Finite Morphisms and Flatness

Exercise 1.6.4.Letf : (C,0)→(C²,0) be a finite morphism of germs such that, for a sufficiently small representativef :U →V, the restrictionf|U\{0}

induces an isomorphismf|U\{0}:U\ {0}−→^∼= C\ {0}, whereC⊂V is a curve.

Prove the following statements:

(1) The Fitting ideal Fitt(f_∗OX)

0 is a principal ideal ofO_C²,0.

(2) The Fitting, annihilator, and reduced structure of the germ of the image off at0coincide.

Hint for (1).Use the Auslander Buchsbaum formula (Corollary B.9.4).

The Weierstraß division Theorem 1.8 yields thatC{x, y}/y²−xis a free, hence flat,C{x}-module of rank 2 (with basis 1, y). Thus, f is flat at 0. On the other hand,gis not flat at0. In fact, to see thatM =C{x, y}/xyis not flat overC{x}, tensor the exact sequence of C{x}-modules

0−→ x −→C{x} −→C−→0

withM. The induced map x ⊗_C{x}M →C{x} ⊗_C{x}M =M is not injec- tive, sincex⊗[y]= 0 is mapped to [xy] = 0.

A priori, flatness is a purely algebraic concept. But it turns out to have a geometrical meaning which can be roughly formulated as a continuous be- haviour of the fibres. For instance, looking at the fibres off andgin Example 1.77.1, we get f⁻¹(x) = (x,√

x),(x,−√ x)

if x= 0, and f⁻¹(0) = (0,0) with multiplicity 2. Hence, the fibres of f behave “continuously” at 0 if we count them with multiplicity. On the other hand,g⁻¹(x) = (x,0)

ifx= 0, andg⁻¹(0) ={0} ×C. In particular, the fibre dimension ofgjumps locally at 0.

For finite maps, flatness has a particularly nice geometric interpretation.

As shown below, the finite coherence Theorem 1.67 implies that all numerical invariants which can be described as the fibre dimension of coherent sheaves behavesemicontinuously in a family. If the family is flat, then the invariants vary even continuously, which means that they are locally constant. Hence, for finite morphisms, flatness is the precise algebraic formulation of what has been, somehow mysteriously, called the “principle of conservation of num- bers”.

The following theorem can be understood as a sheafified version of the flatness criterion (Proposition B.3.5) for finite maps:

Theorem 1.78.Let f :X →Y be a finite morphism of complex spaces and F a coherentOX-module. Then the following conditions are equivalent:

(a)F isf-flat, that is, Fx is a flat OY,f(x)-module for allx∈X.

(b)(f_∗F)_y is a flat OY,y-module for ally∈Y. (c)f_∗F is a locally free sheaf onY.

In particular,f is flat ifff_∗OX is a locally free sheaf on Y. Proof. Sincef is finite, Proposition 1.56 yields that (f_∗F)_y∼=%

x∈f⁻¹(y)Fx, hence the equivalence of (a) and (b). The finite coherence Theorem 1.67 im- plies thatf_∗F is a coherentOY-sheaf, in particular, (f_∗F)y is a finitely gen- eratedOY,y-module for each y∈Y. By the flatness criterion of Proposition B.3.5, (f_∗F)y is flat iff it is a free OY,y-module. The equivalence of (b) and

(c) follows now from Theorem 1.80 (1) below.

Definition 1.79.A subset of a complex spaceX is calledanalytically closed if it is a closed analytic subset ofX; it is calledanalytically open if it is the complement of a closed analytic subset ofX.

Theorem 1.80.LetX be a complex space,F a coherentOX-module, and set F(x) := Fx

mX,xFx, ν(F, x) := dim_CF(x). (1) The following are equivalent

(a)F is locally free onX.

(b)Fx is a freeOX,x-module for each x∈X.

(2) IfX is reduced, then (a) and (b) are also equivalent to (c) The functionx →ν(F, x)is locally constant on X.

