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Flat families of schemes - Algebraic Geometry exam
Carlo Scarpa
15 February, 2017
All rings mentioned are unitary and commutative. To indicate that I ⊆ R is an ideal of the ring R , we will often use the notation I R. For other conventions used throughout, see [Har77Har77].
Contents
11 FlatnessFlatness 1 1.11.1 Flat morphisms of schemesFlat morphisms of schemes................................. 4 1.1.11.1.1 Generic flatness and the open nature of flatnessGeneric flatness and the open nature of flatness.................. 4 1.1.21.1.2 One-parameter familiesOne-parameter families............................... 5 1.21.2 Some properties of flatnessSome properties of flatness.................................. 8 1.2.11.2.1 Flatness and dimension of fibresFlatness and dimension of fibres.......................... 10
22 Hilbert polynomials and flatnessHilbert polynomials and flatness 12
33 AppendixAppendix 15 3.13.1 Algebraic factsAlgebraic facts........................................ 15 3.23.2 Scheme-theoretic closureScheme-theoretic closure................................... 16 3.33.3 Associated pointsAssociated points....................................... 18
44 More examplesMore examples 19
List of Figures
11 The twisted cubic squashed on theThe twisted cubic squashed on the zz = 0 plane (Example= 0 plane (Example 1.211.211.211.21).).............. 7 22 Viviani’s curve squashed on theViviani’s curve squashed on the zz = 0 plane (Example= 0 plane (Example 1.221.221.221.22).)............... 9 33 Two different schemes obtained from ExampleTwo different schemes obtained from Example 4.54.54.54.5...................... 20 44 The scheme of ExampleThe scheme of Example 4.64.64.64.6................................... 21 55 The family of cubics of ExampleThe family of cubics of Example 4.74.74.74.7.............................. 22
1 Flatness
The notion of flatness was introduced by J. P. Serre in his famous paper [Ser56Ser56].
Definition 1.1. Let R be a ring. A R -module M is flat if the functor ⊗ RM : Mod R → Mod R is exact.
Remark. Recall that ⊗ RM is always right exact, so the only thing to test to check flatness of a R -module M is that for any exact sequence 0 → N 1 → N 2 also 0 → N 1 ⊗ R M → N 2 ⊗ R M is exact.
A simple example of a flat R -module is R itself, or more in general any free R -module.
Example 1.2. Let S be a multiplicatively closed subset of the ring R , and consider the R -module S −^1 R , the localization of R aver S. Then S −^1 R is a flat R -module. Indeed, consider an exact sequence of R -modules 0 → M → N , and assume that an element xs of S −^1 M becomes 0 in S −^1 N. By definition this means that there is some s ′^ ∈ S such that s ′ x = 0 in N , but since M is a R -submodule of N , we also have s ′ x = 0 in M. But then xs was already 0 in S −^1 M , thus 0 → S −^1 M → S −^1 N is exact.
Remark. If M is a flat R -module and t ∈ R is not a zero divisor, then M −→· t M is injective; indeed, since 0 → R −→· t R is exact, also 0 → R ⊗ R M −→· t R ⊗ R M is.
Lemma 1.3. For a R-module M , the following are equivalent:
_1. M is a flat R-module;
- for every ideal I_ R, the map I ⊗ R M → M defined by j ⊗ m 7 → jm is injective.
This lemma gives us an equivalent definition of flatness in terms of ideals of R ; as we shall see, this will be an important tool in studying properties of flatness.
Proof. The implication 1 ⇒ 2 is immediate, so we just prove that 2 implies 1. Let f : N 1 → N 2 be an injective morphism of R -modules, and consider f ⊗ id : N 1 ⊗ R M → N 2 ⊗ R M. To prove that f ⊗ id is injective it is of course sufficient to show that it is injective when restricted to any submodule of the form N (^) 1 ′ ⊗ R M , where N (^) 1 ′ is a finitely generated R -submodule of N 1. From now on we may assume then that N 1 and N 2 are finitely generated R -modules, and that f : N 1 → N 2 is the inclusion map. In this context there are n ∈ Z > 0 and R -submodules L 2 ⊆ L 1 ⊆ Rn^ such that N 2 = Rn/L 2 , N 1 = L 1 /L 2. We wish to show that
L 1 L 2 ⊗ R^ M^
Rn L 2 ⊗ R^ M
is injective. Notice that there are natural maps L 2 ⊗ R M → Rn^ ⊗ R M ∼= M n^ and L 1 ⊗ R M → M n ; if we manage to show that these are injective, then also L L^12 ⊗ R M → R n L 2 ⊗ R^ M^ will be injective, since under this hypothesis L L^12 ⊗ R M ∼= LL^12 ⊗⊗ RRMM , R n L 2 ⊗ R^ M^ ∼= M^ n L 2 ⊗ RM and these isomorphisms are such that L 1 L 2 ⊗ R^ M^
Rn L 2 ⊗ R^ M
L 1 ⊗ RM L 2 ⊗ RM
M n L 2 ⊗ RM
is commutative. To conclude then it will be enough to prove that for any R -submodule K of Rn , the natural map K ⊗ R M → M n^ is injective. We do this by induction on n. For n = 1 K is an ideal in R , so K ⊗ R M → M is injective by our hypothesis. Assume now that n > 1 and that the thesis is valid for n − 1. Define K ′^ = K ∩
( { 0 } × · · · × { 0 } ︸ ︷︷ ︸ n − 1 times
× R )
so that K/K ′^ ⊆ Rn −^1. Then we have the following commutative diagram with exact rows:
0 K ′^ K K/K ′^0
0 R Rn^ Rn −^1
tensoring with M we get another diagram with exact rows
K ′^ ⊗ R M K ⊗ R M ( K/K ′) ⊗ R M 0
0 M M n^ M n −^1
where the first and third vertical arrows are injections, by our induction hypothesis and the base case. Then it follows that also the vertical arrow in the middle is an injection, so we have the thesis.
