We give an elementary proof that for a ring homomorphism A -> B, satisfying
the property that every ideal in A is contracted from B, the following property
holds: for every chain of prime ideals p_0 \subset ... \subset p_r in A there
exists a chain of prime ideals q_0 \subset ... \subset q_r in B such that q_i
\cap A = p_i.Comment: to appear in Journal of pure and applied algebr

Brenner, Holger

English

arXiv

AbstractWe give an elementary proof that for a ring homomorphism A→B satisfying the property that every ideal in A is contracted from B the following property holds: for every chain of prime ideals p0⊂⋯⊂pr in A there exists a chain of prime ideals q0⊂⋯⊂qr in B such that qi∩A=pi

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Lifting chains of prime ideals 

Journal of Pure and Applied Algebra 179 (2003) 1–5
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Lifting chains of prime ideals
Holger Brenner
Mathematische Fakultat, Ruhr-Universitat, Universitatsstr. 150, 44780 Bochum, Germany
Received 22 February 2002; received in revised form 17 July 2002
Communicated by C.A. Weibel
Abstract
We give an elementary proof that for a ring homomorphism A → B satisfying the property
that every ideal in A is contracted from B the following property holds: for every chain of prime
ideals p0 ⊂ · · · ⊂ pr in A there exists a chain of prime ideals q0 ⊂ · · · ⊂ qr in B such that
qi ∩ A= pi.
c© 2003 Elsevier Science B.V. All rights reserved.
MSC: 13B24
Let A and B be commutative rings and let ’ :A → B be a ring homomorphism.
This induces a continuous mapping ’∗ : SpecB → SpecA by sending a prime ideal
q ⊂ B to ’−1(q). Properties of the ring homomorphism are then often re6ected by
topological properties of ’∗. For example, if A→ B is integral, then “going up” holds,
and if A → B is 6at, then “going down” holds (see [4, Proposition 4.15 and Lemma
10.11]). If moreover ’∗ : SpecB → SpecA is surjective and going up or going down
holds, then also the following property holds: for every given chain of prime ideals
p0⊂ · · · ⊂ pr in A there exists a chain of prime ideals q0⊂ · · · ⊂qr in B lying over it.
In this note we give a direct and elementary proof showing that this chain lifting
property holds also under the condition that every ideal in A is contracted from B,
i.e. I = ’−1(IB) holds for every ideal I ⊆ A. This result can be found for pure
homomorphisms in Picavet’s paper (see [10, Proposition 60 and Theorem 37]) and is
proved using valuation theory. Our direct method allows to @nd explicitly chains of
prime ideals and characterizes which prime ideals q0 over p0 may be extended to a
chain. We start with the following lemma.
E-mail address: brenner@cobra.ruhr-uni-bochum.de (H. Brenner).
0022-4049/03/$ - see front matter c© 2003 Elsevier Science B.V. All rights reserved.
PII: S0022 -4049(02)00214 -1
2 H. Brenner / Journal of Pure and Applied Algebra 179 (2003) 1–5
Lemma 1. Let B be a commutative ring, let a0; : : : ; ar be ideals and F0; : : : ; Fr mul-
tiplicatively closed systems. De7ne inductively (set Sr+1 = {1}) for i = r; : : : ; 0 the
following multiplicatively closed sets
Si = {s∈B : (s; ai) ∩ Fi · Si+1 	= ∅}:
Then the following are equivalent.
(i) 0 	∈ S0.
(ii) ai ∩ Fi · Si+1 = ∅ for i = 0; : : : ; r.
(iii) There exists a chain of prime ideals q0 ⊆ · · · ⊆ qr such that ai ⊆ qi and
qi ∩ Fi · Si+1 = ∅.
(iv) There exists a chain of prime ideals q0 ⊆ · · · ⊆ qr such that ai ⊆ qi and
qi ∩ Fi = ∅.
Proof. It is clear that the Si are multiplicatively closed and that Si+1 ⊆ Si. (i) ⇔ (ii).
If 0∈ S0, then a0 ∩ F0 · Si+1 	= ∅, and if ai ∩ Fi · Si+1 	= ∅ for some i, then 0∈ Si and
thus also 0∈ S0.
We show (ii) ⇒ (iii) by induction. Since a0 ∩ F0S1 = ∅, there exists [2, Chapter 2,
Section 5, Corollary 2] a prime ideal q0 such that a0 ⊆ q0 and q0 ∩ F0S1 = ∅.
Thus suppose that the chain q0 ⊂ · · · ⊂ qi is already constructed. We have to
look for a prime ideal qi+1 which includes both qi and ai+1 and which is disjoint to
