Geometrie algebrique Algebraic Geometry

Description

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

• Rappoport
• Beauville
• Laszlo
• Isomorphisme
• Assertion

Informations

• Publié par: pefav
• Nombre de lectures: 39
• Langue: English

Extrait

G´eom´etriealg´ebrique/Algebraic Geometry

Unlemmededescente Arnaud B EAUVILLE et Yves L ASZLO

Re´sume´A un anneau,– Soient f,afiulbdeAeeme´le´nilpmistn b lAsee´par´e compl´et´edeApourlatopologie(fdanouqleovsnrpuoefibr´d’unn´eee.quusNo)di-a vectorielsurSpec(A)´equivauta`celled’unfibr´esurl’ouvertf6= 0 de Spec(A) et sur Spec( b uqorpiceo’lrusserseueledr´esagimdtu’inosomprihmstuverA),ef6de Spec(A= 0 b ) .

A descent lemma Abstracta ring,– Let A be fAA , a nonzero divisor in b A 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 b A) , together with an isomorphism of their pull back to the open subsetf6of Spec(= 0 b A) .

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 A f and A f the 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 = M f , an A-module G = A⊗ A M and an A f -isomorphismϕ: A⊗ A F→G f . 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. f M is injective: T HEOREM .−Suppose given: – anA f -moduleF; b – anf-regularA-moduleG; b b – anA f -linear isomorphismϕ: A⊗ A F−→G f . Then there exists af-regularA-moduleMand isomorphismsα: M f −→F, b β: A⊗ A M−→Gsuch thatϕis the composition of −1 β 1⊗α f b b A⊗ A F−−−−→A⊗ A M f −−−−→G f .

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 A→A is not flat.

b Proof: Put B = A f ×A , and letρ: A→The homomor-B be the canonical map. phismρisfaithfulA-module M the B-module, which means that for each nonzero

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