Remark. Notice that condition 2 in Lemma 1.31.3 is actually equivalent to
2 ′. for every finitely generated I R , the map I ⊗ R M → M is injective.
1.1 Flat morphisms of schemes
Definition 1.9. Let f : X → Y be a morphism of schemes, and let F be a sheaf of O X -modules. We say that F is f -flat at x ∈ X if the stalk F x , seen as a O Y,f ( x )-module via f ]^ : O Y,f ( x ) → O X,x , is flat. If F is f -flat at every x ∈ X , we say that F is f -flat. We will also say that F is f -flat at some y ∈ Y if it is f -flat at each x ∈ X such that f ( x ) = y.
We are particularly interested in the case F = O X ; in this case we will simply say that f is flat at x ∈ X or at y ∈ Y. It’s also quite common to use phrases like “let X → Y be a flat family of schemes”, without naming the morphism. The idea behind this condition is that a flat morphism of schemes X → Y describes a family of schemes parametrized by the base Y which is, in some sense, continuous. Why this is the case seems to be a bit of a mystery (these are Hartshorne’s words), but nonetheless flat morphisms exhibit a number of nice properties that we would expect from a continuous family of geometric objects. The previous definition describes actually a relative notion of flatness. The corresponding absolute (i.e. that does not depend on a chosen morphism) definition is the following: a sheaf of O X -modules F is flat if and only if each stalk F x is a flat O X,x -module.
Lemma 1.10. Let X be a Noetherian scheme, and let F be a coherent sheaf of O X -modules. Then F is flat if and only if it is locally free.
As usual, the notion of flatness for affine schemes is completely algebraic.
Lemma 1.11. Let ϕ : A → B be a morphism of rings, and let f : Spec( B ) → Spec( A ) be the corresponding map of schemes. Let M be a B-module. Then M ˜ is f -flat if and only if M is a flat A-module.
As a partial justification for the definition of flatness, consider an affine scheme X = Spec( A ) of finite type over an algebraically closed field k. Any ideal I A defines the closed subscheme Z = Spec( A/I ). Let M ˜ be a quasicoherent sheaf on X ; as usual, we may regard sections of M ˜ as “ M -valued functions” on X. The restriction of M ˜ to Z is the coherent sheaf on Z defined by the A/I -module M ⊗ A A/I , and because M is A -flat we know that M ⊗ A A/I ∼= M/ ( IM ). In other words, a (local) section of M over X restricts to 0 in Z if and only if over Z it took values in IM. Thus we can say that M ˜ is flat when its sections “restrict nicely” to closed subsets of X.
1.1.1 Generic flatness and the open nature of flatness
Now we show that, under some reasonable hypothesis, the failure of flatness can only happen on a “small” subset of the base. This is a consequence of the “Generic freeness Lemma” of Grothendieck [GD65GD65, Lemme 6_._ 9_._ 2].
Theorem 1.12 (Generic freeness). Let A be an integral Noetherian ring, B a finitely generated A- algebra and M a finitely generated B-module. Then there is some f ∈ A \ { 0 } such that Mf is a free Af -module.
As a corollary we easily get that each sufficiently regular morphism of schemes is “generically flat”.
Corollary 1.13 (Generic flatness). Let f : X → Y be a morphism of finite type of schemes, with Y locally Noetherian and integral. For any coherent O X -module F , there is an open set U ⊆ Y such that F f − (^1) ( U ) is flat over U.
Proof. By taking an affine open cover of Y , we see that it will be enough to prove the theorem in the case when Y is affine. Assume then that Y = Spec( A ) for a Noetherian integral domain A. Since f is of finite type, we can cover X by a finite number of open affine sets {Spec( Bi ) | i = 1 ,... , N } such that each Bi is a finitely generated A -algebra. If we prove that for each i = 1 ,... , N there is an open set Ui ⊆ Y such that FSpec( Bi )∩ f − (^1) ( Ui ) is
flat, then it will be enough to take U =
⋂ N i =1 Ui^ to get an open set with the desired property. Hence we may also assume that X = Spec( B ) for some finitely generated A -algebra B.
By assumption there is a finitely generated B -module M such that F = M ˜. Then Theorem 1.121. tells us that there is some nonzero a ∈ A such that Ma is a free Aa -module, and this is precisely what we had to prove.
A result that has a similar flavour is that the set of points at which a coherent sheaf is flat is open, c.f. [GD66GD66, Th´eor`eme 11_._ 1_._ 1].
Proposition 1.14 (Open nature of flatness). Let f : X → Y be a morphism locally of finite type, with Y locally Noetherian, and let F be a coherent O X -module. Then { x ∈ X | F is f − flat at x } is an open subset of X.
Since this is again a local question, it is enough to prove it for X = Spec( B ), Y = Spec( A ), F = M ˜ for a finitely generated B -module M and f ]^ : A → B making B into a finitely generated A -algebra. Then the proposition follows again by an algebraic lemma.
Lemma 1.15. Let A be a Noetherian ring, B a finitely generated A-algebra, M a finitely generated B-module. choose q ∈ Spec( B ) , and let p A be the preimage of q under the natural map A → B. If M q is a flat A p -module, then there is g ∈ B \ q such that for all prime ideals q′^ B with q ⊆ q′^ and g 6 ∈ q′ , M q′ is a flat A p′ -module, where p′^ is the preimage of q′.
For a proof, we refer to [GD66GD66, 11_._ 1_._ 1_._ 1].
1.1.2 One-parameter families
Before studying flatness in full generality, we show that in some special (but nonetheless interesting) cases, flatness can be described as an actual continuity property, in the sense that in a flat family each fibre is the “limit” of all the other neighbouring fibres. To do this we will need the concept of associated points of a scheme, which is explained in some detail in the Appendix (Section 33 ).
