Flatness in algebraic geometry: a family of conics in 3 AC

Jin Hyung To

June 1, 2018

Assume that all rings are commutative with 1.

1 A family of conic curves

Here is the real part of a family of conics defined by the equation x2 + y2 − t = 0 in the affine space 3 = Spec ( [x, y, t]). AC C

x2 + y2 − 2 = 0

x2 + y2 − 1 = 0

• x2 + y2 = 0

x2 + y2 + 1 = 0

x2 + y2 + 2 = 0

C[x,y,t] 2 2 The C-algebra morphism ϕ : C[t] → (x2+y2−t) defined by ϕ(t) = t + (x + y − t) gives the morphism of affine schemes

 [x, y, t]  f : Spec C → Spec( [t]) (x2 + y2 − t) C

−1  C[x,y,t]  defined by f(p) = ϕ (p) for each prime ideal p ∈ Spec (x2+y2−t) . In this section we will show that f is a flat morphism using local criterion of flatness and also other criterion for flatness. First we observe this morphism. The only prime ideals in C[t] are (t + a) or (0) for a ∈ C, so the image of ϕ is (t + a) or (0). For example, p = (x + (x2 + y2 − t), y + (x2 + y2 − t)) ⇒ f(p) = ϕ−1(p) = (0), 2 2 2 2 −1 p = (x + a + (x + y − t), y + b + (x + y − t)) ⇒ f(p) = ϕ (p) = (0) for any a, b ∈ C, p = (x + (x2 + y2 − t), y + (x2 + y2 − t), t + (x2 + y2 − t)) ⇒ f(p) = ϕ−1(p) = (t), p = (t + (x2 + y2 − t)) ⇒ f(p) = ϕ−1(p) = (t), 2 2 −1 p = (t + a + (x + y − t)) ⇒ f(p) = ϕ (p) = (t + a) for a ∈ C.

1 The topological fiber at q is f −1(q), and the fiber at q is

 [x, y, t]  Spec C × Spec( (q)), (x2 + y2 − t) Spec(C[t]) C where q ∈ Spec(C[t]), and C(q) is the residue field C[t]q/m of the local ring (C[t]q, m), where m = qC[t]q, and q = (t − a) or (0) for a ∈ C. ∼ For example, the residue field at q = (t) is C(q) = C[t](t)/(t)C[t](t) = C (The kernel of the map C[t] → C[t](t) → C[t](t)/(t)C[t](t) is (t)), so

 [x, y, t]   [x, y, t]   [x, y]  Spec C × Spec( ) = Spec C ⊗ = Spec C (x2 + y2 − t) Spec(C[t]) C (x2 + y2 − t) C[t] C (x2 + y2)

 C[x,y,t]  We observe the following. Let X = Spec (x2+y2−t) ,Y = Spec (C[t]). We will prove that f is a flat morphism. (1) Every fiber of the map f has the same dimension 1. The fiber at the point corresponding to a prime ideal q ⊂ C[t] is  [x, y, t]   [x, y, t]  Spec C × Spec( (q)) = Spec C ⊗ (q) . (x2 + y2 − t) Spec(C[t]) C (x2 + y2 − t) C[t] C

We compute the tensor product of rings.  C[x,y] if q = (t − a) C[x, y, t]  (x2+y2−a) ⊗ [t] C(q) = (x2 + y2 − t) C C(t)[x,y]  (x2+y2−t) if q = (0).

n n n n n−1 n−2 n−1 n Since t = t − a + a = (t − a)(t + t a + ... + a ) + a , tensoring C((t − a)) is plugging in t = a. At the generic point q = (0) the residue filed is C(q) = C(t), so t is a unit. Hence the dimension of each of the fibers is 1 The fiber at q = (t) is the union of two affine lines, and has singularity at (0, 0). The fiber at q = (t − a) with a 6= 0 is a nonsingular conic curve.

Sine the dimension of fiber is 1, dim Xa = dim X − dim Y at a point a ∈ Y . Here dim X = 2 and dim Y = 1. ( V (x2 + y2) if a = 0 The fiber Xa = . V (x2 + y2 − a) if a 6= 0

The fiber X0 is the union of two affine lines, and has singularity at (0, 0). Xa is nonsingular conic curve for every a 6= 0.

