1 Basic Properties of Complex Spaces and Germs
Theorem 1.21 Inverse function theorem)
1.6 Applications of the Finite Coherence Theorem
Moreover, show that the annihilator structure and the reduced structure are not compatible with the base changeg:T →Y, x →(x,0) (see Remark 1.45.1 (2)).
Exercise 1.5.4.Let f :X →Y be a finite morphism, let A⊂X be an an- alytic set, and let y∈Y. Moreover, let π−1(y) ={x1, . . . , xs} and assume that the germ (A, xi) decomposes intoriirreducible components,i= 1, . . . , s.
Prove that the germ of the imagef(A) aty decomposes into at mosts i=1ri
irreducible components.
Exercise 1.5.5.Let π:A→B be a Weierstraß map and (y, z)∈A. Prove the following statements:
(1)πis anopen map, that is, it maps open sets inA to open sets inB.
(2) To every sequence (yν)ν∈N⊂B converging to y there exists a sequence (zν)ν∈N⊂Csuch that (yν, zν)∈A and (zν)ν∈N converges toz.
Hint for (1). Use Hensel’s lemma to reduce the statement to the case that π−1(y) ={(y, z)}.
Exercise 1.5.6.Prove the uniqueness statement in the general Weierstraß division Theorem 1.60.
In particular,OX,x is a finitely generatedOY,y-module, which is precisely the meaning off:OY,y → OX,x being finite.
By Corollary 1.13, (c) is equivalent to (d). Thus, we are left with the proof of (d)⇒(b). By definition, Of−1(y)=OX
myOX where my⊂ OY is the ideal sheaf of the (reduced) point {y}. Iff:OY,y→ OX,x is quasifinite then dimCOf−1(y),x= dimCOX,x/myOX,x<∞. Nakayama’s lemma implies thatmpX,xOf−1(y),x= 0, hencempX,x⊂myOX,x, for somep >0. It follows that, locally at x, we have an inclusion of setsf−1(y) =V(myOX)⊂V(mpX,x) =
{x}.
Lemma 1.71.Let(f, f) : (X, x)→(Y, y)be a finite morphism of germs such that f:OY,y→ OX,x is injective. Thenf is surjective (that is, has a surjec- tive representative f :U →V).
Proof. By the local finiteness Theorem 1.66, there is a finite representative f :U →V such thatf(U) is closed inV and f∗OU is a coherent OV-sheaf.
Then, for sufficiently smallV andU,
AnnOV(f∗OU) = Ker(f:OV →f∗OU) = 0,
since the stalk at y is zero by assumption, and since the annihilator sheaf is coherent, too (A.7, Fact 5). Therefore,f(U) =V
AnnOV(f∗OU)
=V. Remark 1.71.1.Lemma 1.71 applies, in particular, to a Noether normaliza- tion: let (X,0)⊂(Cn,0) be a complex space germ withOX,0=OCn,0/I, and letϕ:C{y1, . . . , yd}→ OX,0be a Noether normalization (Theorem 1.25). Set- tingf=ϕandf =
ϕ(y1), . . . , ϕ(yd)
, we obtain a finite and surjective mor- phism (f, f) : (X, x)→(Cd,0), which we refer to as aNoether normalization (of complex space germs).
The following theorem, due to R¨uckert, is the analytic counterpart to the Hilbert Nullstellensatz for polynomial rings.
Theorem 1.72 (Hilbert-R¨uckert Nullstellensatz). Let X be a complex space, I ⊂ OX a coherent ideal sheaf. Then
J V(I)
= √ I, where J
V(I)
is the full ideal sheaf of V(I).
Proof. Since, obviously,√ I ⊂ J
V(I)
, and since both sheaves have the same support, we have to show that for each x∈V(I) the inclusion map
√I
x =
Ix → J V(I)
x
is surjective.
Consider a primary decomposition ofIx,Ix=q1∩. . .∩qr, with√ qi=pi
prime ideals. Then
Ix = r i=1
pi, J V(Ix)
= r i=1
J V(qi)
= r i=1
J V(pi) (see Exercise 1.3.5). Thus, it suffices to show that for a prime idealp⊂ OX,x
we haveJ V(p)
=√ p.