(3) The sets

Sd(F) := x∈Xν(F, x)> d

, d∈Z, NFree(F) := x∈XFx is not free are analytically closed inX.

(4) If X is reduced, then Free(F) :=X\NFree(F) is dense in X. If X is reduced and irreducible, then

NFree(F) =Sd0(F) with d₀= min ν(F, x)x∈X .

Note that ν(F, x) = mng(Fx) is the minimum number of generators of the OX,x-moduleFx. Further note that the assumption thatX is reduced in (2) and (4) is necessary: take X the non-reduced pointT_ε, OTε=C[ε]/ε²and F =ε · OTε.

It follows from (3) that the set Free(F) =X\NFree(F) is analytically open inX. On the other hand, Free(F) is the disjoint union of the sets

Freed(F) = x∈XFxis free of rankd .

Thus, each of the sets Freed(F) is analytically open in X, too. Finally, note that Free(F) is also the flat locus ofF (Proposition B.3.5).

Proof. (1), (2): Fbeing locally free means that each pointx₀∈Xhas a neigh- bourhoodU such thatF |U ∼=O^νU for someν. Clearly (a) implies (b) and (c), even ifX is non-reduced.

As F is coherent, for eachx0∈X there exists a connected open neigh- bourhoodU ofx0 and an exact sequence

O^qU

−→ OA U^p

−→ F |π U −→0.

Hence, for each x∈U the sequence O^q_U,x→ O^{A p}_U,x→ Fx→0 is exact, and, after tensoring with C as OU,x-module, we get an exact sequence of finite dimensional C-vector spaces

C^{q A}−−−→^(x) C^p−→ F(x)−→0, with rank(A(x)) =p−ν(F, x).

By Nakayama’s lemma, we may choose finitely many sections in Γ(U,F) which represent a basis of the fibre F(x0) and which generate Fx for all x∈U (shrinkingU if necessary). Hence, we may assumep=ν(F, x₀). In this situation

S_d(F)∩U = x∈Urank(A(x))< p−d

is the zero set of the ideal generated by all (p−d)-minors ofA. In particular, it is analytic. We use this setting in the following.

Supposing (c), we may assume thatν(F, x) is constant onU and, hence, rank(A(x)) = 0 on U, that is, we may assume that each entry aij∈ OU(U) of A satisfies aij(x) = 0 for all x∈U. By Corollary 1.73, this means that eachaij is nilpotent. IfX is reduced, this implies that eachaij is zero. Thus, OU^p ∼=F |U which implies (a).

Now, assume thatX is not necessarily reduced and that (b) is satisfied.

Consider the exact sequence

0−→ Im(A)−→ O^pU

−→ F |π U −→0. (1.7.1) SinceFx0 is free, the induced sequence 0→ Im(A)(x₀)→C^p→ F(x₀)→0 is exact. Hence, ν(Im(A), x₀) =p−ν(F, x₀) = 0, that is, Im(A)(x₀) = 0.

By Nakayama’s lemma, this impliesIm(A)_x0= 0. Since Im(A) is coherent, Im(A) = 0 in a neighbourhood ofx₀, which implies (a).

(3) To show that NFree(F) is analytically closed in X, consider again the exact sequence (1.7.1). The stalkFx₀ is free iff this sequence splits atx0(Ex- ercise 1.7.1), that is, iff there is a morphismσ:Fx₀ → O^pU,x₀withπx₀◦σ= id.

Now consider the map

π: Hom(F|U,O^pU)−→Hom(F|U,F|U), ψ −→π◦ψ .

If the sequence (1.7.1) splits atx0, we haveϕ=πx₀(σ◦ϕ) for each homomor- phismϕ:Fx0 → Fx0. Thus,πx0 is surjective. Conversely, if πx0 is surjective, then the identity map id :Fx0 → Fx0 has a preimageσ:Fx0 → OU,x^p ₀, which is a splitting.

We have shown that the stalk Fx0 is free iff π_x0 is surjective, that is, iff Coker(π_x0) = 0. Since Coker(π) is a coherent OU-sheaf, we get that NFree(F|U) = supp

Coker(π)

is analytically closed inU.

