3 Theorem 1 Theorem Let be a discrete valuation ring with 64257eld of fractions and let be a smooth group scheme of 64257nite type over Let sh be a strict Henselisation of and let sh be its 64257eld of fractions Then admits a N57524eron model over ID: 29272 Download Pdf

126K - views

Published bynatalia-silvester

3 Theorem 1 Theorem Let be a discrete valuation ring with 64257eld of fractions and let be a smooth group scheme of 64257nite type over Let sh be a strict Henselisation of and let sh be its 64257eld of fractions Then admits a N57524eron model over

Download Pdf

Download Pdf - The PPT/PDF document "Smooth morphisms Peter Bruin February ..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Page 1

Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to deﬁne smooth morphisms of schemes, which are one of the main ingre- dients in Neron’s fundamental theorem [BLR, 1.3, Theorem 1]: Theorem. Let be a discrete valuation ring with ﬁeld of fractions , and let be a smooth group scheme of ﬁnite type over . Let sh be a strict Henselisation of , and let sh be its ﬁeld of fractions. Then admits a Neron model over if and only if sh is bounded in We will not explain the boundedness condition (see [BLR, 1.1]),

but this condition is known to be satisﬁed in the case where is proper over . In particular, we get the following result (recall that an Abelian variety of dimension over a scheme is a proper smooth group scheme over with geometrically connected ﬁbres of dimension ): Corollary. Let be a discrete valuation ring with ﬁeld of fractions , and let be an Abelian variety over . Then admits a Neron model over The deﬁnition of smoothness includes two ‘technical’ condi tions: ﬂatness and ‘locally of ﬁnite presentation’. We start by deﬁning these;

then we state the d eﬁnition of smoothness and a criterion for smoothness in terms of diﬀerentials. We also summarise t he diﬀerent notions of smoothness found in EGA. Finally, we give some equivalent deﬁnitions of etale morphisms. Flat modules and ﬂat morphisms of schemes Deﬁnition. A module over a ring is called a ﬂat -module if for every short exact sequence 00 of -modules, the sequence 00 is again exact. Equivalently, since the tensor product is al ways right exact, an -module is ﬂat if and only if the functor is left exact,

i.e. preserves kernels. Proposition. Let is a ring homomorphism. For every ﬂat -module , the -module is also ﬂat. Furthermore, if is ﬂat as an -module, and if is a ﬂat -module, then is also ﬂat as an -module. Proof . Easy. Examples of ﬂat -modules are the locally free (or projective ) modules; in fact, it can be shown that if is a Noetherian ring, the ﬁnitely generated ﬂat -modules are precisely the locally free -modules of ﬁnite rank. Deﬁnition. A module over a ring is called faithfully ﬂat if it is ﬂat and in

addition we have the implication = 0 = 0 for every -module A faithfully ﬂat module has the useful property that a short exact sequence of -modules is exact if and only if it is exact after tensoring with . An example of this is the following lemma with its corollary, which we will need later. Lemma. Let be a ring, let be a faithfully ﬂat -algebra, and let be an -module. Then is ﬂat over if and only if is ﬂat over Proof . If is ﬂat over , then is ﬂat over by the base change property. Conversely, suppose is ﬂat over , and let 00

Page 2

be a short exact sequence of -modules. Then we have to show that 00 is again exact. Because is faithfully ﬂat over , it suﬃces to check this after tensoring with But then we get 00 which is exact; namely, is ﬂat over and is ﬂat over , hence is ﬂat over Corollary. Let be a Noetherian ring, let be a faithfully ﬂat -algebra, and let be a ﬁnitely generated -module. Then is locally free if and only if is locally free. Proof . This follows from the lemma since ‘ﬂat’ and ‘locally free’ a re equivalent for ﬁnitely generated modules over a