Theorem 1.16. Let f : X → Y be a morphism of locally Noetherian schemes, with Y a regular and integral scheme of dimension 1_. Let y_ ∈ Y be a closed point. Then f is flat at y if and only if no associated point of X maps to y.
Note that, under these hypothesis, Y has exactly one non-closed point, its generic point.
Proof. Since Y is regular of dimension 1 and y is a closed point of Y , dim(O Y,y ) = dim( Y ) = 1, so O Y,y is a Noetherian local integral domain of dimension 1: it must be a PID^11. Assume that x ∈ X is such that f ( x ) = y , and that f is flat at x. Let u ∈ O Y,y be a generator of m y. By hypothesis u does not divide zero in O Y,y , and if we look at the map of local rings f (^) y] : O Y,y → O X,x we see that f (^) y] ( u ) ∈ m x is an element that does not divide 0, by the Remark at page 22. Hence x cannot be an associated point of X , by Lemma 3.163.16. For the other direction, assume that f −^1 ( y ) does not contain any associated point of X. Let x ∈ X be such that f ( x ) = y. By Lemma 1.41.4 we know that f is flat at x if and only if O X,x is a torsion-free module, i.e. for all nonzero t ∈ O Y,y , f (^) y] ( t ) is not a zero divisor in O X,x. Assume, on the contrary, that there is some t ∈ O Y,y such that f (^) y] ( t ) is a zero divisor in O X,x. Let u be a generator of m y , and let n ∈ Z≥ 0 be the unique^22 integer such that t = aun^ for some a ∈ O Y,y. Then f (^) y] ( aun ) b = 0 for some nonzero b ∈ O X,x , and f (^) y] ( a ) b 6 = 0; indeed, if f ] ( a ) were a zero divisor, a would be an element of m y , implying that t = a ′ un +1^ for some a ′. Then also f ] ( u ) is a zero divisor in O X,x , and so f ] ( u ) is contained in some associated prime p of O X,x , by Lemma 3.43.4. But this means that there is an associated point x ′^ ∈ X such that f ( x ′) = y , against our assumption.
Corollary 1.17. Let f : X → Y be a morphism of locally Noetherian schemes, with Y a regular and integral scheme of dimension 1_. Then f is flat if and only if every associated point of X maps to the generic point of Y._ (^1) Every regular local domain of dimension 1 is a DVR, and in particular it is a PID[Har77Har77, Theorem 6_._ 2A, chapter 1]. (^2) The uniqueness follows from the fact that O Y,y , being a DVR, is a UFD.
Figure 1: The twisted cubic squashed on the z = 0 plane (Example 1.211.21).
and in particular the fibre over the closed point 0 is defined in A^3 k by the ideal
( z^2 , xz, yz, y^2 − x^2 ( x + 1)).
We see that the resulting scheme is contained in the plane z = 0, and has the same support as a nodal cubic. For x 6 = 0 and y 6 = 0 moreover the local ring of this X 0 has no nilpotents, hence X 0 is reduced away from the node (0 , 0 , 0). At the node instead we have a nilpotent element in the ring of X 0 , since z is not in the defining ideal of X 0 but z^2 is. The general member of this family is a smooth cubic, but the flat limit for t → 0 is singular, with an embedded point.
Example 1.22. A very similar example is obtained by using the same automorphisms of A^3 k, and choosing as the starting point the scheme defined by the ideal (− 2 x + z^2 − 4 , x^2 + y^2 − 2 x ). This scheme is just the curve obtained by intersecting a cylinder and a sphere, it’s known as “Viviani’s curve”. The family of morphisms σa of the previous example “flattens” this curve onto the plane z = 0, intuitively giving a circle for a going to 0. A picture of the process is given in Figure 22.
Doing the same computation we find that the family of schemes is described by the family of ideals
Ia = (− 2 x +
z^2 a^2 − 4 , x^2 + y^2 − 2 x )
and describes a subscheme of Spec(k[ x, y, z, t ]) which is flat over Spec(k[ t ]). Indeed the family Xa for a 6 = 0 is given by the fibres of the morphism
Spec
( k[ x, y, z, t ] t (− 2 x + z 2 t^2 −^4 , x
(^2) + y (^2) − 2 x )
) ⊆ D ( t ) ⊆ Spec(k[ x, y, z, t ]) −→ Spec(k[ t ]).
According to our theory, this family admits a flat limit; this is computed again via Gr¨obner basis, and it is given by the spectrum of the ring
k[ x, y, z, t ] (5 t^4 + 4 t^4 y^2 − 6 t^2 z^2 + z^4 , t^2 + 2 t^2 x − z^2 , 5 t^2 + 4 t^2 y^2 − 5 z^2 + 2 xz^2 , − 2 x + x^2 + y^2 )
We see that again the fibre over the closed point 0 is described by the circle − 2 x + x^2 + y^2 = 0 with some nonreduced structure given by the nilpotent element z ; this time, however, the nonreducedness is everywhere.
One reason for studying in detail one-parameter families is that they can be used to check flatness of much more general families, c.f. [EH00EH00, Lemma II − 30].
Theorem 1.23. Let k be a field, Y a reduced locally Noetherian k -scheme and f : X → Y a morphism of finite type. Then for any closed point y ∈ Y , f is flat at y if and only if for any regular, integral, locally Noetherian k -scheme Y ′ , any closed point y ′^ ∈ Y and any morphism ϕ : Y ′^ → Y such that ϕ ( y ′) = y, the morphism
X ′^ = X × Y Y ′^ f ′ −→ Y ′
is flat at y ′.
Proof. The implication “⇒” is immediate from the previous results and the fact that flatness is stable under arbitrary base changes. The other direction, which is much more complicated, in given by [RG71RG71, Corollaire 4_._ 2_._ 10].
1.2 Some properties of flatness
A very nice property of flatness is that it is stable under arbitrary base changes.