(4) LetX be reduced,x0∈X andU any neighbourhood ofx0. Letd0be the minimum value ofν(F, x) onU. ThenSd₀(F)∩U is analytically closed inU by (3) and its complementU\Sd₀(F) is non-empty. By (2),Fis locally free on U\Sd₀(F). Hence, any neighbourhood ofx0contains points ofX\NFree(F).

Thus, Free(F) =X\NFree(F) is dense inX. If, additionally,X is irreducible then, as a proper closed analytic subset ofX,Sd₀(F) is nowhere dense inX. Hence, its complementX\Sd₀(F) is dense in X. This shows thatF cannot

be locally free at any pointx∈Sd₀(F).

The theorem says, in particular, if the stalk Fx is free thenF is locally free in a neighbourhood of x. Moreover, the open setX\NFree(F) decomposes in connected components and F has constant rank on each component. An alternative proof of the analyticity of NFree(F) is given in Exercise 1.7.5.

As a corollary, we obtain the main result of this section, which provides the promised geometric interpretation of flatness for finite morphisms:

Theorem 1.81 (Semicontinuity of fibre functions). Let f :X→Y be a finite morphism of complex spaces and let F be a coherent OX-module.

(1) The function

y −→ν(f_∗F, y) =

x∈f⁻¹(y)

dim_CFx/m_yFx

is upper semicontinuous⁸ on Y (here, my denotes the maximal ideal of OY,y).

(2) If F isf-flat thenν(f_∗F, y)is locally constant onY.

(3) If Y is reduced then ν(f_∗F, y)is locally constant onY iffF isf-flat.

Statement (2) is called theprinciple of conservation of numbers.

Proof. (1) We get from Proposition 1.56 that (f_∗F)y ∼= %

x∈f⁻¹(y)

Fx/myFx.

Sincef_∗F is coherent onY (by the finite coherence Theorem 1.67), the result follows from Theorem 1.80 (3).

(2) If F isf-flat then f_∗F is locally free by Theorem 1.78, hence locally at y0∈Y of constant rank equal toν(f_∗F, y0).

(3) This follows from Theorem 1.80 (4).

Another corollary is the following

Theorem 1.82 (Openness of flatness). Let f :X →Y be finite andF a coherent OX-module. Then the set of points x∈X where Fx is OY,f(x)-flat is analytically open in X. In particular, the set of points in which f is flat is analytically open.

Proof. By Theorem 1.80, Free(f_∗F) is analytically open inY. Since (f_∗F)_y ∼= %

x∈f⁻¹(y)

Fx

8 A function ϕ:Y →R, Y a topological space, is calledupper semicontinuous if for eachy0∈Y there is a neighbourhood V ofy0 such thatϕ(y)≤ϕ(y0) for all y∈V.

asOY,y-module, (f_∗F)y isOY,y-free iffFxisOY,y-free for allx∈f⁻¹(y). By Proposition B.3.5 this is equivalent toFxbeing OY,y-flat for allx∈f⁻¹(y).

Now, givenx₀∈X there are neighbourhoodsU =U(x₀) andV =V(y₀), y₀=f(x₀), such thatf_U,V :U →V is finite andf_U,V⁻¹(y₀) ={x₀}. If the stalk Fx0 is OY,y0-flat then (f_U,V)_∗F|U is free in a neighbourhood V (y₀)⊂V. Thus,Fx isOY,f(x)-flat for allx∈f⁻¹(V )∩U. Remark 1.82.1.A much stronger theorem due to Frisch says that Theorem 1.82 holds foreach holomorphic mapf :X→Y (see Theorem 1.83).

Remarks and Exercises

UsingExt sheaves, we can give a more conceptual description of the non-free locus NFree(F).

Let (X,OX) be a ringed space. AnOX-moduleJ is calledinjective if the functorF →Hom_OX(F,J) is exact on the category ofOX-modules. It is a fact that eachOX-moduleF has an injective resolution

0→ F → L⁰(F)−→ L^{ϕ0 1}(F)−→ L^{ϕ1 2}(F)−→^ϕ2 . . . (that is, the sequence is exact and the modulesLⁱ(F) are injective).