Noetherian ring. Deﬁnition. A morphism of schemes is called ﬂat at a point if the local ring X,x is ﬂat as a module over the local ring Y,f , and is called ﬂat if it is ﬂat at every point of Proposition. Open immersions are ﬂat morphisms. If and are ﬂat, then is ﬂat. Flatness is preserved under base change in the sense t hat in a Cartesian diagram with ﬂat, is also ﬂat. If is a locally Noetherian scheme, it follows from the above rem ark about ﬂat modules that ﬁnite morphism is ﬂat if and only if it is

locally free of ﬁnite rank. (Locally) ﬁnitely presented morphisms Let be a morphism of rings. Recall that is ﬁnitely generated as an -algebra if there exists a surjective homomorphism , . . . , x of -algebras. We say that is ﬁnitely presented as an -algebra if there exists such a homo- morphism with the property that its kernel is a ﬁnitely gener ated , . . ., x ]-ideal. If is Noetherian, this condition is automatic; in other words; nitely presented’ and ‘ﬁnitely generated are equivalent for algebras over a Noetherian ring. Deﬁnition. A morphism

of schemes is called locally of ﬁnite presentation at a point if there exist aﬃne neighbourhoods = Spec and = Spec of ) and , respectively, such that and is a ﬁnitely presented -algebra. The morphism is called locally of ﬁnite type (resp. locally of ﬁnite presentation ) if it is locally of ﬁnite type (resp. locally of ﬁnite presentation) at every point of . It is of ﬁnite type if it is locally of ﬁnite type and quasi- compact, and it is of ﬁnite presentation if it is locally of ﬁnite presentation, quasi-compact and

quasi-separated. Obviously, every morphism of ﬁnite presentation is of ﬁnite type. It can be checked that for locally Noetherian, the two notions are equivalent [EGA IV 1.6].

Page 3

Regular schemes Deﬁnition. Let be a Noetherian local ring with maximal ideal and residue class ﬁeld . Then is regular if and only if dim ) = Krull dim Fact. The localisation of a regular ring at a prime ideal is again re gular. Deﬁnition. Let be a locally Noetherian scheme. Then is regular if all its local rings are regular, or equivalently (by the above fact) if all the

local rings at closed points of the aﬃne open subschemes in some aﬃne open cover are regular. Smooth morphisms Smooth morphisms are the things which Neron models are all a bout. We use the deﬁnition from EGA since it is more general than Hartshorne’s deﬁnition. Deﬁnition. Let be a morphism of schemes. Then is called smooth at a point if the following conditions hold: a) is ﬂat at b) is locally of ﬁnite presentation at c) the ﬁbre is geometrically regular at , i.e. all the localisations of the (semi-local) ring X,x Y,f )) are

regular, where )) denotes an algebraic closure of the residue ﬁeld of Y,f We say that is smooth if it is smooth at every point of , i.e. if a) is ﬂat; b) is locally of ﬁnite presentation; c) the ﬁbres of are geometrically regular. It is important to note that geometric regularity is a proper ty of schemes over a ﬁeld , whereas regularity is a property of schemes. In general, if is a property of schemes (such as regular, reduced, irreducible, connected, integral), we say that a s cheme over a ﬁeld is geometrically if is , where is an algebraic closure

of The property of a morphism being smooth is preserved under an y base change, i.e. if is smooth and is any morphism of schemes, then is again smooth. This follows from the fact that if is a geometrically regular scheme over a ﬁeld , then is geometrically regular for any ﬁeld extension . It would not be true if we had only required ‘regular’ instead of ‘geometrically regular’ for the ﬁbres Notice that if is locally Noetherian, we can replace ‘locally of ﬁnite pres entation’ by ‘locally of ﬁnite type’ in the deﬁnition of smoothness.

Page 4

Kahler diﬀerentials Deﬁnition. Let be a morphism of rings (as always, rings are supposed to be com mutative with 1). A derivation from over is an -linear map with -module, such that the following product rule holds: bb ) = b db db for all b, b B. universal derivation is a -module B/A together with a derivation B/A as above such that for every other derivation there is a unique -linear map : B/A making the diagram B/A commutative. The module B/A is also called the module of Kahler diﬀerentials Because of the universal property, a universal derivation i s

unique up to unique isomorphism, if it exists. Let us give two constructions of it: 1) Choose a presentation for as an -algebra. This means to choose a set of generators for as an -algebra, so that we have a surjective homomorphism B, where ] is the polynomial algebra over with generators labelled by ; write for the ideal of ] which is the kernel of this homomorphism, and choose a set ] of generators of as an ]-module. For each ] and each , we let denote the partial derivative of the polynomial with respect to . Then we put B/A .D mod the homomorphism 7 is a derivation and induces a

derivation B/A which is universal, as one can easily check. In particular, i , . . . , x ] is a ﬁnitely generated polynomial algebra, we see (taking , . . . , x and ) that B/A =1 B dx 2) Consider as an algebra over the tensor product via the multiplication homomorphism 7 bb