Proposition 1.24. Let f : X → Y be a morphism of schemes, and let F be a f -flat O X -module. Then
1. for any base change
X × Y Y ′^ X
Y ′^ Y
ϕ
f ′^ f g
also ϕ ∗F is f ′ -flat;
2. if h : Y → Z is a flat morphism of schemes, then F is ( h ◦ f ) -flat.
Before going on, we define a relative version of the cohomology of a sheaf.
Definition 1.25. Let f : X → Y be a morphism of schemes. The higher direct image functors Rif ∗ are the right derived functors of f ∗ : Ab( x ) → Ab( Y ).
This definition is a bit obscure, but it is not too hard to check some properties that help the intuition. For the details we refer to [Har77Har77, §8, chapter III].
Proposition 1.26. Let f : X → Y be a morphism of schemes, and let F be a sheaf of O X -modules. Then for every i ≥ 0 , Rif ∗F is the sheaf associated to the presheaf { V 7 → Hi ( f −^1 ( V ) , F f − (^1) ( V ))
} .
Moreover, if X is Noetherian, F is quasi-coherent and Y is affine, Rif ∗F ∼= Hi ˜( X, F ).
First nice property of flatness: cohomology commutes with flat base changes.
Proposition 1.27. Let f : X → Y be a separated morphism of finite type of Noetherian schemes, and let F be a quasi-coherent sheaf on X. Let u : Y ′^ → Y be a flat morphism, with also Y ′^ Noetherian, and consider the fibred product
X ⊗ Y Y ′^ X
Y ′^ Y
v
f ′^ f u
Then there is a natural isomorphism between u ∗ Rif ∗F and Rif (^) ∗′( v ∗F ) , for all i ≥ 0_._
Proof. Since affine open sets form a basis for the topology on Y ′, we may assume that Y ′^ and Y are affine, Y ′^ = Spec( A ′) and Y = Spec( A ). By Proposition 1.261.26 then we have to exhibit an isomorphism between Hi ( X ′ , v ∗F ) and Hi ( X, F ) ⊗ A A ′, where X ′^ := X ⊗ Y Y ′. Notice that X is separated (since it is separated over an affine scheme), and since F is quasi-coherent we can compute Hi ( X, F ) by Cech cohomology. For the same reasons we can computeˇ Hi ( X ′ , v ∗F ) by Cech cohomology, since byˇ base extension also f ′^ is separated and of finite type (hence X ′^ is also separated and Noetherian). Let U be an open cover of X by affine sets; since the fibred product of affine schemes is again affine, v −^1 ( U ) is again affine, and so v −^1 (U) is an open cover of X ′. We compute Cech cohomologies withˇ respect to these two covers. By definition, the Cech complexˇ C·( v −^1 (U) , v ∗F ) is just C·(U , F ) ⊗ A A ′; indeed, for every U ∈ U we have v ∗F ( v −^1 ( U )) = F ( U ) ⊗ A A ′^ (to check this, simply notice that U is affine). So we have to compute the cohomology of the complex
C^0 (U , F ) ⊗ A A ′^ C^1 (U , F ) ⊗ A A ′^ C^2 (U , F ) ⊗ A A ′^...
but since A ′^ is a flat A -module, tensoring with A ′^ preserves the cohomology of any sequence of A - modules.
1.2.1 Flatness and dimension of fibres
This theorem tells us that, when considering flat families, the dimension of the fibres of the family behaves according to our “geometric intuition”. As it is given here, it is a slight generalization of Proposition 9_._ 5 in [Har77Har77, chapter III].
Theorem 1.28. Let f : X → Y be a flat morphism of locally Noetherian schemes. Choose x ∈ X, and let y = f ( x ). Then dim(O X,x ) = dim(O Xy ,x ) + dim(O Y,y )
where Xy is the fibre over y.
Proof. We can make a base change Y ′^ → Y , with Y ′^ = Spec(O Y,y ), and consider the new morphism f ′^ : X ′^ = X ⊗ Y Y ′^ → Y ′. Then f ′^ is still flat, and if x ′^ ∈ X ′^ is a preimage of x under the map X ′^ → X , we have dim(O X ′ ,x ′ ) = dim(O X,x )
and also
dim(O X ′ y ,x ′ ) = dim(O Xy ,x ) dim(O Y ′ ,y ) = dim(O Y,y ).
Indeed, making the base change Y ′^ → Y just means that we are restricting the morphism f : X → Y to an arbitrarily small neighbourhood of y in Y , and since all the numbers involved in the theorem are can be computed locally, they are left untouched by this restriction. So we may assume that Y = Spec( A ) for a local Noetherian ring A , with y being the maximal ideal of A. We proceed by induction on dim( Y ). If dim( Y ) = 0, it means that every element of y is nilpotent. Now, choose an affine neighbourhood Spec( B ) of x in X , and notice that the fibre Xy around x is defined by Spec( B ⊗ A k( y )). The ideal sheaf of Xy in X is thus defined by the kernel of the map B → B ⊗ A k( y ), and this can be computed as follows: consider the short exact sequence
0 → y → A → k( y ) → 0
and tensoring with B we get by flatness another short exact sequence
0 → yB → B → k( y ) ⊗ A B → 0
hence the ideal is yB B. But since each element of y is nilpotent this ideal is contained in the nilradical of B , hence dim(O Xy ,x ) = dim X,x and the thesis follows. Assume now that dim( Y ) > 0. We can make a base extension to Y red, and the numbers in question do not change; hence we may assume that Y is reduced. Since its dimension is greater than 1, there is a prime ideal p ⊆ A such that { 0 } ⊆ p ⊂ y , so that we can find some t ∈ y that is not contained in any minimal prime of A by the Prime Avoidance Lemma 3.63.6; notice that, since Spec A is Noetherian, it has a finite number of irreducible components (hence A has finitely many minimal primes). In particular t is not a zero divisor, by Lemma 3.23.2. Let Y ′^ = Spec( A/ ( t )), and make the base extension Y ′^ → Y , obtaining
X ′^ X
Y ′^ Y
f ′^ f
and notice as usual that f ′^ is also a flat morphism. Moreover, if Spec( B ) is, as before, an open affine neighbourhood of x in X , we have that x has a preimage in X ′^ and that this preimage has an open affine neighbourhood isomorphic to Spec( B/ ( f ] ( t ))). This follows readily by noticing that the following diagram is a push-forward
A B
A/ ( t ) B ⊗ A A/ ( t )
f ]
and that f ] ( t ) ⊆ x , B ⊗ A A/ ( t ) ∼= B/ ( f ] ( t )). Since f is flat, also f ] ( t ) is not a zero divisor in O X,x , and so by Lemma 3.33.3 and Lemma 3.13.1 we have
dim(O Y ′ ,y ) = dim(O Y,y ) − 1 dim(O X ′ ,x ) = dim(O X,x ) − 1_._
By the induction hypothesis then we find
dim(O X ′ ,x ) = dim(O X ′ y ,x ) + dim(O Y ′ ,y )
and to get the thesis it is then enough to notice that fibres of X ′^ and X over y are naturally isomorphic, hence dim(O X y ′ ,x ) = dim(O Xy ,x ).
