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KRULL 1. Krull Dimension: the Main Theorems Let R Be a Commutative

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KRULL 1. Krull Dimension: the Main Theorems Let R Be a Commutative KRULL E. BALLICO 1. Krull dimension: the main theorems Let R be a commutative ring with unit 1, R 6= 0. We recall that the Krull dimension dim R of R is the supremum of all integers n ≥ 0 such that there is an increasing chain P0 ( P1 ( ··· ( Pn (1) of prime ideals of R. There are noetherian rings R such that dim R = +1. A chain in (1) is said to have length n. Note that in a noetherian ring any strictly increasing chain of ideals is finite, but the supremum may be +1. As an exercise prove that in Z the supremum of the lengths of chains of ideals (NOT of PRIME ideals) is +1. We will see in the next sections large classes of Noetherian rings (e.g. finitely generated algebras over a field or over Z) which have finite dimension and many other nice properties. For the moment we do not assume noetherianity. Fix P 2 Spec(R). There are two interesting rings associated to P : the domain R=P and the local ring RP . The dimension of RP is called the height ht(P ) ([8, 9]) or the codimension codim of P ([5]). Exercise 1.1. Prove that dim RP + dim R=P ≤ dim R (2) and discuss the informations you get if you are told that equality holds for all prime ideals of R. In 1928 W. Krull proved the following key theorem, usually called with a German name whose English translation is Principal Ideal Theorem. Theorem 1.2. (PIT) Let R be a Noetherian ring. Let I ( R be an ideal generated by c elements. Then ht(P ) ≤ c for any minimal prime containing I. We present the proof of PIT and its consequences from [5, pp. 232-233] (see [7, Theorem 152] for another proof). Theorem 1.7 is also [7, Theorem 153]. You need noetherianity and to know the primary decomposition to make sense of the words \ minimal prime ", because in the definition of noetherianity never you met minimality (maximality, yes, minimality, no). The minimal primes containing I are the minimal, i.e. the non-embeddedp ones; more elementary, they are the prime ideals appearing in a minimal description of I = P1\· · ·\Ps as a finite intersection of prime ideals. Sometimes one call ht(I) the height of I, sometimes it is called the little height of I. Theorem 1.2 implies that ht(I) ≤ c, but in general (for certain rings) it is stronger, because minimal primes of I may have different heights. Proof of the case c > 1 of PIT, assuming that it is true when c = 1: By induction on c we may as- sume that the theoremp is true (for any noetherian ring) for ideals generated by at most c − 1 elements. Write I = P \ P1 \···\ Ps, s ≥ 0 with P; P1;:::;Ps the minimal primes of I. Note that the height of P is equal to the height of the maximalp ideal of RP . Thus taking RP instead of R (which is still Noetherian) we may assume that I = P with P maximal. If ht(P ) = 0, then the theorem is true. Thus we may assume that P strictly contains another prime. Let P1 be a prime such that P ) P1 and there is no prime Q with P ) Q ) P1 (to prove he existence of such P1 use the ascending condition property). To prove that ht(P ) ≤ c it is sufficient to prove that ht(P1) ≤ c − 1. By the inductive assumption it is sufficient to prove that P1 is a minimal prime containing an ideal generated by c − 1 elements. Write I = (x1; : : : ; xc) with xi 2 R. Since P is a minimal prime containing I and P1 ( P , there is i 2 f1; : : : ; cg such that xi 2= I. With no loss of generality we may assume i = 1, i.e. x1 2= P1. Since there is no prime strictly between P1 and P , P is a minimal prime of the ideal (P1; x1) := P1 + Rx1. Thus (since we assuming that P is p the maximal ideal of R), (P1; x1) = P . Since the ring is noetherian, P is finitely generated, say 1 2 E. BALLICO p ki P = (a1; : : : ; am). Since (P1; x1) = P , there is an integer ki > 0 such that ai 2 (P1; x1). Prove n the existence of an integer n > 0 such that P ⊆ (P1; x1) (you may take n = k1 + ··· + km − 1 if m ≥ 2 and n = k1 if m = 1). Write n xi = aix1 + yi; i = 2; : : : ; c