Fi+1 · Si+2. If such a prime ideal would not exist, then (qi + ai+1)∩Fi+1 · Si+2 	= ∅, say
q+ a= f · s, where q∈ qi, a∈ ai+1, f∈Fi+1 and s∈ Si+2. Then by de@nition q∈ Si+1
contradicting the induction assumption.
(iii) ⇒ (iv) and (iii) ⇒ (ii) are clear, so we have to show (iv) ⇒ (iii). We show this
by descending induction, the beginning for i= r is clear. Suppose that qi−1∩Fi−1Si 	= ∅,
and let q=fs be an element in the intersection, q∈ qi−1, f∈Fi−1, s∈ Si. Since Fi−1
is disjoined to the prime ideal qi−1, it follows that s∈ qi−1. On the other hand, since
s∈ Si we have an equation bs + q′ = f′s′, where b∈B, q′ ∈ qi, f′ ∈Fi, s′ ∈ Si+1, and
this contradicts the induction hypothesis.
Remark 2. The referee (whom I thank for his careful reading) pointed out that there
exists a similar and more general result in a preprint of Bergman (see [1]). Bergman
studies for a partially ordered set I and ideals ai and multiplicatively closed subsets Si in
a commutative ring the existence of prime ideals pi, ai ⊆ pi, pi∩Si=∅ such that pi ⊂ pj
holds for i6 j. Bergman [1, Proposition 9] gives a characterization for the existence
of such prime ideals for a tree order I in terms of an inductively de@ned system of
equations which is related to our characterization in Lemma 1(ii). It is possible that
using Bergman’s result one may obtain a stronger version of the following theorem.
Theorem 3. Let A and B be commutative rings and let ’ :A → B be a ring homo-
morphism such that I =’−1(IB) holds for every ideal I ⊆ A. Then for every chain of
prime ideals p0 ⊂ · · · ⊂ pr in SpecA there exists a chain of prime ideals q0 ⊂ · · · ⊂ qr
in B such that pi = qi ∩ A for i = 0; : : : ; r.
H. Brenner / Journal of Pure and Applied Algebra 179 (2003) 1–5 3
Proof. Let a chain of prime ideals p0 ⊂ · · · ⊂ pr in A be given. We shall apply the
preceding lemma to the ideals ai=piB and the multiplicatively closed sets Fi=A−pi ⊂
B. Note that the @ber over p consists of the prime ideals q for which pB ⊂ q and
q ∩ ’(A − p) = ∅ hold. De@ne Si ⊆ B as before and suppose that 0∈ S0. This means
that there exists an element a0 ∈ a0 such that a0 =f0 · s1, where f0 ∈F0, s1 ∈ S1. This
means by de@nition that we have an equation
b1s1 + a1 = f1s2; where b1 ∈B; a1 ∈ a1; f1 ∈F1 and s2 ∈ S2:
Going on recursively we @nd equations
bjsj + aj = fjsj+1; where bj ∈B; aj ∈ aj; fj ∈Fj and sj+1 ∈ Sj+1
and eventually
brsr + ar = fr; where br; sr ∈ Sr; ar ∈ ar ; fr ∈Fr:
We multiply the last equation by fr−1 · · ·f0 and get
br(srfr−1)fr−2 · · ·f0 + arfr−1 · · ·f0 = frfr−1 · · ·f0:
We may replace br(srfr−1)fr−2 · · ·f0 by
br(br−1sr−1 + ar−1)fr−2 · · ·f0 = brbr−1(sr−1fr−2) · · ·f0 + brar−1fr−2 · · ·f0
and so going on we @nd that
fr · · ·f0 = br · · · b1a0 + br · · · b2a1f0 + br · · · b3a2f1f0 + · · ·+
brar−1fr−2 · · ·f0 + arfr−1 · · ·f0:
This equation shows that
fr · · ·f0 ∈ (p0 + p1f0 + p2f1f0 + · · ·+ pr−1fr−2 · · ·f0 + prfr−1 · · ·f0)B
and this yields an equation in A (here we apply the condition that every ideal is
contracted),
p0 + p1f0 + p2f1f0 + · · ·+ pr−1fr−2 · · ·f0 + prfr−1 · · ·f0 − fr · · ·f0 = 0;
where pi ∈ pi. We may write this as
p0 =−f0(p1 + p2f1 + · · ·+ pr−1fr−2 · · ·f1 + prfr−1 · · ·f1 − fr · · ·f1)
and therefore p1 + p2f1 + · · · + pr−1fr−2 · · ·f1 + prfr−1 · · ·f1 − fr · · ·f1 ∈ p0 ⊂
p1. Then again we may multiply out f1 and so on until we @nd pr−1 + prfr−1 −
frfr−1 ∈ pr−2 ⊂ pr−1 and then prfr−1 − frfr−1 ∈ pr−1, hence pr − fr ∈ pr−1 and
fr ∈ pr , which is a contradiction.
Remark 4. The condition that every ideal is contracted is ful@lled for example if
’ :A → B is a pure homomorphism. This means that for every A-module M the
natural mapping M → M ⊗A B is injective. If B contains A as a direct summand, then
A ⊆ B is pure. Direct summands arise often in invariant theory: if a linearly reductive
group acts on a ring B, then the ring of invariants A = BG is a direct summand in B
4 H. Brenner / Journal of Pure and Applied Algebra 179 (2003) 1–5