Write X = Proj(k[ x 0 ,... , xr ]), and let x ∈ H^0 ( X, O(1)) be such that the hyperplane H = { x = 0} does not contain any irreducible component of supp(F ), so that H ∩ supp(F ) has dimension strictly less than d. Let μ : O X (−1) → O( X ) be the map given by multiplication by x ; this map fits into an exact sequence of sheaves on X
0 O X (−1)^ μ O X O H 0
where O H is the structure sheaf of H considered as a sheaf on X. Tensoring with F gives another exact sequence
0 K F (−1) F R 0 μ ⊗ 1
and since O X (1) preserves exact sequences (it is locally free) we find for all n an exact sequence
0 K ( n ) F ( n − 1) F ( n ) R( n ) 0_._
It is immediate to check that the additivity property of χ implies that χ (F ( n )) − χ (F ( n − 1)) = χ (R( n )) − χ (K ( n )), so it will be enough to prove that there is a numerical polynomial Q ∈ Q[ z ] such that χ (R( n )) − χ (K ( n )) = Q ( n ) to have the thesis (c.f. [Har77Har77, Proposition 7_._ 3, chapter I]). This last part follows from the inductive hypothesis, since the supports of K and R have dimension strictly smaller than d. Indeed, Q p is zero whenever p 6 ∈ supp(F ) or x is a unit in O X,p , so supp(Q p ) ⊆ H ∩ supp(F ). the same thing holds for K : if x is a unit in O X,p , the multiplication by x cannot have a nontrivial kernel in F p.
Remark. The requirement of k to be algebraically closed is just so we can guarantee that k is infinite. This is needed to make sure that there is an hyperplane in P n k not containing any irreducible component of supp(F ). However, we can avoid making this additional hypothesis: indeed, for an arbitrary field k, its algebraic closure ¯k is a flat k-module. Then the morphism of schemes Spec(¯k) → Spec(k) is flat; let the following diagram be the base change via this morphism
X ′^ X
Spec(¯k) Spec(k)
π
and recall that Proposition 1.271.27 tells us that for all i ≥ 0
Hi ( X, F ) ⊗k ¯k ∼= Hi ( X ′ , π ∗F ).
Hence for the dimensions of the cohomology groups (vector spaces, in this case) we have
dim¯k Hi ( X ′ , π ∗F ) = dim¯k
( Hi ( X, F ) ⊗k k¯
) = dimk Hi ( X, F )
so Theorem 2.22.2 holds also for k not algebraically closed.
Remark. Recall that for a projective scheme X over a Noetherian scheme and a coherent sheaf F , there is some N ∈ Z≥ 0 such that dim(F ( n )) = 0 for all n ≥ N. Then the previous theorem tells us that there is a polynomial P ( z ) with rational coefficients such that dim(F ( n )) = P ( n ) for all sufficiently big n. Traditionally this is known as the Hilbert polynomial of F. The Hilbert polynomial of X is defined as the Hilbert polyomial of the sheaf O X (1); it carries a lot of informations about X , for example the dimension, the degree and the arithmetic genus.
Lemma 2.3. Consider an exact sequence of coherent modules on X = P nA,
F 1 → F 2 → · · · → F n − 1 → F n.
Then, for all m sufficiently large we have an exact sequence
Γ ( X, F 1 ( m )) → Γ ( X, F 2 ( m )) → · · · → Γ ( X, F n − 1 ( m )) → Γ ( X, F n ( m )).
Proof. By induction on n. For n = 3, expand the sequence as
0 → ker 1 → F 1 → ran 1 → 0 → ran 1 → F 2 → ran 2 → 0 → ran 2 → F 3 → Q → 0_._
From each short exact piece we can apply Serre’s vanishing Theorem [Har77Har77, Theorem 5_._ 2, chapter III] to find exact sequences in cohomology, like
0 → H^0 ( X, ker 1 ( m )) → H^0 ( X, F 1 ( m )) → H^0 ( X, ran 1 ( m )) → 0_._
Putting together the various pieces so obtained, we get the thesis. For n > 3, the same reasoning can be applied: just split the sequence at the last element.
Theorem 2.4. Let T be an integral Noetherian scheme, and let X be a closed subscheme of P nT. Then X is flat over T if and only if the Hilbert polynomials Pt of the fibres Xt (seen as closed subschemes of P n k( t ) ) do not depend upon t ∈ T.
Proof. By considering O X as a coherent sheaf on P nT , we see that it will be enough to prove that for any coherent sheaf F on X = P nT , F is flat over T if and only if for all t ∈ T the Hilbert polynomial of F t equals the Hilbert polynomial of F 0 , where 0 ∈ T is the generic point. Up to performing a base change Spec(O T,t ) → T we may also assume that T = Spec( A ) for a local Noetherian domain A. We will show that the following are equivalent:
- F is flat over T ;
- H^0 ( X, F ( m )) is a free A -module of finite rank, for all sufficiently large m ;
- the Hilbert polynomial Pt of F t on Xt = P n k( t ) is independent of t.
