Geometrie algebrique Algebraic Geometry
8 pages
English

Geometrie algebrique Algebraic Geometry

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8 pages
English
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Geometrie algebrique/Algebraic Geometry Un lemme de descente Arnaud BEAUVILLE et Yves LASZLO Resume – Soient A un anneau, f un element simplifiable de A , A? le separe complete de A pour la topologie (f)-adique. Nous prouvons que la donnee d'un fibre vectoriel sur Spec(A) equivaut a celle d'un fibre sur l'ouvert f 6= 0 de Spec(A) et sur Spec(A?) , et d'un isomorphisme de leurs images reciproques sur l'ouvert f 6= 0 de Spec(A?) . A descent lemma Abstract – Let A be a ring, f a nonzero divisor in A , A? the completion of A for the (f)-adic topology. We prove that the data of a vector bundle on Spec(A) is equivalent to the data of a vector bundle on the open subset f 6= 0 of Spec(A) and on Spec(A?) , together with an isomorphism of their pull back to the open subset f 6= 0 of Spec(A?) . Abridged English Version: Let A be a ring, f a nonzero divisor in A , A? the completion of A for the (f)-adic topology. We denote by Af and A?f the ring of fractions A[1/f ] and A?[1/f ] .

  • isomorphisme

  • a?

  • a? a??a

  • donnee de descente sur f?g

  • gf ?

  • mf ??

  • homomorphisme fidele

  • assertion

  • a?f


Sujets

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Publié par
Nombre de lectures 26
Langue English

Exrait

G´eom´etriealg´ebrique/Algebraic Geometry
Unlemmededescente Arnaud BEAUVILLEet Yves LASZLO
Re´sume´A un anneau,– Soient f,aulbdeAeeme´le´nilpmistnblAsee´par´e compl´et´edeApourlatopologie(fdanouqleovsnrpuoebr´dunn´eee.quusNo)di-a vectorielsurSpec(A)´equivauta`celledunbr´esurlouvertf6= 0 de Spec(A) et sur Spec(buqorpiceolrusserseueledr´esagimdtuinosomprihmstuverA),ef6de Spec(A= 0 b) .
A descent lemma Abstracta ring,– Let A be fAA , a nonzero divisor in bA forthe completion of the (fSpec(A) is equivalent)-adic topology. We prove that the data of a vector bundle on to the data of a vector bundle on the open subsetf6on Spec(of Spec(A) and = 0 bA) , together with an isomorphism of their pull back to the open subsetf6of Spec(= 0 bA) .
b Abridged English Version:Let A be a ring,fA thea nonzero divisor in A , b completion of A for the (f)-adic topology. We denote by Afand Afthe ring of b fractions A[1/f] and A[1/fA-module we can associate an ATo any ] . f-module b b b b F = Mf, an A-module G = AAM and an Af-isomorphismϕ: AAFGf. Conversely, can we recover an A-module from the data (F,G, ϕ) ? This turns out to be true under the assumption that M isf-regularthat the homothety, i.e. fM is injective: THEOREM.Suppose given: – anAf-moduleF; b – anf-regularA-moduleG; b b – anAf-linear isomorphismϕ: AAF−→Gf. Then there exists af-regularA-moduleMand isomorphismsα: Mf−→F, b β: AAM−→Gsuch thatϕis the composition of 1 β 1αf b b AAF−−−−→AAMf−−−−→Gf.
The triple(M, α, β)is uniquely determined up to a unique isomorphism. IfFandGare finitely generated(resp.flat,resp.projective and finitely generated), thenMhas the same property. Though this looks like a classical descent problem, it does not seem to follow directly from Grothendieck’s faithfully flat descent theory: if A is not noetherian, b the map AA is not flat.
b Proof: Put B = Af×A , and letρ: AThe homomor-B be the canonical map. phismρisfaithfulA-module M the B-module, which means that for each nonzero
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