with ai 2 R and yi 2 P1. We claim that P1 is minimal among the primes containing y2; : : : ; yc (which would be sufficient to prove he theorem by the inductive assumption). Assume the existence of a prime P2 such that P1 ) P2 ⊇ (y2; : : : ; yc). By assumption P is nilpotent modulo (x1; y2; : : : yc). Thus a power of P=(y2; : : : ; yc) is contained in (x1; y2; : : : ; yc)=(y2; : : : ; yc). Thus P=(y2; : : : ; yc) is a minimal prime containing the principal ideal (x1; y2; : : : ; yc)=(y2; : : : ; yc) of the noetherian ring R=(y2; : : : ; yc). By the case c = 1 of PIT, P=(y2; : : : ; yc) has at most eight 1. Thus either P2=((y2; : : : ; yc) = P or P2=((y2; : : : ; yc) = P1. Hence either P2 = P or P2 = P1, a contradiction. Remark 1.3. Let R be a noetherian ring and P a prime ideal. Let j : R ! RP be the localization map. The ideal µ := RP P is the maximal ideal of RP . For any positive integer k we have k k k µ = RP P . We know (notes on the primary decomposition) that µ is a µ-primary ideal. Hence P (k) := j−1(µk) is a P -primary ideal (same notes). The P -primary ideal P (k) is called the k-th symbolic power of P . We have P k ⊆ P (k) and equality holds if and only if P k is a primary ideal. By construction in a primary decomposition of P k the ideal P (k) is the P -primary component of P k. Proof of the case c = 1 of the PIT (Theorem 1.2) assuming a part of Theorem 2.3: It is sufficient to prove that if Q ( P is a prime ideal, then Q is a minimal prime of R. Replacing R with RP we may assume that R is local with P as its maximal ideal. Since P is minimal over x, the noetherian ring R=(x) has dimension 0. Thus it is artinian (Theorem 2.3). Thus the descending chain (x) + Q(n), n > 0, stabilizes, say with Q(n) ⊆ (x) + Q(n+1). Thus for each f 2 Q(n) there are a 2 R and g 2 Q(n+1) such that f = ax + g. Thus ax 2 Q(n). Since P is a minimal prime over (x), we have x2 = Q and hence a 2 Q(n). This implies that Q(n) = (x)Q(n) + Q(n+1). Since (n) (n+1) x 2 P , Nakayama's lemma implies Q = Q . Applying again Nakayama's lemma in RQ we n get (RQQ) = 0. Thus dim RQ = 0, i.e. Q has height = codimension 0. The following lemma is often called the prime avoidance lemma ([5, Lemma 3,3], [8, Lemma 1.B at p. 2]). Lemma 1.4. Fix an integer r ≥ 2 and take r + 1 ideals I1;:::;Ir;J. Suppose that at least r − 2 of the ideals I1;:::;Ir are prime and that J ⊆ I1 [···[ Ir then there is j 2 f1; : : : ; rg such that J ⊆ Ij Proof. Since the lemma is obvious for r = 1, we may assume r ≥ 2 and use induction on r. In particular we may assume that for any S ⊂ f1; : : : ; rg such that #S = r − 1 we have J * [j2SIj. Since J ⊆ I1 [···[ Ir, for each S = f1; : : : ; rg n fjg there is xj 2 Ij \ J such that xj 2= [h6=jIj. If r = 2 use that x1 + x2 2 J, but x1 + x2 is neither in I1 nor in I2. If r > 2 and, say I1 is a prime, Q x1 + j>1 xj cannot be in any Ij, but it is in J, a contradiction Exercise 1.5. Lemma 1.4 is obvious without assuming that at least r − 2 of the ideals Ij are primes if R contains an infinite field K, because (since K is infinite) no K-vector space V 6= 0 is a finite union of proper subspaces. Check that PIT (see the proof of Theorem 1.7 below for some help) that in a noetherian ring the prime ideals satisfies the descending chain condition . Corollary 1.6. ([5, Corollary 10.3]) The prime ideals in a noetherian ring satisfy the descending chain condition, with the length of a chain descending from a prime P upper bounded by the number of generators of P . Theorem 1.7. (Converse of PIT). Take R noetherian. Then any prime P of R has finite codi- mension (= height) and the codimension, c, is the minimal number of generators of an ideal I such that P is minimal over I. 3 Proof. Since R is noetherian, P is finite generated, say P = (a1; : : : ; am). By PIT we have ht(P ) ≤ m. By PIT we also have that for any ideal J such that P is minimal over J, ht(P ) is at most the number of generators of J. Take an ideal I such that P is minimal over I and that I has the minimal number of generators for any such ideals, say I = (y1; : : : ; yc) with c ≥ 0.
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