Page 5

of -algebras, and let be its kernel. Then is a -module, and I/I (( /I ) = is a -module. We put B/A I/I and B/A 7 b. Again, one can check that this has the required universal pro perty. The construction of the module of Kahler diﬀerentials is co mpatible with localisation. This implies

that if is a morphism of schemes, there exists a sheaf B/A on and a morphism of -modules X/Y having the expected universal property. Proposition. For every composed morphism of schemes, there is an canonical exact sequence Y/Z X/Z X/Y of -modules. Furthermore, the formation of X/Y is compatible with base change in the sense that for every Cartesian diagram there is a canonical isomorphism /Y = ( X/Y

Page 6

Smoothness and regularity for schemes over a ﬁeld In practice, it is convenient to have a more explicit conditi on for geometric regularity. Such a condition is provided by

the Kahler diﬀerentials which we h ave just seen. We start with the algebraic analogue of what we want to prove. Theorem. Let be a local ring containing a ﬁeld isomorphic to its residue ﬁeld. Assume furthermore that is a localisation of a ﬁnitely generated -algebra. Then B/k is a free module of rank equal to dim if and only if is a regular local ring. Proof . [Hartshorne, Theorem II.8.8] (according to a remark of Bas Edixhoven during the talk, Hartshorne’s assumption that is perfect is unnecessary). Proposition. Let be a ﬁeld, let be a scheme which is

locally of ﬁnite type over . Let be a closed point of , and let be the dimension of at (i.e. the Krull dimension of X,x ). Then the following are equivalent: (1) is geometrically regular at (2) the stalk X/k,x is free of rank Proof . Fix an algebraic closure of , and put X,x k. We have to prove that Spec is a regular scheme if and only if X/k,x is free of rank If is geometrically regular, then for all maximal ideals the local ring is regular by the deﬁnition of geometric regularity, and its dimension equals [Hartshorne, Exercise II.3.20]. By the previous theorem we see that B/ is

locally free of rank . Therefore B/ is a ﬂat -module. Furthermore, it follows from the fact that is faithfully ﬂat over that is faithfully ﬂat over X,x . The above lemma implies that X/k,x is ﬂat over X,x . Now we are done, because a ﬁnitely generated ﬂat module over a Noetherian ring is proj ective, and a projective module over a local ring is free. Conversely, suppose X/k,x is free of rank . Then B/ = X/k is also free of rank so by the previous theorem we see that is regular for all maximal ideals . This implies that Spec is a regular scheme.

Example. Let be an imperfect ﬁeld of characteristic , where is an odd prime number, and let be an element which is not a -th power. We put x, y which is a 1-dimensional -algebra, and we consider the morphism of schemes : Spec Spec k. Let be the maximal ideal ( t, y ) of . Then A/ x, y , x ) is a -vector space of dimension + 1, with basis , x, . . ., x , y , and is the 1-dimensional subspace generated by . A similar computation at the other maximal ideals shows tha t Spec is regular. Now consider the base extension ), where . Denoting by the )-algebra ), we have )[ x, y For the maximal

ideal = ( u, y ) we see that B/ is of dimension 3 over ), with basis , x, y , and is the 2-dimensional subspace generated by and . Therefore Spec is not regular at ( u, y ), so Spec is not geometrically regular over and is not smooth. Using the above proposition, we can show this more easily by comput ing A/k , the module of Kahler diﬀerentials; with , this gives A/k = ( dx dy dx dy A/ This module is locally free of rank 1 outside = ( t, y ), whereas A/k A/k is isomorphic to A/ A/

