Complete Intersections in Regular Local Rings ∗ J. K. Verma Department of Mathematics Indian Institute of Technology Powai, Mumbai 400 076 E-mail: [email protected] Let R be a 3-dimensional regular local ring. Let p be a dimension one prime of R: We are concerned with two problems: (1) when is p a complete intersection and (2) when is p a set theoretic complete intersection ? We prove a result of Cowsik and Nori to answer (1) and several results answering (2). Cowsik observed that if the symbolic Rees algebra of p is Noetherian then p is a set-theoretic complete intersection. Thus it becomes important to know when the symbolic Rees algebra is Noetherian. We will present Huneke's elegant criterion for its Noetherian property. 1 Reductions of Ideals D.G. Northcott and D. Rees [NR] introduced the concept of reduction of an ideal. An ideal J contained in an ideal I of a commutative ring R is called a reduction of I if JIn = In+1 for some n IN: This relationship is preserved under ring homorphisms and ring extensions. If I is a2 zero dimensional ideal of a local ring then the reduction process simplifies I without changing its multiplicity. Proposition 1.1. Let J I be m-primary ideals of a local ring (R; m): If J is a reduction of I then e(I) = e⊆(J): Proof. Let JIn = In+1: Then for all m `(R=In+m) `(R=J m) `(R=Im): ≥ ≥ Hence PI (n + m) PJ (m) PI (m) where P denotes the Hilbert polynomial. This ≥ ≥ shows that PI (m) and PJ (m) have equal degrees and leading coefficients. Definition 1.2. A reduction J of I is called a minimal reduction of I if no ideal properly contained in J is a reduction of I: ∗Notes of lectures given in the NBHM Instructional School on Complete Intersections at IISc, Ban- galore May 20 { June 8, 1996. 1 2 Lemma 1.3. If K is a reduction of J and J is a reduction of I then K is a reduction of I: Proof. Let KJ m = J m+1 and JIn = In+1: Then KIm+n = KJ mIn = Im+n+1: Lemma 1.4. Let (R; m) be a local ring and J I be ideals of R: Then J is a reduction of I iff J + Im is a reduction of I: ⊆ Proof. Let JIn = In+1: Then JIn + mIn+1 = In+1 , hence (J + mI)In = In+1: Conversely let (J + Im)In = In+1: By Nakayama's lemma, JIn = In+1: Definition 1.5. For an ideal I of a local ring (R; m); put F (I) = 1 In=Inm: We say M n=0 F (I) is the fiber ring of I: The Krull dimension of F (I) is called the analytic spread of I: This will be denoted by `(I): Theorem 1.6. Let I be an ideal of a local ring (R; m) with residue field k: For a I put 2 a∗ = residue class of a in I=mI: Let a1; a2; : : : ; as I: Then the following are equivalent: 2 (i) (a1∗; a2∗; : : : ; as∗) is a zero dimensional ideal of F (I): (ii) J = (a1; : : : ; as) is a reduction of I: n 1 n n Proof. The nth homogeneous component of K := (a1∗; : : : ; as∗) is (JI − + mI )=mI : n 1 n n Thus K is zero dimensional iff for all n large JI − + mI = I : By (1.4) the last equation holds iff J is a reduction of I: Corollary 1.7. Every reduction J of I contains a minimal reduction of I: Let a1; a2;:::; as be chosen from J such that the following hold: (a) a1∗; : : : as∗ are k-linearly independent (b) dim F (I)=(a1∗; : : : ; as∗) = 0 (c) s is minimal with respect to (b). Then a1; a2; : : : ; as is a minimal system of generators of a minimal reduction of I con- tained in J: Proof. Put K = (a1; a2; : : : ; as): Observe that K mI = mK: This is equivalent to Ker(K=mK I=mI) = 0: This is a consequence of\ (a). By (1.6), (b) implies that K −! is a reduction of I: Suppose that K 0 K is a reduction of I: Then K 0 + mI = K + mI ⊂ by (c). Hence K (K 0 + mI) K = K 0 + mI K = K 0 + mK: By Nakayama's lemma ⊂ \ \ K = K 0 : It is clear that a1; : : : ; as minimally generate K: In fact a1; : : : ; as are part of a minimal basis of I: Proposition 1.8.Let (R; m) be a local ring with infinite residue field k: Let a1; : : : ; as 2 I; an ideal of R: Then a1∗; : : : ; as∗ form a