For a second OX-module M, we have an induced sequence of sheaves which is a complex

0→Hom_OX(M,F) → Hom_OX

M,L⁰(F) ϕ₀

−→Hom_OX

M,L¹(F)

ϕ₁

−→Hom_OX

M,L²(F) ϕ₂

−→. . . . ThenExt⁰_O

X(M,F) :=Hom_OX(M,F) and, for alli≥1, Extⁱ_OX(M,F) :=Hⁱ

Hom_OX

M,L^•(F)

:=Ker(ϕ_i+1)

Im(ϕ_i). For details and further properties ofExt, in particular for the long exactExt sequences, we refer to [God].

If it happens thatMhas a resolution by locally free sheavesMi of finite rank,. . .→ M2→ M1→ M0→ M →0, then

Extⁱ_OX(M,F)∼=Hⁱ

Hom_OX

M_•,F . However, such a locally free resolution ofMmay not exist.

Now, let (X,OX) be a complex space and M a coherent OX-module.

Then M has locally a locally free resolution and, if F is coherent, then Extⁱ_O

X(M,F) is coherent, too (Exercise 1.7.7). Moreover, for allx∈X, we

have then

Extⁱ_O

X(M,F)

x∼= Extⁱ_OX,x(Mx,Fx),

which can be computed by a free resolution of the OX,x-module Mx. This allows us to compute NFree(M) via Ext (see Exercise 1.7.8).

Exercise 1.7.1.Let A be a ring, and let π:F →M be a surjection of A- modules withF free. Prove that the following are equivalent

(a)M is projective.

(b) There is a morphismσ:M →F withπ◦σ= idM.

(c) The map HomA(M, F)→HomA(M, M),ψ →π◦ψ, is surjective.

Exercise 1.7.2.LetAbe a ring and

0−→M −→^α F −→^β M −→0 (1.7.2) an exact sequence of A-modules with F free. Show that the following are equivalent:

(a) The sequence (1.7.2) is left split, that is, there exists a morphism τ:F →M withτ◦α= id_M.

(b) The sequence (1.7.2) is right split, that is, there exists a morphism σ:M →F withβ◦σ= idM.

(c)F ∼=M ⊕M . (d)M is projective.

(e)M is projective.

Exercise 1.7.3.Let A be a Noetherian local ring, and let N, M be finite A-modules. Denote by mng(M), the minimal number of generators ofM (see Definition 1.19). Show that the following holds:

(1) mng(M⊗AN) = mng(M)·mng(N) and mng(*p

M) =_mng(M)

p

. (2)M is free of rankdiff*d

M is free of rank 1.

(3)M is free of rank 1 iff the canonical map

φ:M^∗⊗AM →A , ϕ⊗x →ϕ(x), is an isomorphism.

Exercise 1.7.4.LetX be a complex space and E a locally free OX-module of finite rankn. LetE^∗=Hom_OX(E,OX) denote thedualOX-module.

(1) Show that E^∗ is a locally freeOX-module of rank n.

(2) Prove that there is a canonical isomorphism (E^∗)^∗∼=E.

(3) Prove thatHom_OX(E,F)∼=E^∗⊗_OX F for an arbitrary OX-moduleF. Exercise 1.7.5.Let F be a coherent sheaf on a complex space X and let φ:F^∗⊗_OX F → OX be given byϕ⊗f →ϕ(f). Prove that

X\Freed(F) = supp(Ker(φ))∪supp(Coker(φ)).

Conclude that Freed(F) is analytically open and that NFree(F) is analytically closed inX.

Exercise 1.7.6.Letf :X →Y be a finite morphism of complex spaces which is flat. Prove thatf is open (see also Theorem 1.84).

Exercise 1.7.7.LetX be a complex space andF,Gcoherent OX-modules.

