The Generic Freeness Lemma

Part III Computational Methods

10.1 The Generic Freeness Lemma

Chapter 10

Fibers and Images of Morphisms Revisited

In this chapter we will continue the investigation that was started in Sec- tion 7.2. First we use Gr¨obner basis theory to prove the generic freeness lemma. This leads to an algorithm for computing the image of a morphism of affine varieties. Then we will draw more theoretical consequences on the images of morphisms and the dimension of fibers. Finally, we will apply our results to the topic of invariant theory. As mentioned before, the results of this chapter will not be used anywhere else in the book, so there is an option to skip it.

Proof of Lemma 10.1. Let B ⊂ R[x₁, . . . , x_n] be the set of all monomials which are not divisible by any leading monomial LM(g) withg∈G. Sinceψ is injective onR, we have 1∈B. Moreover,ψ(B)⊆Sis linearly independent over R, since if Pm

i=1a_iψ(t_i) = 0 witht₁, . . . , t_m ∈B pairwise distinct and a_i ∈R, then h:= Pm

i=1a_it_i ∈ ker(ψ), so h= 0 since no monomial of h is divisible by an LM(g) forg∈G.

Let M := (ψ(B))_R ⊆S be the (free) R-module generated by ψ(B). We claim that for every s∈S there exists au∈U such thatus∈M. To prove this, takef ∈R[x1, . . . , xn] withs=ψ(f). By the modification of the Normal Form Algorithm9.8discussed in Remark9.11, there existsf^∗∈R[x1, . . . , xn] which is a normal form w.r.t.Gof someu·f, whereuis a product formed from leading coefficients of elements ofG. By multiplyingu with further leading coefficients of elements ofG, we can achieve thatuis a power ofQ

g∈GLC(g), so u∈U. The definition of a normal form implies that f^∗ lies in (B)_R and u·f −f^∗∈(G)_R[x

1,...,x_n]⊆ker(ψ). So

u·s=u·ψ(f) =ψ(u·f) =ψ(f^∗)∈M, which proves the claim.

Now it is straightforward to check thatBe :=n_ψ(t)

1 |t∈Bo

⊆U⁻¹S is a basis ofU⁻¹S as aU⁻¹R-module. This completes the proof. ut By combining Lemma10.1with Proposition9.18and Lemma7.16, we get an algorithm for computing the image of a morphism between the spectra of affine algebras (which, in the case of an algebraically closed ground field, comes down to computing the image of a morphism of affine varieties). Be- fore we state the algorithm, we derive the “existence-version” of the generic freeness lemma.

Corollary 10.2 (Generic freeness lemma). LetRbe an integral domain and letS be a ring extension ofRthat is finitely generated as anR-algebra. Then there exists a non-zero elementa∈Rsuch that for every multiplicative subset U ⊆Rwitha∈U, the localizationU⁻¹Sis free as aU⁻¹R-module, and there exists a basis containing 1∈U⁻¹S.

Proof. We have an epimorphismψ:R[x1, . . . , xn]→S. LetI:= ker(ψ),K:=

Quot(R), and J := (I)_K[x1,...,xn] = K·I. Choose a monomial ordering on K[x1, . . . , xn], and letG⊆J\{0}be a Gr¨obner basis ofJ. By multiplying the polynomials inGwith suitable non-zero elements of R, we can achieve that Gis contained inI, soGis a Gr¨obner basis ofI. With a:=Q

g∈GLC(g)∈ R\ {0}, the result follows from Lemma10.1. ut Loosely speaking, Corollary 10.2says that freeness holds “almost every- where” (assuming the hypotheses of the lemma). More precisely, it holds after localization at allP ∈Spec(R) witha /∈P. TheseP form an open, dense sub- set of Spec(R). (Proof of density: We havea /∈ {0} ∈Spec(R), and the closure

10.1 The Generic Freeness Lemma 149 of{0} is Spec(R).) The generic freeness lemma is due to Grothendieck, and it has been traditionally referred to as the generic flatness lemma. In fact, in its original version, the assertion is flatness of the map U⁻¹R → U⁻¹S, a property which is weaker than freeness and is not treated in this book. In Exercise 10.1 we explore the necessity of the hypotheses of Corollary 10.2.

