Deformation theory of representable morphisms of algebraic stacks

Olsson, Martin

doi:10.1007/s00209-005-0875-9

Cited by 43 publications

(38 citation statements)

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“…induced by β is an isomorphism (this differs from the definition in ( [13], 17.4 (3)); see the discussion of this in ( [15], 2.15)). If A has enough injectives, then for any I ∈ A and K ∈ D b (A), one can define groups Ext j (K, I ), and a distinguished triangle (1.8.6) in D b (A) induces in a natural way a long exact sequence …”

Section: Notation and Prerequisitesmentioning

confidence: 87%

“…In this section we record some consequences of the results of ( [15]) and the transitivity triangle (1.1 (v)) about the relationship between the logarithmic cotangent complex and deformation theory generalizing ( [8] Log version of ([8], III.1.2.3)). Let f : X → Y be a morphism of fine log schemes and let I be a quasi-coherent sheaf on • X.…”

Section: Deformation Theory Of Log Schemesmentioning

confidence: 99%

“…There is a tautological equivalence of categories (see ( [15], 2.2) for the meaning of the right hand side)…”

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confidence: 99%

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The logarithmic cotangent complex

Olsson

2005

Math. Ann.

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Section: Notation and Prerequisitesmentioning

confidence: 87%

Section: Deformation Theory Of Log Schemesmentioning

confidence: 99%

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The logarithmic cotangent complex

Olsson

2005

Math. Ann.

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“…Suppose further given a 1-morphism a : S A → S over S, corresponding to an object in (f * S)(A). Let L S/S denote the cotangent complex of the morphism of S → S ( [17]). …”

Section: (Deformation Modules)mentioning

confidence: 99%

Hom-stacks and restriction of scalars

Olsson

2006

Duke Math. J.

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Abstract. Fix an algebraic space S, and let X and Y be separated Artin stacks of finite presentation over S with finite diagonals (over S). We define a stack Hom S (X , Y) classifying morphisms between X and Y. Assume that X is proper and flat over S, and fppf-locally on S there exists a finite finitely presented flat cover Z → X with Z an algebraic space. Then we show that Hom S (X , Y) is an Artin stack with quasi-compact and separated diagonal. Statements of resultsFix an algebraic space S, let X and Y be separated Artin stacks of finite presentation over S with finite diagonals. Define Hom S (X , Y) to be the fibered category over the category of S-schemes, which to any T → S associates the groupoid of functors X T → Y T over T , whereTheorem 1.1. Let X and Y be finitely presented separated Artin stacks over S with finite diagonals. Assume in addition that X is flat and proper over S, and that locally in the fppf topology on S there exists a finite and finitely presented flat surjection Z → X from an algebraic space Z. Then the fibered category Hom S (X , Y) is an Artin stack locally of finite presentation over S with separated and quasi-compact diagonal. If Y is a DeligneMumford stack (resp. algebraic space) then Hom S (X , Y) is also a Deligne-Mumford stack (resp. algebraic space). Remark 1.2. If S is the spectrum of a field and X is a Deligne-Mumford stack which is a global quotient stack and has quasi-projective coarse moduli space, then by ([15], 2.1) there always exists a finite flat cover Z → X . Remark 1.3. When S is arbitrary and X is a twisted curve in the sense of ([1]), it is also true thatétale locally on S there exists a finite flat cover Z → X as in (1

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“…Aoki montre en effet dans [11] que si X et Y sont deux champs algébriques, alors sous de bonnes hypothèses le champ H om (X , Y ) est un champ algébrique au sens d'Artin. La démonstration fait appel à un certain nombre de résul-tats non triviaux concernant les déformations de morphismes d'espaces algébriques [30], les déformations de morphismes représentables de champs algébriques [42], les déformations de champs algébriques [9] et la théorie du complexe cotangent [30,34,40]. Le cas du champ de Picard s'en déduit en prenant pour Y le champ BG m .…”

Section: Introductionunclassified

Foncteur de Picard d’un champ algébrique

Brochard

2008

Math. Ann.

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RésuméIn this article we study the Picard functor and the Picard stack of an algebraic stack. We give a new and direct proof of the representability of the Picard stack. We prove that it is quasi-separated, and that the connected component of the identity is proper when the fibers of X are geometrically normal. We study some examples of Picard functors of classical stacks. In an appendix, we review the lisse-étale cohomology of abelian sheaves on an algebraic stack.

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Deformation theory of representable morphisms of algebraic stacks

Abstract: We study the relationship between the deformation theory of representable 1-morphisms between algebraic stacks and the cotangent complex defined by Laumon and Moret-Bailly.

Cited by 43 publications

References 14 publications

The logarithmic cotangent complex

The logarithmic cotangent complex

Hom-stacks and restriction of scalars

Foncteur de Picard d’un champ algébrique

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