(1) Show that for anyx0∈X andi∈Nthere exists an open neighbourhood U ofx0 and an exact sequence

0→ R → Li−1→. . .→ L1 ϕ1

−→ L0 ϕ0

−→ F|U →0

with Lj∼=Oⁿ_U^j and R a coherent OU-module. This module R is called the i-th syzygy module of F|U and denoted by Syz_i(F|U). The sheaves Syz_i(F|U) can be glued to the i-th syzygy sheaf Syz_i(F) which is a co- herentOX-module.

(2) Prove that, fori≥0, theOX-moduleExtⁱ_O

X

F,G) is coherent by showing that there is an exact sequence

Hom_OU(Li−1,G|U)→Hom_OU(R,G|U)→ Extⁱ_OU

F|U,G|U)→0. (3) Use (2) to show that, forx∈X,

Extⁱ_O

X(M,F)

x∼= Extⁱ_OX,x(Mx,Fx).

Exercise 1.7.8.LetF be a coherent sheaf on the complex spaceX. (1) Show that F is locally free iffExt¹_OX

F,Syz₁(F)

= 0.

Hint.Use the argument in the proof of Theorem 1.80.

(2) Let J ⊂ OU denote either the 0-th Fitting ideal or the annihilator ideal ofExt¹_OX

F,Syz₁(F)

. Show that NFree(F) = supp

Ext¹_O

X

F,Syz₁(F)

=V(J). 1.8 Flat Morphisms and Fibres

The aim of this section is to collect some of the most important properties of flat morphismsf :X →S of complex spaces to provide an easy reference, in particular, for the sections dealing with deformation theory.

Recall from Section 1.7 that, for finite morphisms, flatness implies locally the constancy of the total multiplicity of the fibres. If, additionally, the base is reduced then flatness can even be characterized by this numerical condi- tion (Theorem 1.80). Moreover, we proved that flatness is an open property (Theorem 1.82).

In the following, we do not assume that f is finite. As we shall see, also in this general situation, flatness implies strong continuity conditions on the fibres. Moreover, flatness is used to study the singular locus of an arbitrary morphism of complex spaces.

Before we state geometric consequences of the algebraic properties of flat- ness treated in Appendix B, let us cite the following two important theorems.

The first one is due to Frisch and generalizes Theorem 1.82:

Theorem 1.83 (Frisch). Let f :X →S be a morphism of complex spaces.

Then theflat locusof f, that is, the set of all pointsx∈X such thatf is flat atx, is analytically open inX.

Proof. See [Fri].

The second theorem is due to Douady. It provides another openness result for flat morphisms (generalizing Exercise 1.7.6):

Theorem 1.84 (Douady). Every flat morphism f :X→S of complex spaces isopen, that is, it maps open sets inX to open sets in S.

Proof. See [Dou] or [Fis, Prop. 3.19].

Together with Frisch’s Theorem 1.83, Douady’s theorem implies that if a mor- phismf :X→Sis flat atx∈X, then it is locally surjective onto some neigh- bourhood off(x) inS. In particular, closed embeddings of proper subspaces are never flat.

The next proposition is the geometric version of Theorems B.8.13 and B.8.11:

Proposition 1.85.Let f :X →S be a morphism of complex spaces, and let x∈X. Then, fors=f(x)andXs=f⁻¹(s), the following holds:

(1)dim(X, x)≤dim(Xs, x) + dim(S, s) with equality iff is flat at x.

(2) If S=C^d and f = (f1, . . . , fd), then f is flat at x iff f1, . . . , fd is an OX,x-regular sequence.

(3) IfX is a complete intersection atx, or, more generally, Cohen-Macaulay atx,⁹ andS=C^d, thenf is flat at xiffdim(X, x) = dim(X_s, x) +d.

Proposition B.5.3 yields a criterion for checking whether a morphism of com- plex space germs is an isomorphism.

Lemma 1.86.Let

(X, x) ^f

φ

(Y, y)

ψ

(S, s)

be a commutative diagram withφflat. Thenf is an isomorphism ifff induces an isomorphism of the special fibres,

f: (φ⁻¹(s), x)−→^∼= (ψ⁻¹(s), y).