Exercises Problems:

C[x,y,t] (2) Prove that f(D(h)) is an open set for every h ∈ (x2+y2−t) , i.e., f is an open morphism.

(3) Is f smooth of relative dimension 1? In other words, prove or disprove that (i) f is flat; (ii) if X0 ⊆ X and Y 0 ⊆ Y are irreducible components such that f(X0) ⊆ Y 0, then dim X0 =

2 dim Y 0 + 1 ; (iii) for each point x ∈ X (closed or not),

dimk(x)(ΩX/Y ⊗ k(x)) = 1.

C[x,y,t] (4) Is f faithfully flat? In other words, prove or disprove that for every prime ideal p ⊂ (x2+y2−t) , p C[x,y,t] Op,X is a flat Of(p),Y -module, and for every prime ideal ⊂ (x2+y2−t) Op,X ⊗Of(p),Y M 6= 0 for every nonzero Of(p),Y -module M.

C[x,y,t] The C-algebra morphism ϕ makes (x2+y2−t) a C[t]-algebra, and ϕ induces a local morphism #  C[x,y,t]   C[x,y,t]  f : C[t]f(p) → 2 2 for each p ∈ Spec 2 2 . X is flat over Y via the morphism f (x +y −t) p (x +y −t)  C[x,y,t]  (or f is a flat morphism) if OX is flat over Y , where X = Spec (x2+y2−t) and Y = Spec(C[t]), i.e., Op,X is a flat Of(p),Y -module for all p ∈ X. Clearly, Op,X is a finitely generated Op,X -module for all p ∈ X. So, by the local criterion for C[t] flatness, Op,X is a flat Of(p),Y -module if and only if Tor1 (Op,X , C(f(p))) = 0 for every p ∈ X.

 C[x,y,t]  We know that Op,X = 2 2 and Of(p),Y = C[t]f(p). (x +y −t) p

 C[t] Tor1 (Op,X , C(t)) if f(p) = (0), C[t]  Tor1 (Op,X , C(f(p))) =  C[t] Tor1 (Op,X , C) if f(p) 6= (0) (i.e., f(p) = (t + a)). Case 1: f(p) = (0)

  ! C[t] C[t] C[x, y, t] Tor1 (Op,X , C(f(p))) = Tor1 2 2 , C[t](0)/(0)C[t](0) (x + y − t) (0)   ! C[t] C[x, y, t] = Tor1 2 2 , C[t](0) (x + y − t) (0) = 0

C[t]  C[x,y,t]  C[t]  C[x,y,t]  since Tor1 (x2+y2−t) , C[t] = Tor1 C[t], (x2+y2−t) = 0.

Case 2: f(p) 6= (0)(f(p) = (t + a)) We resolve C = C[t](t+a)/(t + a)C[t](t+a) as a C[t]-module.

·t 0 → C[t] −→ C[t] → C → 0

 C[x,y,t]  Tensoring with 2 2 gives the following complex of C[t]-modules: (x +y −t) p        C[x,y,t]  1⊗(·t)  C[x,y,t]   C[x,y,t]  0 → 2 2 ⊗ [t] C[t] −−−−→ 2 2 ⊗ [t] C[t] → 2 2 ⊗ [t] C → 0 (x +y −t) p C (x +y −t) p C (x +y −t) p C

C[t] Since 1 ⊗ (·t) is injective, Tor1 (Op,X , C) = 0. Thus, X is flat over Y .

We also use the following criterion (Theorem 2) for flatness to show that X is flat over Y .

3 Lemma 1. If M is a nonzero module over a ring A, then there exists a maximal ideal I ⊂ A such Q that MI := M ⊗A AI 6= 0. More generally, the homomorphism m 7→ (m ⊗ 1)I from M to MI . I

Proof. Let ϕ : M → MI := M ⊗A AI (ϕ(m) = m ⊗ 1). Then m ∈ ker(ϕ) if and only if m ⊗ 1 = 0 if and only if r(m ⊗ 1) = rm ⊗ 1 = 0 for some r∈ / I if and only if rm = 0 for some r∈ / I, i.e., r ∈ Ann(m) for some r∈ / I. Note that Ann(m) = A if and only if m = 0. By assumption, M 6= 0, so Ann(m0) 6= A for some m0 6= 0. There is a maximal ideal I0 ⊂ A containing Ann(m0). So, m0 ∈/ ker(M → MI0 ), i.e., m0 ⊗ 1 6= 0, so MI0 6= 0. Theorem 1. (Proposition 9.2, [Hartshorne1]) Let ϕ : A → B be a homomorphism of rings, and let M be a B-module. Let f : X → Y be the corresponding morphism of affine schemes, where X = Spec(B) and Y = Spec(A), and let F = Mf. Then F is flat over Y if and only if M is a flat A-module.