Choose a Noether normalization
ϕ:C{y}=C{y1, . . . , yd}→ OX,x
J V(p)
and a liftingϕ:C{y}→ OX,x, which induces a morphismC{y}→ OX,x/p.
Since V(p) =V J
V(p)
as topological spaces, the induced morphism of germsV(p)→(Cd,0) is finite. By Proposition 1.70, it follows that OX,x/p is finite overC{y}, in particular,OX,x/p is integral over C{y} (via ϕ). Thus, eachf ∈ J
V(p)
⊂ OX,x satisfies a relation (of minimal degree) fr+a1fr−1+. . .+ar ∈ p
with ai ∈ϕ(C{y}). Since p⊂ J V(p)
, we have ar∈ J V(p)
∩ϕ( C{y}) which is 0 asϕis injective. It follows that
f ·(fr−1+a1fr−2+. . .+ar−1) ∈ p
and fr−1+a1fr−2+. . .+ar−1∈p, because we started with a relation of minimal degree. Aspis a prime ideal, we getf ∈p, which proves the theorem.
Corollary 1.73.Let F be a coherent sheaf on X, and let f ∈Γ(X,OX).
Suppose thatf, considered as a morphismf :X →Csatisfies f|supp(F)= 0.
Then, for each x∈X, there exists a neighbourhood U of x and a positive integerrsuch that frF|U = 0.
In particular, iff(x) = 0for allx∈X then all germsfx∈ OX,x are nilpo- tent.
Proof. Apply the Hilbert-R¨uckert Nullstellensatz toI=AnnOX(F). For the
second statement takeF=OX.
Corollary 1.74.Let F be a coherent sheaf on X, x∈supp(F). Then the following are equivalent:
(a)xis an isolated point of the support ofF. (b)mrX,xFx= 0for some r >0.
(c)dimCFx<∞.
Proof. (a)⇒(b) If xis an isolated point of the support of F, then for each f ∈mX,x there exists a neighbourhoodU ofxsuch thatf|supp(F)∩U = 0. By Corollary 1.73, there exists somer >0 such that frFx= 0. Since mX,x is a finitely generatedOX,x-module, we easily deduce (b).
(b)⇒(c) IfmrX,xFx= 0, then
dimCFx= dimCFx/mrX,xFx= r i=1
dimCmiX,x−1Fx/miX,xFx,
which is finite asOX,xis Noetherian andFxa finitely generatedOX,x-module.
(c)⇒(a) Let dimCFx<∞. Then, by Nakayama’s lemma there exists an integer s >0 such thatmsX,xFx= 0, that is, msX,x⊂AnnOX,xFx. Hence, lo- cally atx, supp(F) =V(AnnOXF)⊂V(msX,x) ={x}. We close this section with Cartan’s theorem on the coherence of the full ideal sheaf. Since the proof is slightly more involved than the proofs of the previous fundamental coherence Theorems 1.63 and 1.67, we only sketch the proof given by Grauert and Remmert. For details, we refer to [GrR2, Section 4.2]
or [DJP, Theorem 6.3.2].
Theorem 1.75 (Coherence of the full ideal sheaf ). LetAbe an analytic set in the complex spaceX. Then the full ideal sheafJ(A)of all holomorphic functions on X vanishing onA is a coherentOX-sheaf.
In view of Oka’s coherence Theorem 1.63, Cartan’s theorem may be rephrased as follows: let f1, . . . , fr∈ OX(U), U ⊂X open, x∈U, represent a set of generators for the stalkJ(A)x⊂ OX,x. Thenf1, . . . , fr generate J(A) on a whole neighbourhood of x in X (A.7, Fact 1). In other words, locally at x, the full ideal sheaf ofAcoincides with theOU-moduleI=r
i=1fiOU. Note that, a priori, it is clear that Ix ⊂ J(A)x for all x∈U; but it is not clear that the opposite inclusion holds, that is, thatIx is a radical ideal (Hilbert-R¨uckert Nullstellensatz).