Page 7

Equivalent deﬁnitions of smoothness Besides the

‘ﬁbre-by-ﬁbre’ criterion for smoothness given in the previous section, there are several equivalent deﬁnitions of smoothness to be found in EGA IV. Th e ﬁrst one is related to the property of formal smoothness [EGA IV , deﬁnition 17.1.1.] Deﬁnition. Let be a morphism of schemes. Then is said to be formally smooth , or to possess the inﬁnitesimal lifting property , if for every ring , every nilpotent ideal of and every morphism Spec , the canonical map Hom (Spec A, X Hom (Spec( A/J , X is surjective. It can be shown that formal smoothness

of a morphism can be che cked on open coverings of or , so in this sense it is a local property. For the proof, see [EG A IV , proposition 17.1.6]. Notice that formal smoothness can be seen as a property of the functor on -schemes which the scheme represents. This means that we can in principle check whethe r a functor, if it is representable, will be represented by a smooth scheme, befo re we even know that it is representable. Deﬁnition. Let be a morphism of schemes which is locally of ﬁnite type. The relative dimension of at , denoted by dim , is the dimension of the

topological space underlying the ﬁbre at the point [EGA IV , deﬁnition 14.1.2]. Proposition. Let be a morphism of schemes, and let be a point of . Then the following are equivalent: (1) is smooth at (2) is locally of ﬁnite presentation at , and there is an open neighbourhood of such that is formally smooth; (3) is ﬂat at , locally of ﬁnite presentation at , and the -module X/Y is locally free in a neighbourhood of in , of rank dim at Proof . The equivalence of (1) and (2) is [EGA IV , corollaire 17.5.2]. The equivalence of (2) and (3) is [EGA IV ,

proposition 17.15.15]. Corollary. The following are equivalent for a morphism (1) is smooth; (2) is locally of ﬁnite type and formally smooth; (3) is ﬂat and locally of ﬁnite presentation, and the -module X/Y is locally free of rank equal to the relative dimension of at all points of Because the rank of a locally free module is locally constant , we see from the last characteri- sation of smoothness that the relative dimension of a smooth morphism is locally constant. This is not the case for arbitrary ﬂat morphisms which are locally of ﬁnite presentation.

Example. To see how the inﬁnitesimal lifting property fails for a non- smooth morphism, consider again the example Spec Spec , where is a ﬁeld of characteristic 3 and x, y for some . Let be the -algebra ξ, t, and let be the ideal ; then = 0 and B/J ξ, t, ). We claim that the homomorphism B/J given by 7 and 7 cannot be lifted to a homomorphism Namely, such a homomorphism has to satisfy 7 a 7 b a, b but then maps to and maps to 0, a contradiction since in and = 0 in

Page 8

Etale morphisms Deﬁnition. Let be a morphism of schemes. Then is called

etale if it is smooth with ﬁbres of dimension 0. Let be an etale morphism. Let be a point of , let ) be its residue class ﬁeld, and let ) be an algebraic closure of ). Then the geometric ﬁbre Spec ) is a regular scheme of dimension 0, i.e. a disjoint union of spec tra of ﬁelds. These ﬁelds have to be ﬁnite extensions of ), but since ) is algebraically closed this means that is etale if and only if is ﬂat, locally of ﬁnite presentation and the geometric ﬁbre over every point of is a disjoint union of copies

of ). Equivalently, a morphism is etale if and only if it is ﬂat, locally of ﬁnite presentati on and X/Y = 0. Yet another deﬁnition: is etale if and only if it is ﬂat and unramiﬁed, where ‘unramiﬁed’ means that is locally of ﬁnite presentation and for every point , we have X,x and /k )) is a ﬁnite separable ﬁeld extension. References [BLR] S. Bosch, W. Lutkebohmert and M. Raynaud, Neron Models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21 . Springer-Verlag, Berlin, 1990. [EGA] A.

Grothendieck, Elements de geometrie algebrique IV ( Etude locale des schemas et des morphismes de schemas), 1, 4 (rediges avec la collaboration de J. Dieudonne). Publications mathematiques de l’IH ES 20 (1964), 32 (1967). [Hartshorne] R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics 52 . Springer- Verlag, New York, 1977.

Â© 2020 docslides.com Inc.

All rights reserved.