A VALUATION THEOREM FOR NOETHERIAN RINGS ANTONI RANGACHEV Abstract. Let A ⊂ B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra. Denote by A the integral closure of A in B. We show that A is determined by finitely many unique discrete valuation rings. Our result generalizes Rees' classical valuation theorem for ideals. We obtain a variant of Zariski's main theorem and then we give a simple proof of his result. 1. Introduction Let A ⊂ B be integral domains. Denote the integral closure of A in B by A. Suppose there exist valuation rings V1;:::; Vr in Frac(A) such that r (1) A = \i=1Vi \B; where the intersection takes place in Frac(B). We say that (1) is a valuation decomposition of A. We say the decomposition is irredundant or minimal if dropping any Vi violates (1). The main result of this paper is the following valuation theorem. Theorem 1.1. Suppose A is Noetherian and B is a finitely generated A-algebra. Then either AssA(B=A) = f(0)g, or A = B, or there exist unique discrete valuation rings V1;:::; Vr in r Frac(A) such that A = \i=1Vi \B is minimal. Furthermore, if A is locally formally equidi- mensional, then each Vi is a divisorial valuation ring with respect to a Noetherian subring of A. It's well-known that A may fail to be Noetherian [SH06, Ex. 4.10]. The proof of Thm. 1.1 rests upon three key observations. First, we show that A is generically Noetherian. Then we use this to prove that AssA(B=A) is finite by results of [Ran20]. We set each Vi to be the localization of A at a prime in AssA(B=A), which is a DVR by [Ran20, Thm. 1.1 (i)]. Finally, to get the equality in (1) we show that the minimal primes of an ideal in A which is the annihilator of an element of B=A are in AssA(B=A). As another application of these observations we obtain a variant of Zariski's main theorem. We also provide a simple proof of Zariski's result. 1 1 Let R be a Noetherian domain. Suppose A = ⊕i=0Ai ⊂ B = ⊕i=0Bi is a homogeneous inclusion of graded Noetherian domains with A0 = B0 = R. Suppose B is a finitely generated A-algebra. For each n denote by An the integral closure of An in Bn. It's the R-module consisting of all elements in Bn that are integral over A. For the discrete valuations Vi in Thm. 1.1 set Vi := Vi \ Frac(R). Define AnVi \Bn to be the set of elements in Bn that map to AnVi as a submodule of BnVi. The following is a corollary to our main result. 2010 Mathematics Subject Classification. 13A18, 13B22, 13A30, 14A15, (14B05). Key words and phrases. Integral closure of rings, discrete valuation rings, Rees valuations, Chevalley's constructability result, Zariski's main theorem. 1 2 ANTONI RANGACHEV Corollary 1.2. Suppose AV = AV for each valuation V in Frac(R). Then either A = B, or AssA(B=A) = f(0)g, or r (2) An = \i=1AnVi \Bn for each n. Furthermore, if Vi 6= Frac(R) for i = 1; : : : ; r, then (2) is minimal and the Vis in (2) are unique. Let I be an ideal in a Noetherian domain R. Let t be a variable. The graded algebra R[It] := R ⊕ It ⊕ I2t2 ⊕ · · · is called the Rees algebra of I. It's contained in the polynomial ring R[t] := R ⊕ Rt ⊕ Rt2 ⊕ · · · . For each n denote by In the integral closure of In in R. Set A := R[It] and B := R[t] in Cor. 1.2. Note that for each valuation ring V in Frac(R) we have AV = V [t] or AV = V [at] where IV = (a) for some a 2 I. Thus AV is integrally closed, and so AV = AV . Cor. 1.2 recovers a classical result due to Rees [R56]. Corollary 1.3. [Rees' valuation theorem] Let R be a Noetherian domain and I be a nonzero ideal in R. There exists unique discrete valuations V1;:::;Vr in the field of fractions of R n r n such that I = \i=1I Vi \ R for each n. In the setting of Cor. 1.2 assume additionally that R is locally formally equidimensional. We can give a geometric interpretation of the centers of the Vis in R using Chevalley's con- structability result as follows. Consider the structure map c: Proj(A) ! Spec(R). For