Proof. We must show that Mq := M ⊗B Bq is a flat Aϕ−1(q)-module for every prime ideal q ⊂ B if and only if M is a flat A-module. Suppose M is a flat A-module. Then Mp := M ⊗A Ap is a flat Ap-module for every prime ideal p ⊂ A. So, for every prime ideal q ⊂ B,Mϕ−1(q) := M ⊗A Aϕ−1(q) is a flat Aϕ−1(q)-module. The ring homomorphism ϕ : A → B induces the ring homomorphism A −1 → B . By the base extension theorem, M := M ⊗ B = M −1 ⊗ B is a flat ϕ (q) q q B q ϕ (q) Aϕ−1(q) q Aϕ−1(q)-module. Conversely, suppose that for some exact sequence 0 → L → N of A-modules, L ⊗A M → N ⊗A M has a nonzero kernel K, i.e.,

0 → K → L ⊗A M → N ⊗A M.

This is an exact sequence of B-modules, so by Lemma 1, there exists a prime ideal q ⊂ B such that Kq 6= 0. So,

0 → K ⊗B Bq → L ⊗A (M ⊗B Bq) → N ⊗A (M ⊗B Bq). = = = Kq Mq Mq

L ⊗ M = (L ⊗ A −1 ) ⊗ M and N ⊗ M = (N ⊗ A −1 ) ⊗ M . A q A ϕ (q) Aϕ−1(q) q A q A ϕ (q) Aϕ−1(q) q

Let ϕ : A → B be a homomorphism of rings. Let X = Spec(B) and Y = Spec(A). Recall that the corresponding morphism f : X → Y of affine schemes is flat (or we say X is flat over Y ) if OX is flat over Y . In other words, Oq,X is a flat Of(q),Y -module for all q ∈ X, where Oq,X is # a Of(q),Y -module through the local morphism f : Of(q),X → Oq,X . Note that Of(q),Y = Aϕ−1(q) −1 and Oq,X = Bq. Here, f(q) = ϕ (q).

Corollary 1. Let ϕ : A → B be homomorphism of rings. Then a corresponding morphism f : Spec(B) → Spec(A) of affine schemes is flat if and only if B is a flat A-module.

Proof. We need to show that OSpec(B) is flat over Spec(A) if and only if B is a flat A-module. Let

M = B. Then Mf = OSpec(B). By Theorem 1, Mf is flat over OSpec(A) if and only if M = B is a flat A-module.

4 pn Example 1. For a commutative ring A with 1 and a prime number p, let µpn (A) := {a ∈ A|a −1 = pn 0} be the group scheme with ring A[x]/(x − 1) over Spec(A), i.e., µpn (A) = Spec(O(µn(A))) = pn n Spec(A[x]/(x − 1)). Since O(µn(A)) is a free module of rank p over A, by Corollary 1, the scheme µn(A) is flat over Spec(A). In fact, µpn is a functor from commutative rings to abelian groups for every integer n ≥ 0.

Example 2. The abelian group Z2 := Z/2Z is a Z-module. In fact, every Z-module is the same Z as an abelian group. Since Tor1 (Z/2Z, Z2) 6= 0, Z2 is not a flat Z-module. This can be interpreted as a ring (a field). Then Spec(Z2) is not flat over Spec(Z).

2 Example 3. The group scheme µ2(Z) = Spec(Z[x]/(x − 1)) is flat over Spec(Z). Theorem 2. Let M be a module over a Noetherian ring A. Then M is a flat A-module if and A only if Tor1 (M, A/p) = 0 for every prime ideal p ⊂ A. Proof. See [To] for full proof or [Hartshorne2].