Exercise 10.2 contains a version of the generic freeness lemma for modules overS. Exercise10.3contains a surprising application of the generic freeness lemma: A special case of this application says that a subalgebra of an affine K-domain has a localization which is again an affineK-domain.

As announced above, we now get to the algorithm for computing the im- age of a morphism ϕ^∗: Spec(A) → Spec(B) of spectra of affine algebras.

DefiningB as a quotient ring of a polynomial ring is the same as giving an embedding Spec(B),→Spec (K[x1, . . . , xn]). Therefore we may assume that B is a polynomial ring. In the following,K[x1, . . . , xn] andK[y1, . . . , ym] are polynomial rings over a field.

Algorithm 10.3 (Image of a morphism of spectra).

Input: An ideal I ⊆ K[y₁, . . . , y_m] defining an affine algebra A :=

K[y₁, . . . , y_m]/I, and polynomials g₁, . . . , g_n ∈ K[y₁, . . . , y_m] defining a K-algebra-homomorphismϕ:K[x₁, . . . , x_n]→A, x_i 7→g_i+I.

Output: Ideals J1, . . . , Jl ⊆ K[x1, . . . , xn] and polynomials f1, . . . , fl ∈ K[x1, . . . , xn] such that the image of the induced morphismϕ^∗: Spec(A)→ Spec (K[x1, . . . , xn]) =:Y is

im(ϕ^∗) =

l

[

i=1

VY(Ji)\ VY(fi)

(10.1) and the image closure is

im(ϕ^∗) =VY(J₁). (10.2)

(1) Choose monomial orderings “≤_x” on K[x₁, . . . , x_n] and “≤_y” on K[y₁, . . . , y_m], and let “≤” be the block ordering on K[x₁, . . . , x_n, y1, . . . , ym] with “≤y” dominating.

(2) Form the ideal J :=

I∪ {g1−x₁, . . . , g_n−x_n}

K[x₁,...,x_n,y₁,...,y_m]

and compute a Gr¨obner basisGofJ w.r.t. “≤”.

(3) Set

Gx:=K[x1, . . . , xn]∩G, Gy:={NFG_x(g)|g∈G} \ {0}, and

M :={LCy(g)|g∈Gy} \K⊆K[x1, . . . , xn].

Here LC_y(g) denotes the leading coefficient w.r.t. “≤y” of g considered as a polynomial in they_i-variables.

(4) Initialize the listsJ₁, . . . , J_landf₁, . . . , f_lby settingl= 1, J₁:= (G_x)_K[x

1,...,xn], and f₁:= Y

f∈M∪{1}

f.

(5) For allf ∈M, perform Step (6).

(6) Apply the algorithm recursively with (I∪ {f(g₁, . . . , g_n)})_K[y

1,...,y_m] as first argument andg₁, . . . , g_n as second argument. Append the resulting lists of ideals and polynomials to the current listsJ₁, . . . , J_landf₁, . . . , f_l. Theorem 10.4. Algorithm10.3terminates after finitely many steps and cal- culates the image of ϕ^∗ and its closure correctly.

Proof. We use the notation from the algorithm. By way of contradiction, assume that there exists an idealI⊆K[y1, . . . , ym] such that the algorithm applied to Idoes not terminate after finitely many steps. By Hilbert’s Basis Theorem 2.13, we may assume I to be maximal with this property. Since all steps except (6) clearly terminate after finitely many steps, there exists f = LCy(g) ∈ M (with g ∈ Gy) such that Step (6) does not terminate for f. By the maximality of I, this implies f(g1, . . . , gn) ∈ I, so f ∈ J by the definition of J. Since f ∈ K[x1, . . . , xn]\ {0}, Theorem 9.16 yields a g⁰ ∈ G_x whose leading monomial divides LM(f). Since “≤” is a block ordering, LM(g) = LM_y(g)·LM(f), so LM(g⁰) divides LM(g), too. This contradicts the fact that all g ∈ G_y are in normal form w.r.t.G_x, showing the termination of the algorithm.