(1 ⇒ 2). Let U be the usual open affine cover of projective space. We can compute the cohomology groups Hi ( X, F ( m )) by means of the Cech complexˇ C·(U , F ( m )), and since F is flat each term of the complex is a flat A -module. Moreover, for m >> 0 all the higher cohomology groups Hi ( X, F ( m )) vanish, for i > 0. In other words, by taking m large enough we have an exact sequence
0 H^0 ( X, F ( m )) C^0 (U , F ( m ))... C n (U , F ( m )) 0
where all the terms except the first are A -flat. But then also H^0 ( X, F ( m )) must be A -flat, by Proposition 1.61.6. Since it is also finitely generated and A is a local Noetherian ring, from Proposition 1.81.8 we find that H^0 ( X, F ( m )) must be a free A -module. (2 ⇒ 1). Let S = A [ x 0 ,... , xn ], so that X = Proj( S ). Since 2 holds, there is some m 0 such that H^0 ( X, F ( m )) is free for m ≥ m 0. If we define M =
⊕ m ≥ m 0 H (^0) ( X, F ( m )), notice that M equals
Γ ∗(F ) =
⊕ m ∈Z Γ^ ( X,^ F^ ( m )) for degrees^ m^ ≥^ m^0 , so^ M ˜^ =^ Γ ˜∗(F^ ). On the other hand, we know that Γ^ ˜∗(F ) = F. Since by hypothesis M is a free A -module, this means that F is flat. (2 ⇒ 3). The Hilbert polynomial Pt of the fibre over t ∈ T is characterized by the property Pt ( m ) = dimk( t ) H^0 ( Xt, F t ( m )) for large enough m ; then it will be enough to prove for every t ∈ T that, for m large enough, Pt ( m ) = rank AH^0 ( X, F ( m )). This will follow from equation ( 11 ), which we prove without assuming 2. We propose to show that
H^0 ( Xt, F t ( m )) ∼= H^0 ( X, F ) ⊗ A k( t ). (1)
Choose t ∈ T , t = p for a certain p ∈ Spec( A ). Since A p is a flat A -module, by performing the base change Spec( A p) → T we reduce to the case in which t is a closed point of T. Indeed, recall from Proposition 1.271.27 that cohomology commutes with flat base changes. Since A is Noetherian, the maximal ideal of t is finitely generated, say by r elements. Then we can find an exact sequence
A ⊕ r^ A k( t ) 0 (2)
from which we can get an exact sequence
Lemma 3.6 (Prime avoidance). Let I 1 ,... , In be prime ideals of the ring R, and let J R be an ideal such that J 6 ⊆ Ii for all i = 1 ,... , n. Then there exists an element x ∈ J such that x 6 ∈ ⋃ n i =1 Ii.
Remark. Lemma 3.63.6 actually holds even if at most two of the ideals I 1 ,... , In are not prime, but we will not need it in this stronger form.
Proof. By induction on n. For n = 1 the claim is obvious. Assume then that n > 1 and the thesis holds for n − 1; without loss of generality, we may also assume that there are no inclusions among I 1 ,... , In. By the inductive hypothesis we can find x ∈ J \ ( I 1 ∪ · · · ∪ In − 1 ). If x 6 ∈ In we are done, so assume that x ∈ In. Notice that if the product ideal JI 1_... In_ − 1 were contained in In then we would have J ⊆ In , against our hypothesis; then there is some y ∈ JI 1_... In_ − 1 \ In , and x + y is an element of J not contained in In.
3.2 Scheme-theoretic closure
Let X be a scheme, and let V ⊆ X be a closed subset. We define the reduced induced subscheme structure on V as follows: for X = Spec( A ) affine, consider
a :=
⋂ {p ∈ Spec( A ) | p ∈ V }
which is the largest ideal of A such that V = V (a). Then a defines a scheme structure on V , and with this structure V (a) is a reduced scheme: indeed, a = ⋂ {p ∈ Spec( A ) | a ⊆ p}. When X is not affine, we take a covering U = { Ui | i ∈ I } of V consisting of open affines of X. We should check that the scheme structures on each V ∩ Ui glue together to define a global scheme structure on V. By affine communication, it is enough to check that for any distinguished open set D ( f ) of Ui = Spec( A ), the reduced structure on V ∩ Ui induced by Spec( A ) gives the reduced structure on V ∩ D ( f ) induced by Spec( Af ), when restricted to D ( f ). This is quite easy to check: let a be the ideal defining the reduced structure on V ∩ Ui , and let Zi := Spec( A/ a). Then Zi ∩ D ( f ) ∼= Spec( Af / (a Af )). Notice also that the reduced structure on V ∩ D ( f ) is defined by the ideal b =
⋂ {p ∈ Spec( Af ) | p ∈ V ∩ D ( f )}, which is equal to a Af ; this proves the claim, so the reduced induced structure on V is well-defined. From the definition of the reduced induced structure, we find immediately that it is the “smallest” closed subscheme structure on the given closed subset.
Lemma 3.7 (Universal property of the reduced induced structure). Let V be a closed subset of a scheme X, and consider Y as a closed subscheme of X by endowing it with the reduced induced scheme structure. Then for any other closed subscheme Y ′^ of X with the same underlying topological space as Y , the inclusion Y ↪ → X factors through Y ′.
These ideas can be generalized to define a “scheme-theoretic” notion of image (sometimes also called range ) of a morphism of schemes.
Definition 3.8. Let f : X → Y be a morphism of schemes, and let I be the largest quasi-coherent subsheaf of ker( f ]^ : O Y → f ∗O X ). We define the scheme-theoretic image of f to be the closed subscheme of Y defined by I. We denote it by f ( X ).