homogeneous system of parameters of F (I) if and only if J = (a1; : : : ; as) is a minimal reduction of I: 3 Proof. Since k is infinite, it is possible to choose a homogeneous system of parameters of F (I) from the degree one component of F (I): Hence every minimal reduction of I is minimally generated by dim F (I) = `(I) elements. If a1∗; : : : ; as∗ form a homogeneous sys- tem of parameters of F (I) then s = dim F (I) and F (I)=(a1∗; : : : ; as∗) is zero dimensional. Hence a1; : : : ; as generate a minimal reduction of I: Conversely if J = (a1; : : : ; as) is a minimal reduction of I then dim F (I)=(a1∗; : : : ; as∗) = 0 and s is minimal with respect to this property. Hence a1∗; : : : ; as∗ constitute a homogeneous system of parameters. Theorem 1.9. For an arbitrary ideal I of a local ring (R; m) the following inequalities are satisfied. altI := sup htp p is a minimal prime of I `(I) µ(I): f j g ≤ ≤ Proof. We may assume that R=m is infinite. Let J be a minimal reduction of I: Then the equation JIn = In+1 holds for some n which gives V (I) = V (J): Therefore by Krull's altitude theorem altI = altJ µ(J) = `(I): Since dim F (I) dim I=Im; we get `(I) µ(I): ≤ ≤ ≤ 2 Reductions and integral dependence Definition 2.1. Let R be a commutative ring, I an R-ideal. An element x R is called n n 1 2 integral over I if there exist elements a1; a2; : : : ; an so that x + a1x − + + an = 0; i ··· where ai I for i = 1; 2; : : : ; n: 2 Proposition 2.2. The set of integral elements, I;¯ over I is an ideal of R: Proof. Consider the Rees algebra R(I) = 1 Intn of I; where t is an indeterminate. M n=0 n n 1 Let x be integral over I satisfying the equation x + a1x − + + an = 0; for some i n n 1 i ···n i n ai I ; i = 1; 2; : : : ; n: Then (xt) + (a1t)(xt) − + + (ait )(xt) − + + ant = 0: Hence2 xt is integral over R(I): If x; y I¯ then xt;··· yt are integral over··· R(I): Thus xt + yt is integral over R(I): Let u R and2 ut be integral over R(I): Then there exist n2 n 1 b1; b2; : : : ; bn R(I) such that (ut) + b1(ut) − + + bn = 0: Equating coefficient of tn we obtain2un + b un 1 + + b = 0 where b···are defined by b = b tj where 1n − nn ij i X ij j ··· ¯ ¯ ¯ bij I for i = 1; 2; : : : ; n: This shows that u I: In particular x + y I: If x I and c 2R; it is easy to see that cx I:¯ Hence I¯ is2 an ideal. 2 2 2 2 Proposition 2.3. Let I be an ideal of a commutative ring R: Then x I¯ iff I is a reduction of (I; x): 2 n n 1 i n Proof. Suppose x + a1x − + + an = 0 for some ai I ; i = 1; 2; : : : ; n: Then x n 1 n ···1 n 2 2 I(I; x) − which yields I(I; x) − = (I; x) : Conversely suppose that I is a reduction of n 1 n n m n 1 (I; x) and I(I; x) − = (I; x) : Then x = i=1 aibi where ai I and bi (I; x) − : n 1 n 1 j j P 2 2 Thus bi = − aijx − − for some aij I ; j = 0; 1; : : : ; n 1 and i = 1; 2; : : : ; m: Hence Pj=0 n m n 1 n 1 j 2 ¯ − x − aiaijx − − = 0: Thus x I: − Pi=1 Pj=0 2 4 Proposition 2.4. Let I J be ideals of a commutative ring such that J is finitely generated. Then I is a reduction⊆ of J iff J I:¯ ⊆ ¯ Proof. Let J = (I; x1; x2; : : : ; xm): Let J I: Then x1 is integral over I; hence I is a ⊆ reduction of (I; x1) by (2.3). Now apply induction on m to see that I is a reduction of n 1 n J: Conversely let I be a reduction of J: Then for an indeterminate t; (It)(Jt) − = (Jt) for some n: Therefore R[Jt] is a finite R[It]-module. Hence xt is integral over R[It] for any x J: Therefore x I:¯ 2 2 3 Analytic spread of monomial ideals and certain determinantal ideals The Cowsik-Nori theorem on complete intersections requires one to calculate the analytic spread of an ideal I in a regular local ring (R; m). If R is a power series ring or a localisation of a polynomial ring at a prime ideal then the fiber ring of I can be presented as a quotient of polynomial ring. This presentation can be effectively calculated by using the package Macaulay.