9 X is called acomplete intersection (resp.Cohen-Macaulay)atxif the local ring OX,x is a complete intersection (resp. Cohen-Macaulay). We also say that the germ (X, x) is acomplete intersection singularity (resp. Cohen-Macaulay). Note that smooth germs and hypersurface singularities are complete intersection sin- gularities, hence Cohen-Macaulay. Further, every reduced curve singularity and every normal surface singularity are Cohen-Macaulay (see Exercise 1.8.5).

Proof. We only need to show the “if” direction. For this consider the induced maps of local rings,φ:OS,s→ OX,x andf:OY,y→ OX,x.

Sincef induces an isomorphism of the special fibres andy∈ψ⁻¹(s), the germ off⁻¹(y) consists of the pointxonly. Hence,f is a finite morphism of germs (Theorem 1.70) and, therefore, OY,y is a finite OX,x-module. That f induces an isomorphism of the special fibres means algebraically that

f⊗id : OY,y⊗_OS,sOS,s/mS,s−→ OX,x⊗_OS,sOS,s/mS,s

is an isomorphism. Therefore, the assumptions of Proposition B.5.3 are ful-

filled andf is an isomorphism.

We are now going to prove that flatness is preserved under base change. The proof in analytic geometry is slightly more complicated than in algebraic ge- ometry where it follows directly from properties of the tensor product.

Proposition 1.87 (Preservation of flatness under base change). If Z ^g

f

X

f

T _g S

is a Cartesian diagram of morphisms of complex spaces withf flat, thenfis also flat.

Since the fibre product reduces to the Cartesian product if S={pt} is a (reduced) point, and since a map to {pt} is certainly flat, we deduce the following corollary:

Corollary 1.88 (Flatness of projection). IfX, T are complex spaces then the projectionX×T →T is flat. Equivalently, for everyx∈X andt∈T, the analytic tensor productOX,x⊗ O T ,t is a flat OT ,t-module.

For the proof of Proposition 1.87 we need two lemmas:

Lemma 1.89.Letf :X →Sandg:Y →Sbe morphisms of complex spaces.

Moreover, letgbe finite. Then, using the notations of Definition 1.46 and A.6, there is a natural isomorphism

π_X⁻¹OX⊗_π⁻¹

Y g⁻¹OSπ_Y⁻¹OY

∼=

−→ OX×SY, induced by the mapa⊗b →ab:=π_X(a)·π_Y(b).

In particular, forS={pt}the reduced point and forY a fat point, we get thatOX,x⊗ O Y =OX,x⊗_COY.

Proof. We have to show that the morphism is stalkwise an isomorphism at each point p= (x, y)∈X×Y such that f(x) =s=g(y) (cf. A.5). Further, we may suppose thatX ⊂Cⁿ,Y ⊂C^m,S⊂C^kand thatx, y, sare the origins ofCⁿ,C^m,C^k, respectively. Hence, what we actually have to show is that the map

C{x}/I_X⊗_C{s}/I_S C{y}/I_Y −→C{x,y} J ,

J =IXC{x,y}+IYC{x,y}+f1−g1, . . . , fk−gkC{x,y},

induced by the multiplication ψ:C{x} ⊗_CC{y} →C{x,y}, a⊗b →ab, is an isomorphism.

Let J0⊂C{x} ⊗CC{y} denote the ideal generated by h⊗1, h∈IX, 1⊗h,h ∈IY, and the differencesfi⊗1−1⊗gi,i= 1, . . . , k. Then we have to show thatψinduces an isomorphism

(C{x} ⊗CC{y})/J0

∼=

−→C{x,y}/J .

The latter map is always injective (even if g is not finite): it is faithfully flat by Propositions B.3.3 (5),(8) and B.3.5, applied to

(C{x} ⊗_CC{y})/J₀→C{x,y}/J→C[[x,y]]/JC[[x,y]]. By Proposition B.3.3 (10)(ii),ψ⁻¹(J) =J₀. Hence, we get injectivity.

