Smooth morphisms Peter Bruin  February  Introduction The goal of this talk is to dene smooth morphisms of schemes which are one of the main ingre dients in Nerons fundamental theorem BLR
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Smooth morphisms Peter Bruin February Introduction The goal of this talk is to dene smooth morphisms of schemes which are one of the main ingre dients in Nerons fundamental theorem BLR

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

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Smooth morphisms Peter Bruin February Introduction The goal of this talk is to dene smooth morphisms of schemes which are one of the main ingre dients in Nerons fundamental theorem BLR




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Presentation on theme: "Smooth morphisms Peter Bruin February Introduction The goal of this talk is to dene smooth morphisms of schemes which are one of the main ingre dients in Nerons fundamental theorem BLR"— Presentation transcript:


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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 Nerons 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 Neron 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 Neron 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
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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].
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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 Neron models are all a bout. We use the definition from EGA since it is more general than Hartshornes 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.
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Kahler 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 Kahler 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
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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 Kahler 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
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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 Kahler 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, Hartshornes 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 Kahler 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/
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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 , definition 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 , definition 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
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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. Lutkebohmert and M. Raynaud, Neron Models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21 . Springer-Verlag, Berlin, 1990. [EGA] A.

Grothendieck, Elements de geometrie algebrique IV ( Etude locale des schemas et des morphismes de schemas), 1, 4 (rediges avec la collaboration de J. Dieudonne). Publications mathematiques de lIH ES 20 (1964), 32 (1967). [Hartshorne] R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics 52 . Springer- Verlag, New York, 1977.