each integer l ≥ 0 set S(l) := fp 2 Spec(R): dim Proj(A ⊗R k(p)) ≥ lg: By Chevalley's [EGAIV, Thm. 13.1.3 and Cor. 13.1.5] S(l) is closed in Spec(R). For i = 1; : : : ; r denote by mi the center of Vi in R. Set e := dim Proj(A ⊗R Frac(R)). Theorem 1.4. Suppose R is locally formally equidimensional. If ht(mi) > 1 for some i, then mi is a minimal prime of S(ht(mi) + e − 1). Acknowledgements. I thank Madhav Nori and Bernard Teissier for stimulating and helpful conversations. I was partially supported by the University of Chicago FACCTS grant \Conormal and Arc Spaces in the Deformation Theory of Singularities." 2. Proofs The proof of Thm. 1.1 is based on three key propositions. Proposition 2.1. Suppose A ⊂ B are integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra. Then there exists f 2 A such that Af is Noetherian. Proof. Denote by E the algebraic closure of Frac(A) in Frac(B). By Zariski's lemma E is a finite field extension of Frac(A). Because E = Frac(A), there exist f1; : : : ; fk 2 A such that 0 0 0 E = Frac(A)(f1; : : : ; fk). Set A := A[f1; : : : ; fk]. Then A is Noetherian and Frac(A ) = 0 0 0 Frac(A). By [Ran20, Prp. 2.1] AssA0 (B=A ) is finite. But AssA0 (A=A ) ⊂ AssA0 (B=A ). So 0 0 AssA0 (A=A ) is finite, too. Select f 2 A from the intersection of all minimal primes in 0 0 AssA0 (A=A ). Then Af = Af ; hence Af is Noetherian. The next proposition strengthens [Ran20, Thm. 1.1 (ii)] in the domain case. Proposition 2.2. Suppose A ⊂ B are integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra. Then AssA(B=A) and AssA(B=A) are finite. A VALUATION THEOREM FOR NOETHERIAN RINGS 3 Proof. By Prp. 2.1 there exists f 2 A such that Af is Noetherian. Let q 2 AssA(B=A). If f 62 q, then q 2 Ass (B =A ). The last set is finite by [Ran20, Prp. 2.1]. Suppose f 2 q. As Af f f before, denote by E the algebraic closure of Frac(A) in Frac(B). It's a finite field extension of Frac(A). Denote by L the integral closure of A in E. By the Mori{Nagata Theorem L is a Krull domain ([Bour75, Prp. 12, pg. 209] and [SH06, Ex. 4.5]). But L is also the integral closure of A in its field of fractions. Let q0 be a prime in L that contracts to q. We have Aq ⊂ Lq0 . By Thm. 1.1 (i) Aq is a DVR. As A and L have the same field of fractions, 0 Aq = Lq0 . Thus ht(q ) = 1. Because L is a Krull domain, there are finitely many height one prime ideals in L containing f. Thus there are finitely many q 2 AssA(B=A) containing f. This proves the finiteness of AssA(B=A). Alternatively, apply directly [Ran20, Thm. 1.1 (ii)] for A0 and B noting that A and A0 have the same integral closure in B. Let p 2 Ass (B=A). If f 62 p, then p is a contraction from a prime in Ass (B =A ) which A Af f f is finite by [Ran20, Prp. 2.1]. If f 2 p, then the proof of [Ran20, Thm. 1.1 (ii)] shows that p 2 AssA(A=fA) which is finite because A is Noetherian. The proof is now complete. Proposition 2.3. Suppose A ⊂ B are integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra. Let b 2 B be such that J := (A :A b) is a nonunit ideal in A. Then the minimal primes of J are in AssA(B=A). Proof. If J = (0), then clearly J 2 AssA(B=A). Suppose J 6= (0). Select a nonzero h 2 J. Then J := ((h):A hb). Thus the minimal primes of J are among the minimal primes of (h) each of which is of height one. Denote by L the integral closure of A in Frac(A). Because L is a Krull domain, then there are finitely many minimal primes of hL. But L is integral over A. So by incomparability each minimal prime of (h) is a contraction of a prime of height one in L which has to be a minimal prime of hL. Therefore, (h) has finitely many minimal primes, and so does J. Denote by q1;:::; ql the minimal primes of J. First, we want to show that for each 1 ≤ i ≤ l si there exists a positive integer si such that qi ⊂ JAqi . We proceed as in the proof of [Ran20, Thm. 1.1 (i)]. Set pi := qi \A. We can assume that A is local at pi.