Mumford stacks
- Kresch
- 2009
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Citation Context ...s section we discuss the notion of rigidification, where a subgroup G of inertia is “removed”. This was studied in [2, 27, 5] when G is in the center of inertia, the general case briefly mentioned in =-=[17]-=-. In what follows, when we refer to a sheaf on a scheme T this will be a sheaf on the fppf site of T . Let S be a scheme, or an algebraic space, and let X → S be a locally finitely presented algebraic... |

2 |
Gromov–Witten theory of Deligne
- Abramovich, Graber, et al.
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Citation Context ... H through φ, and the action does not intervene in exactness. (2) If H is a subgroup scheme of G, then Φ is finite, and in particular affine; hence φ∗ is exact. In this case we denote it by Ind G H . =-=(3)-=- If we think of the structure morphism π : G → S as a homomorphism to the trivial group scheme and F is a G-equivariant quasi-coherent sheaf on S, then we denote π∗F by F G . This quasi-coherent sheaf... |

2 |
Sheaves on Artin
- Olsson
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Citation Context ...any such morphism ρn+1 is automatically an isomorphism since Gn+1 and G ′ n+1 are flat over Sn+1. Let LBGs ∈ Dcoh(OBGs) denote the cotangent complex of BGs over s as defined in [19] (and corrected in =-=[22]-=-). Lemma 2.18. We have LBGs ∈ D [0,1] coh (OBGs) Proof. Consider the map p : s → BGs corresponding to the trivial torsor. The map p is faithfully flat so it suffices to show that p ∗ LBGs has cohomolo... |

2 |
and restriction of scalars
- Hom-stacks
(Show Context)
Citation Context ...e going to need some preliminaries. Suppose that X and Y are algebraic stacks over a scheme S; consider the stack HomS(X , Y) whose sections over an S-scheme T are morphisms of T -stacks XT → YT (see =-=[23]-=-). We will denote by Hom rep S (X , Y) the substack whose sections are representable morphisms XT → YT . Lemma 3.8. Let G and H be finite and finitely presented flat group schemes over a locally noeth... |