 C[x,y,t]  Alternatively, we use Theorem 2 to show that the morphism f : Spec (x2+y2−t) → Spec(C[t]) C[x,y,t] of schemes is flat, where f is induced by the morphism ϕ : C[t] → (x2+y2−t) of rings defined 2 2  C[x,y,t]  by ϕ(t) = t + (x + y − t). By Corollary 1, f is a flat morphism (or Spec (x2+y2−t) is flat C[x,y,t] over Spec(C[t])) if and only if (x2+y2−t) is a flat C[t]-module. C[t] is a Noetherian ring, so this is C[t]  C[x,y,t]  equivalent to Tor1 (x2+y2−t) , C[t]/p = 0 for every prime ideal p ⊂ C[t].

   TorC[t] C[x,y,t] , [t] if p = (0),  [x, y, t]   1 (x2+y2−t) C TorC[t] C , [t]/p = 1 2 2 C   (x + y − t)  C[t] C[x,y,t] Tor1 (x2+y2−t) , C if p 6= (0) (i.e., p = (t + a)).

Case 1: p = (0)

 [x, y, t]  TorC[t] C , [t] = 0 because the functor TorC[t]( ., [t]) is exact. 1 (x2 + y2 − t) C 1 C since C[t] is free over C[t].

Case 2: p 6= (0) (p = (t + a)) We resolve C = C[t](t+a)/(t + a)C[t](t+a) as a C[t]-module.

·t 0 → C[t] −→ C[t] → C → 0.

C[x,y,t] Tensoring with (x2+y2−t) gives the following complex of C[t]-modules:

C[x,y,t] 1⊗(·t) C[x,y,t] C[x,y,t] 0 → (x2+y2−t) ⊗C[t] C[t] −−−−→ (x2+y2−t) ⊗C[t] C[t] → (x2+y2−t) ⊗C[t] C → 0

C[t]  C[x,y,t]   C[x,y,t]  Since 1 ⊗ (·t) is injective, Tor1 (x2+y2−t) , C = 0. Thus, Spec (x2+y2−t) is flat over Spec(C[t]).

5 2 History of flatness

In [Hartshorne2], “The notion of flatness is due to [Serre], who showed that there is a one-to-one correspondence between coherent algebraic sheaves on a projective variety over C and the coherent analytic sheaves on the associated complex analytic space. He observed that the algebraic and analytic local rings have the same completion, and that this makes them a “flat couple.” The observation that localization and completion both enjoy this property, and that flat modules are those that are acyclic for the Tor functors, explained and simplified a number of situations by combining them into one concept. Then in the hands of Grothendieck, flatness became a cen- tral tool for managing families of structures of all kinds in algebraic geometry. The local criterion of flatness is developed in [Grothendieck2] IV, §5. Our statement is [loc. cit., 5.5]. A note before [loc. cit. 5.2] says “La proposition suivante a ´et´ed´egag´eeau moment du S´eminairepar Serre; elle permet des simplifications substantielles dans le pr´esent num´ero.”(“The next proposition was cleared at the time through Seminar by Serre; it allows the substantial simplification into the present version.”) The infinitesimal study of the Hilbert scheme is in Grothendieck’s Bourbaki seminar [Grothendieck1] expos´e221.” In [Eisenbud], Chapter 6 “The notion of flatness was first isolated by Serre [1955 - 1956] and was then systematically developed and mined by Grothendieck. It is now a central theme in algebraic geometry and commutative algebra.”

References

[Eisenbud] Eisenbud, David: Commutative Algebra with a View Toward Algebraic Ge- ometry, Springer, 2004.

[Grothendieck1] Grothendieck, Alexander: Fondements de la g´eom´etriealg´ebrique [Extraits du S´eminaireBourbaki 1957 - 1962], Secr. Math. 11, rue Pierre Curie, Paris (1962).

[Grothendieck2] Grothendieck, Alexander: Revˆetements ´etales et groupe fondamental, Springer, LNM 224, 1971.

[Hartshorne1] Hartshorne, Robin: Algebraic Geometry, Springer, 1976.

[Hartshorne2] Hartshorne, Robin: Deformation Theory, Springer, 2010.

[Sernesi] Sernesi, Edoardo: Deformations of algebraic schemes, Springer, Grundlehren 334, 2006.

[Serre] Serre, J-. P-.: G´eom´etriealg´ebrique et G´eom´etrieanalytique , Ann. Inst. Fourier 6 (1956), 1 - 42.

[To] To, Jinhyung: Deformation Theory

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