To see the surjectivity, we use that C{s}/IS →C{y}/IY, si →gi(y), is finite. The finiteness implies that y^m⊂IY +g1, . . . , gk form sufficiently large. This further implies that in C{x,y}/J we can replace high powers of y by polynomials in the fi (since fi≡gi mod J). Hence, each element of C{x,y}/J can be represented as a finite sum

iai(x)bi(y) with ai∈C{x}

andbi∈C{y}. This completes the proof.

Lemma 1.90 (Finite-submersive factorization lemma). Each mor- phism f: (X, x)→(Y, y) of complex germs factors through a finite map and a submersion, that is, there exists a commutative diagram

(X, x) ^ϕ

f

(Y, y)×(Cⁿ,0)

p

(Y, y)

with n= dim(f⁻¹(y), x), ϕ finite and p the projection on the first factor.

Moreover, if (f⁻¹(y), x)is smooth, thenϕcan be chosen such that it induces an isomorphism (f⁻¹(y), x)−→^∼= (Cⁿ,0).

Proof. Choose a Noether normalizationϕ : (f⁻¹(y), x)→(Cⁿ,0) of the fibre (f⁻¹(y), x) (Theorem 1.25). Thenϕ is finite and, by the lifting Lemma 1.14, we can extendϕ to a mapϕ : (X, x)→(Cⁿ,0). Setting

ϕ:=f×ϕ : (X, x)→(Y, y)×(Cⁿ,0),

we getϕ⁻¹(y,0) =f⁻¹(y)∩ϕ ⁻¹(0) =ϕ ⁻¹(0) ={x}, henceϕis finite and

the result follows.

Proof of Proposition 1.87.The statement is local inT, hence we may consider morphisms of germs.

Since the base change mapg: (T, t)→(S, s) factors through a finite map and a submersion (Lemma 1.90) we have to show that flatness is preserved by finite and submersive base changes. For this, we consider the dual base change diagram on the level of local rings

OZ,z OX,x

g

OT ,t f

OS,s ,

g

f

whereOZ,z∼=OX×ST ,(x,t).

Ifgis finite, then Lemma 1.89 yieldsOX×ST ,(x,t)∼=OX,x⊗OS,sOT ,t, and Proposition B.3.3 (3) implies thatOZ,z isOT ,t-flat.

Now, letgbe a submersion, that is,g is the projection

g: (T, t) = (S, s)×(Cⁿ,0)→(S, s), n= dim(g⁻¹(s), t).

Let (S, s)⊂(C^r,0), and denote by f₁, . . . , f_r and g₁, . . . , g_r the component functions off andg, respectively. Then, set theoretically,

X×ST ={(x,s,y)∈X×S×Cⁿ|f_i(x) =g_i(s,y) for all i}

where X,Cⁿ, S are small representatives of the corresponding germs. Since gi(s,y) =si, we haveX×ST =Γ(f)×Cⁿ, and

OZ =OX×ST =OX×S×Cⁿ/fi−si=OΓ(f)×Cⁿ.

Finally, tensoring the left-hand side of the dual base change diagram by OCⁿ,0/m^k+1_Cn,0, we get the diagram

OΓ(f)×Cⁿ,(x,s,0)/m^k+1_Cn,0 OX,x

OS×Cⁿ,(s,0)/m^k+1_Cn,0 f_(k)

OS,s .

g_(k)

f

Sinceg_(k)is finite, the above reasoning gives thatf_(k) is flat.

This holds for eachk≥0. Hence,fis flat by the local criterion for flatness

(Theorem B.5.1 (4)).