An alternative terminology that can also be used is schematic image , but we will stick to the former one.
Remark. The scheme-theoretic image is indeed a closed subscheme of Y , see [Har77Har77, chapter II, Proposition 5_._ 9]. However, in general it may fail to have some desirable properties; for example, it may not have as underlying topological space the closure of ran( f ) (for an example of this behaviour see [VakVak, Example 4, § 8_._ 3]). However, it can be easily seen that under some reasonable hypothesis, the scheme-theoretic image of f behaves exactly as we would expect.
Example 3.9. Let f : X → Y be a morphism of schemes, and assume that Y = Spec( B ) is affine. Then f ( X ) is the closed subscheme of Y defined by the ideal
{ b ∈ B
∣∣ ∣ f ] ( b ) = 0 ∈ O X ( X )
} .
To make an explicit computation, consider the map ϕ : k[ x, y ] → k[ t, t −^1 ] defined by ϕ ( p ( x, y )) = p ( t, 0). This corresponds to the inclusion of the scheme Spec(k[ t, t −1]) (the line without a point) into A^2 k, and the scheme-theoretic image in A^2 k is defined by the ideal { p ( x, y ) ∈ k[ x, y ] | p ( t, 0) = 0}, i.e. the ideal ( y ). The scheme-theoretic image is just the whole line. As a slight variation, we could consider the inclusion of the line without a point into the projective plane P^2 k. The target scheme is not affine, but we could easily make the computation affine set by affine set and then glue toghether the results to obtain a quasi-coherent sheaf of ideals on P^2 k.
There are some conditions that allow us to compute the scheme-theoretic image of a morphism affine-locally. In this case, scheme-theoretic images exhibit many pleasant properties. For a proof of the following theorem, see [VakVak, Theorem 8_._ 3_._ 4].
Theorem 3.10. Let f : X → Y be a morphism of schemes. Assume that either X is reduced or f is quasi-compact; then ker( f ]^ : O Y → f ∗O X ) is a quasi-coherent sheaf of ideals on Y , and the underlying set of f ( X ) is the closure of { f ( x ) | x ∈ X }.
We are interested in the scheme-theoretic image mostly because it allows us to talk about the scheme-theoretic closure of a subscheme.
Definition 3.11. Let X be a scheme. A locally closed subscheme of X is a morphism of schemes f : Z → X that can be factored as
Z U X f 1 f 2
where f 1 is a closed immersion and f 2 is an open immersion. In other words, a locally closed subscheme is a closed subscheme of an open subscheme. If f : Z → X is a locally closed subscheme, we define the closure of Z in X to be the scheme-theoretic image of f.
The following lemma tells us that, when considering locally closed subschemes of a locally Noethe- rian scheme, the hypothesis of Theorem 3.103.10 are satisfied.
Lemma 3.12. Let X be a locally Noetherian scheme, and let f : Z → X be a locally closed subscheme of X. Then f is quasi-compact.
Proof. Write f as the composition of an open immersion and a closed one, Z ↪ → U ↪ → X. We know that a closed immersion is quasi-compact and that the composition of quasi-compact morphisms is again quasi-compact. Then to prove the lemma it will be enough to show that U ↪ → X is quasi-compact. Let V = Spec( A ) be an open affine subscheme of X ; notice that A is a Noetherian ring, since X is locally Noetherian. We can write V ∩ U =
⋃ i ∈ I D ( fi ) for some collection of^ fi^ ∈^ A , so that V \ ( V ∩ U ) = ⋂ i ∈ I V^ ( fi ).^ Since^ A^ is Noetherian the ideal^ J^ generated by^ { fi^ |^ i^ ∈^ I }^ is finitely generated, and so there are f 1 ,... , fn ∈ J such that U ∩ V =
⋃ n j =1 D ( fj^ ).
Exercise 3.13_._ It will be useful to see an explicit description of the schematic closure, and by the previous results it is enough to do this on an affine scheme. Consider a locally closed subscheme Z ↪ → U ↪ → Spec( A ) where A is a Noetherian ring, and let f 1 ,... , fn be such that U =
⋃ n i =1 D ( fi ). Since Z is a closed subscheme of U , for each i = 1 ,... , n there is an ideal a i Afi such that Z ∩ D ( fi ) = Spec( Afi / a i ). Moreover a i and a j define the same ideal in Afifj , for all i 6 = j. Let ¯a i be a i ∩ A , and set a :=
⋂ n i =1 ¯a i. Then the scheme-theoretic closure of^ Z^ in Spec( A ) is Z ¯^ := Spec( A/ a). It is easy to
check that for all i , a Afi = a i. Indeed, consider (^) fx k i ∈ a i. Since
xf (^) jk ( fifj ) k^ ∈^ a j^ Afifj^ by hypothesis, for all
j there is some kj such that xf (^) ik j ∈ ¯a j. Then if m := max{ k 1 ,... , kn } we have that xf (^) im ∈ a, and so a Afi = ai.
4 More examples
Example 4.1 (Computation of an affine closure). Let X be the crossing of two affine lines, X = Spec( A ) with A = k[ x, y ] / ( xy ), and consider the scheme Z = Spec(k[ x, x −^1 ]). The map k[ x, y ] / ( xy ) → k[ x, x −^1 ] defined as f ( x, y ) + ( xy ) 7 → f ( x, 0) induces on Z a structure of open subscheme of X , since it is an isomorphism between Z and D ( x + ( xy )) ⊂ X. What is the closure of Z in X? We just have to compute the ideal a A of all the elements α ∈ A that are sent to zero by the map A → Ax +( xy ). For all α ∈ A , [ α, 1] = [0 , 1] in Ax +( xy ) if and only if there is some n such that ( x + ( xy )) nα = 0. This happens precisely when α ∈ ( y + ( xy )) A , so a = ( y + ( xy )). Then the closure of Z in X is Spec( A/ ( y + ( xy ))) = Spec(k[ x ]); this is just the affine line.