Sketch of proof. We may use general facts on coherent sheaves (see Appendix A.7) to reduce the proof to the case thatX =D⊂Cn is an open neighbour- hood of 0and to showing coherence locally at 0. Using the existence of an irreducible decomposition of analytic set germs and Exercise 1.3.5 (3), we may assume additionally thatA⊂D isirreducible at 0.
The proof requires now a closer analysis of the structure of locally ir- reducible analytic sets as given by [GrR2, Lemmas 3.3.4, 3.4.1]: locally at 0, there is a finite and open surjection h:A→B, with B⊂Cd open, d= dim(A), which is locally biholomorphic outside (the preimage of) some analytic hypersurface V(Δ)B, the discriminant of h, where Δ is a holo- morphic function onB. The proof of this fact uses the Weierstraß preparation theorem and Hensel’s lemma and it gives a precise description of h(and its local inverse) onX\h−1(V(Δ)). From this description, we get that there are Weierstraß polynomials f1, . . . , fn−d at 0 vanishing on (A,0) such that for x∈D\h−1(V(Δ)) close to0,J(A)x=n−d
i=1 fiOD,x. For the fibre at0, we know thatJ(A)0⊃n−d
i=1 fiOD,0. We complement f1, . . . , fn−d to a generating set f1, . . . , fr of J(A)0. After shrinking D, we may assume thatf1, . . . , frconverge onDand consider the finitely generated
(hence, coherent) ideal sheafI =r
i=1fiOD. From our construction, we know thatJ(A)x=Ix forx=0and for allx∈D\h−1(V(Δ)).
It remains to extend this statement to x∈h−1(V(Δ))\ {0}. For this, let Δ=Δ◦hand consider the ideal quotient
I:Δ:= Ker
OD ·Δ
−→ OD/I
,
which is a coherentOD-sheaf (A.7, Fact 3). Since I0=J(A)0 is prime and Δ∈ I0, we may assume that I:Δ=I (shrinkingD is necessary). Now, let g∈ J(A)x for x close to 0. Then, locally at x, the ideal quotient I :g is coherent and V(I:g)⊂h−1(V(Δ)) =V(Δ). By the Hilbert-R¨ uckert Null- stellensatz, this implies that Δr∈ I:g for some r≥0. If r >0, this means that Δr−1g∈ I:Δ=I, that is, Δr−1∈ I:g. By induction on r we obtain
that 1 =Δ0∈ I:g, that is,g∈ Ix.
Theorem 1.76 (Coherence of the radical). Let I be a coherent ideal sheaf on the complex spaceX. Then the radical√
I is coherent. In particular, the sheafNil(OX)of nilpotent elements ofOX is coherent.
Proof. Since A⊂X is analytic, there exists a coherent ideal sheaf I ⊂ OX
such thatA=V(I). By the Hilbert-R¨uckert Nullstellensatz,J(A) =√ I and
the result follows from Cartan’s Theorem 1.75.
Exercises
Exercise 1.6.1.Let (f, f) : (X, x)→(Y, y) be a finite morphism of complex space germs and assume that (Y, y) isreduced. Show thatf is surjective (that is, has a surjective representativef :U →V) ifff:OY,y → OX,xis injective.
Show that this statement does not generalize to morphisms of non-reduced complex space germs.
Exercise 1.6.2.Let f : (X, x)→(Y, y) be a finite morphism of complex space germs. Prove the following statements:
(1) dim
f(X), y
= dim(X, x).
(2) If f is open, that is, if it has an open representative f :U →V, then dim
Y, y
= dim(X, x).
Hint:Use Exercise 1.3.1.
Exercise 1.6.3.Let f : (X, x)→(Y, y) be a morphism of reduced complex space germs and assume that (Y, y) is irreducible. Prove the following state- ments:
(1) Iff is open, then all elements of the kernel off:OY,y→ OX,x are nilpo- tent.
(2) Iff is finite and if (f∗OX)y is a torsion freeOY,y-module thenf is open.
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 ofOC2,0.
(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).