We state now a theoreticallyand computationally useful criterion for flatness due to Grothendieck:

Proposition 1.91 (Flatness by relations). Let I=f1, . . . , fk ⊂ OCⁿ,0

be an ideal,(S, s)a complex space germ andI=F1, . . . , Fk ⊂ O_Cⁿ_×S,(0,s) a lifting of I, that is,Fi is a preimage offi under the surjection

O_Cⁿ_×S,(0,s)O_Cⁿ_×S,(0,s)⊗OS,sC=OCⁿ,0. Then the following are equivalent:

(a)O_Cⁿ_×S,(0,s)/IisOS,s-flat;

(b) any relation (r1, . . . , rk)among f1, . . . , fk lifts to a relation (R1, . . . , Rk) amongF1, . . . , Fk. That is, for each (r1, . . . , rk)satisfying

k i=1

rifi= 0, ri∈ O_Cⁿ,0, there exists(R1, . . . , Rk)such that

k i=1

RiFi= 0, with Ri∈ O_Cⁿ_×S,(0,s)

and the image ofRi in O_Cⁿ,0 isri; (c) any free resolution ofO_Cⁿ,0/I

. . .→ O^p_C²n,0→ O^p_C¹n,0→ O_Cⁿ,0→ O_Cⁿ,0/I→0 lifts to a free resolution ofOCⁿ×S,(0,s)/I,

. . .→ O_C^p2n×S,(0,s)→ O^p_C¹n×S,(0,s)→ O_Cⁿ_×S,(0,s)→ O_Cⁿ_×S,(0,s)/I→0. That is, the latter sequence tensored with⊗_OS,sCyields the first sequence.

Proof. SetO=OCⁿ,0,O=O_Cⁿ_×S,(0,s)and consider the commutative diagram 0 Ker(d₁) O^{k d1} O O/I 0

0 Ker(d1) O^{k d1} O O/I 0,

where d1 (respectively d1) maps the i-th canonical generator of O^k (respec- tively O^k) to Fi (respectively fi) and the vertical maps are the canonical surjections.

The set of all relations among F1, . . . , Fk (respectively f1, . . . , fk) is the submodule K := Ker(d1) (respectively K:= Ker(d1)) and, hence, condition (b) is equivalent to the canonical mapK →K being surjective.

By the local criterion for flatness (Theorem B.5.1) applied to OS,s→O andO/Iand sinceO is flat overOS,s by Corollary 1.88, we get thatO/Iis OS,s-flat iff Tor^O₁^S,s(O/I,C) = 0. Moreover, tensoring the exact sequence

0−→I−→O −→ O/I−→0

with ⊗_OS,sC, we deduce that O/Iis OS,s-flat iff I⊗C→ O is injective or, equivalently, thatI⊗C→I is an isomorphism.

Note that, moreover, the flatness of O/Iimplies Tor^O_i^S,s(O/I,C) = 0 for i≥1, hence Tor^O₁^S,s(I,C) = 0, and therefore thatIisOS,s-flat.

After these preparations, we can show the equivalence of (a),(b) and (c). To see (a)⇐⇒(b) consider the diagram with exact rows (the tensor products being overOS,s)

0 K O^k I 0

K⊗C O^k⊗C

∼=

I⊗C 0

0 K O^k I 0.

If O/I is flat, then by the above arguments Iis flat, which implies that K ⊗C→O^k⊗Cis injective, andI⊗C→I is bijective. ThenK ⊗C→K is bijective andK →K surjective, which is equivalent to (b).

IfK →Kis surjective, thenK⊗C→Kis surjective and a diagram chase shows thatI⊗C→I is bijective, which is equivalent to (a).

Since (b) is a special case of (c), we have to show that (b) implies (c). As the diagram chase arguments work for any number of generators ofI we may assumek=p1. ThenK →Ksurjective implies that any surjectionO^p2 →K lifts toO^p2 →K. That is, we have a commutative diagram with exact rows

O^{p2 d2} O^p1 I 0

O^{p2 d2} O^p1 I 0

andIis flat overOS,s. Then, by the same arguments as above we obtain that Ker(d2)→Ker(d2) is surjective and statement (c) follows by induction.

Exercises

Exercise 1.8.1.LetF,GbeOX-modules. Show thatF ⊕ G is flat if and only ifF andG are flat.

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