Example 4.2 (Failure of Lemma 3.153.15 for non-Noetherian rings). Consider the ideal
I = ( xiyi | i ∈ Z≥ 0 ) k[ x 1 , y 1 , x 2 , y 2 , x 3 ,... ].
Then A := k[ x 1 , y 1 , x 2 , y 2 ,... ] /I is not a Noetherian ring, and in this ring we can see that Lemma 3.153. fails. Indeed, consider the prime ideal p = (¯ x 1 , x ¯ 2 , ¯ x 3 ,... ), where for any a ∈ k[ x 1 , y 1 ,... ] we denote by ¯ a its image in A under the canonical projection. Then A p is a field, and p A p = 0; in particular, p A p = Ann( α ) for any α ∈ A p \ { 0 }, while p is not an associated prime of A. Indeed, assume that m ∈ A is such that p ⊆ Ann( m ), and let r be such that m ∈ (¯ x 1 , y ¯ 1 ,... , x ¯ r, y ¯ r ). Then it is easy to check that ¯ xr +1 m = 0 if and only if m = 0.
Example 4.3 (Failure of Lemma 3.163.16 for non-Noetherian rings). Let k be an algebraically closed field, let I be the ideal
( x^20 , x 0 − x^21 , x 1 − x^22 , x 2 − x^23 ,...
) k[ x 0 , x 1 , x 2 , x 3 ,... ] and consider the ring
R =
k[ x 0 , x 1 , x 2 , x 3 ,... ] I
If we let y 0 := xi + I for i ≥ 0 then we have that y^20 = 0, y^2 k +1 = yk and y^2 k k =^ y^0 for all^ k^ ∈^ Z≥^0. The ring R satisfies the following properties:
- R is not Noetherian;
- R is a local ring of finite (Krull) dimension;
- if m is the maximal ideal of R , then every element of m is a zero divisor, but m is not an associated prime of R.
1_._ The increasing sequence of ideals ( y 0 ) ⊂ ( y 1 ) ⊂ ( y 2 ) ⊂... does not stabilize. Indeed, if this were not the case there would be some f ∈ R and some k ∈ Z > 0 such that f yk = yk +1. But ( f yk )( k +1) = 0,
while y (
k +1) k +1 =^ y^0 6 = 0. 2_._ Let m be the ideal of R generated by y 0 , y 1 , y 2 ,.... Each element of m is nilpotent, and all the elements in R \ m are invertible. Then R is indeed a local ring, with maximal ideal m. From the fact that each element of m is nilpotent it follows readily that if p ⊆ m is a prime ideal of R , p must equal m. 3_._ Suppose that x ∈ R is such that mx = 0 for all m ∈ m. Since x ∈ m, we can write it as
x = x 0 ya k^0 + · · · + xnya kn
for some k , with 1 ≤ a 0 < · · · < an ≤ 2 k^ and x 0 ,... , xn ∈ k. Thus multiplying x by y (
k (^) − a 0 ) k we obtain
0 = xy (
k (^) − a 0 ) k =^ x^0 ε
so x 0 = 0. Then we may multiply x by y (
k (^) − a 1 ) k to find^ x^1 = 0, and recursively we get^ xj^ = 0 for all j , i.e. x = 0.
Figure 3: Two different schemes obtained from Example 4.54.5.
Example 4.4 (Failure of Lemma 3.163.16 for non-Noetherian rings). A very similar reasoning can be carried out with the ring k[ x 1 , x 2 , x 3 ,... ] ( x^21 , x^22 , x^23 ,... )
Example 4.5 (A flat morphism). Let k be an algebraically closed field, and let f ( x, y ) ∈ k[ x, y ]. Consider the ideal I = ( z^2 − f ( x, y )) k[ x, y, z ], the scheme X = Spec(k[ x, y, z ] /I ) and the map X → A^2 k induced by the inclusion k[ x, y ] → k[ x, y, z ] /I. Then X is flat over A^2 k; indeed, if we write R := k[ x, y ], the R -module k[ x, y, z ] /I is freely generated by 1 and z , so that k[ x, y, z ] /I ∼= R ⊕ R as a R -module. For a couple of particular cases, see Figure 33.
Example 4.6 (A morphism that is not flat). Let k be an algebraically closed field, and consider the affine scheme X = Spec(k[ x, y, z ] / ( xz − y )) together with the map f : X → A^2 k defined by the inclusion f ]^ : k[ z, y ] → k[ x, y, z ] / ( xz − y ). See also Figure 44. Then X is irreducible and of dimension 2, so by Corollary 1.291.29 the fibre over each p ∈ A^2 k should be 0-dimensional. But the fibre over p = ( x, y ) ∈ A^2 k is a whole copy of k[ z ], hence f cannot be flat.
Example 4.7 (Another flat family of cubics). Consider the twisted cubic C given by the 3-uple immersion of P^1 k in P^3 k,
P^1 k → P^3 k [ s : u ] 7 → [ s^3 : s^2 u : su^2 : u^3 ].
Then C is defined in P^3 k by the homogeneous ideal
Ih = ( x 0 x 3 − x 1 x 2 , x 1 x 3 − x^22 , x 0 x 2 − x^21 ).
Consider the family of curves C = { Ca | a 6 = 0} described by the family of homogeneous ideals
Ih ( a ) = ( x 0 x 3 −
a
x 1 x 2 , x 1 x 3 −
a^2
x^22 ,
a
x 0 x 2 − x^21 ).
Each member of the family is a closed subscheme of the fibre, over the closed point ( t − a ) ∈ Spec(k[ t ]), of the natural morphism of schemes
P^3 k × (A^1 k \ {( t )}) = P^3 k × Spec(k[ t, t −^1 ]) −→ A^1 k = Spec(k[ t ]).