(see [5, Chapter 1, Section 1]). Example 7 shows that for a direct summand neither
going up nor going down hold in general.
Picavet studies in [10] the property of a ring homomorphism that over every chain
of prime ideals p ⊂ q there lies a chain of prime ideals above. He calls a ring homo-
morphism with this property subtrusif and shows that a homomorphism ’ :A → B is
universally subtrusif if and only if for every valuation domain A→ V the corresponding
homomorphism V → B⊗A V is pure.
Picavet proved the theorem for universally subtrusive morphisms [10, Proposition
60 in connection with Theorem 37] using several facts from valuation theory: that for
a chain of prime ideals p0 ⊂ · · · ⊂ pr in a domain A there exists a valuation ring
A ⊆ V ⊆ Q(A) and a chain of prime ideals r0 ⊂ · · · ⊂ rr in V with ri ∩ A = pi,
see [6, Corollary 19.7] (see also [8,3] for recent developments in the lifting of chains
to valuation rings), and that a valuation domain is a Bezout domain and hence a
torsion free module over it is 6at (see [9, Theorem 63] and [2, Chapter 1, Section 4,
Proposition 3]).
Corollary 5. Let A and B be commutative rings and let ’ :A → B be a ring homo-
morphism such that I = ’−1(IB) holds for every ideal I ⊆ A. Then dim B¿ dim A.
Proof. This is clear from the theorem.
Corollary 6. Let A be a commutative Noetherian ring and let B be an A-Algebra of
7nite type such that every ideal of A is contracted from B. Then g : SpecB→ SpecA
is submersive, i.e. SpecA carries the quotient topology.
Proof. We have to show that a subset W ⊆ SpecA is open if its preimage is open.
Since g is surjective, we know that W = g(g−1(W )), hence W is constructible by [7,
ThLeorMeme 7.1.4]. For the openness it is therefore enough to show that it is closed under
generalization, and this follows directly from our property: let p′ ∈W and let p ⊂ p′
be a generalization. Let q ⊂ q′ be prime ideals lying over them. Then q′ ∈ g−1(W ) and
since g−1(W ) is open it is closed under generalization, hence q∈ g−1(W ), and this
means p∈W .
It is easy to give an example of a direct summand such that SpecB→ SpecA ful@lls
neither the going down nor the going up property.
Example 7. Let K be a @eld and let the polynomial ring B = K[X; Y; Z] be Z-graded
by degX = deg Y = 1, deg Z =−1. Then the ring of degree zero is
A= B0 = K[XZ; YZ] ∼= K[U; V ]:
A is a direct summand in B, hence the chain lifting property holds.
We consider the chain (XZ) ⊂ (XZ; YZ) in A. The principal prime ideal ZB maps to
(XZ; YZ), but no prime ideal ⊂ ZB maps to (XZ), hence going down does not hold.
The prime ideal (X; Y 2Z−1)B maps to (XY ). But a prime ideal lying over (XZ; YZ)
must contain either ZB or (X; Y )B, hence also going up fails to hold.
H. Brenner / Journal of Pure and Applied Algebra 179 (2003) 1–5 5
Remark 8. A surjective (even bijective) mapping between aNne varieties may not
ful@ll the chain lifting property, since there exist bijective mappings which are not
homeomorphisms.
References
[1] G. Bergman, Arrays of prime ideals in commutative rings, preprint, http://math.berkeley.
edu/gbergman/papers, 2002.
[2] N. Bourbaki, AlgMebre commutative, Hermann, Paris, 1961.
[3] D.E. Dobbs, Lifting chains of prime ideals to paravaluation rings, Rend. Circ. Mat. Palermo, II. Ser.
49 (2) (2000) 319–324.
[4] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer, Berlin, 1995.
[5] J. Fogarty, D. Mumford, Geometric Invariant Theory, Springer, Berlin, 1982.
[6] R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.
[7] A. Grothendieck, J. DieudonnLe, ElLements de gLeomLetrie algLebrique I, Springer, Berlin, 1971.
[8] B.G. Kang, D.Y. Oh, Lifting up an in@nite chain of prime ideals to a valuation ring, Proc. Am. Math.
Soc. 126 (3) (1998) 645–646.
[9] I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
[10] G. Picavet, Submersion et descente, J. Algebra 103 (1986) 527–591.


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Lifting chains of prime ideals

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