We proceed with showing the correctness. The correctness of (10.2) follows from (9.10), Proposition9.17and Theorem9.16. To show (10.1), consider the decomposition

im(ϕ^∗) =

im(ϕ^∗)\ VY(f1)

∪ [

f∈M

im(ϕ^∗)∩ VY(f)

, (10.3)

which follows from the definition off1in Step (4). We claim that

im(ϕ^∗)\ VY(f1) =VY(J1)\ VY(f1). (10.4) Indeed im(ϕ^∗)\ VY(f₁) ⊆ im(ϕ^∗)\ VY(f₁) = VY(J₁)\ VY(f₁), where we used (10.2). Conversely, take P ∈ VY(J₁)\ VY(f₁). In the following we use the notation from Proposition9.18. By (9.12), we obtain

Y

g∈Gy

LC (Φ(g)) = Y

g∈Gy

ϕ(LC_y(g)) =ϕ(cf₁)∈ϕ(K[x₁, . . . , x_n]\P) =:U,

wherec∈K\{0}. Moreover, the definition ofG_yin Step (3) impliesΦ(G_y) = Φ(G\Gx)\ {0}, so Proposition9.18(c) tells us thatΦ(Gy) is a Gr¨obner basis

10.1 The Generic Freeness Lemma 151 of the kernel of ψ. So by Lemma 10.1, U⁻¹A is free as a U⁻¹R-module, and there exists a basis containing 1. By Lemma7.16, this implies that the map Spec(U⁻¹A)→Spec(U⁻¹R) is surjective. By Theorem6.5, this means that for every p ∈Spec(R) withU ∩p =∅, there exists Q∈ Spec(A) with R∩Q=p. Particularly,p:=ϕ(P)∈Spec(R) satisfies the conditionU∩p=∅, since otherwise there would exist h ∈ P and u ∈ K[x1, . . . , xn]\P with ϕ(h) =ϕ(u), leading to the contradictionu= (u−h) +h∈ker(ϕ) +P =P since ker(ϕ) = J1 ⊆P. So we haveQ ∈Spec(A) with R∩Q=p =ϕ(P), which is equivalent to P =ϕ⁻¹(Q). So P ∈im(ϕ^∗)\ VY(f1), and (10.4) is proved.

By induction on the recursion depth, we can assume that for every f ∈M the recursive call of the algorithm computes the image ofϕ^∗_f, where ϕf:K[x1, . . . , xn]→A/(ϕ(f))Ais given byxi7→ϕ(xi) + (ϕ(f))A. So in view of (10.3) and (10.4), it suffices to show that

im(ϕ^∗)∩ VY(f) = im(ϕ^∗_f). (10.5) Indeed, im(ϕ^∗_f) consists of allP∈Spec (K[x1, . . . , xn]) such thatP =ϕ⁻¹_f (q) with q∈A/(ϕ(f))A. This condition is equivalent toP =ϕ⁻¹(Q) with Q∈ Spec(A), ϕ(f)∈ Q. This in turn is equivalent to P =ϕ⁻¹(Q) and f ∈ P, i.e.,P ∈im(ϕ^∗)∩ VY(f). This shows (10.5), and the proof is complete. ut Exercise 10.4contains an explicit example to which the algorithm is ap- plied.

Algorithm10.3also computes the image of a morphismf:X →Y of affine varieties over an algebraically closed fieldK. In fact,Y (just likeX) is embed- ded in someKⁿ, so for computing the image one may assumeY =Kⁿ. The morphism f induces a homomorphismϕ: K[x₁, . . . , x_n]→K[X] =:A. Ap- plying Algorithm10.3toϕyieldsJ₁, . . . , J_l⊆K[x₁, . . . , x_n] andf₁, . . . , f_l∈ K[x₁, . . . , x_n] such that

im(ϕ^∗) =

l

[

i=1

V_{Spec(K[x1,...,xn])}(Ji)\ V_{Spec(K[x1,...,xn])}(fi) . Using the algebra geometry lexicon, it is easy to see that this implies

im(f) =

l

[

i=1

(VKⁿ(Ji)\ VKⁿ(fi)).