A Valuation Theorem for Noetherian Rings 1

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) = {(0)}, 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 equidimensional, 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. ∞ ∞ 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) = {(0)}, 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 ∈ 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 constructability result as follows. Consider the structure map c: Proj(A) → Spec(R). For each integer l ≥ 0 set

S(l) := {p ∈ Spec(R): dim Proj(A ⊗R k(p)) ≥ l}. 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 ∈ 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 ∈ 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 ∈ 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 ∈ A such that Af is Noetherian. Let q ∈ AssA(B/A). If f 6∈ q, then q ∈ Ass (B /A ). The last set is finite by [Ran20, Prp. 2.1]. Suppose f ∈ 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 ∈ 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 ∈ Ass (B/A). If f 6∈ 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 ∈ p, then the proof of [Ran20, Thm. 1.1 (ii)] shows that p ∈ 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 ∈ 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 ∈ AssA(B/A). Suppose J 6= (0). Select a nonzero h ∈ 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. Let Ab be the completion 0 0 0 0 of A with respect to pi. Set A := A ⊗A Ab and B := B ⊗A Ab. Replace Ab, A and B by their reduced structures. Because Ab is a reduced complete local ring and B0 is a finitely generated Ab-algebra, then by [Stks, Tag 03GH] A0 is module-finite over Ab. In particular, A0 0 0 si 0 0 is Noetherian. Clearly, JA is primary to qiA . Thus there exists si such that qi A ⊂ JA . si 0 si si si Hence qi b ∈ A . But qi b ∈ B. Thus by [Ran20, Prp. 2.2] qi b ∈ Api , and so qi b ∈ Aqi . This si implies qi ⊂ JAqi by [AtM69, Prp. 3.14] applied for (A, b)/A. Assume that the si defined above are the minimal possible. Fix 1 ≤ j ≤ l. For each i 6= j si sj −1 by prime avoidance we can select ci ∈ qi and ci 6∈ qj. Let cj ∈ qj with cj 6∈ JAqj . Set c := c1 ··· cl. Then qj = (A :A cb) and thus qj ∈ AssA(B/A).

Proof of Theorem 1.1

We can proceed with the proof of Thm. 1.1. Suppose AssA(B/A) 6= {(0)} and A= 6 B. Then by Prp. 2.2 AssA(B/A) contains finitely many nonzero prime ideals which we denote by q1,..., qr. By [Ran20, Thm. 1.1 (i)] Vi := Aqi is a DVR for each i = 1, . . . , r. Obviously, r r A ⊆ ∩i=1Vi ∩ B. Let b = x/y ∈ ∩i=1Vi ∩ B. Set J := (yA :A x). We have JVi = Vi for each i. Thus J * qi for each i. But Jb ∈ A and J 6= (0). So by Prp. 2.3 if J is a nonunit ideal, its 4 ANTONI RANGACHEV minimal primes are among q1,..., qr which is impossible. Thus J has to be the unit ideal, which implies that b ∈ A. To prove minimality of the valuation decomposition, suppose we can drop Vj in (1) for some j. Then localizing both sides of (1) at qj we obtain that Aqj = Bqj which contradicts with qj ∈ Supp(B/A). Thus (1) is minimal. We are left with proving the uniqueness of the Vis. Suppose s 0 (3) A = ∩j=1Vj ∩ B 0 0 is a minimal discrete valuation decomposition. Set B := Frac(A) ∩ B. We have Aqi 6= Bqi for each i = 1, . . . , r. As the intersection in (3) takes place in B0 we can replace in it B by 0 0 B . Localizing both sides of (3) at q1 we obtain that there exists a valuation Vl such that 0 0 0 0 0 (Vl )q1 6= Frac(A). But V1 = Aq1 ⊂ (Vl )q1 . Thus V1 = (Vl )q1 . Also, Vl ⊂ (Vl )q1 , and so 0 0 0 Vl = (Vl )q1 . Therefore, V1 = Vl . Continuing this process we obtain that each Vi appears in 0 (3). As (3) is minimal, we obtain that s = r and after possibly renumbering we get Vi = Vi for i = 1, . . . , r. Let A0 be the module-finite A-algebra defined in the proof of of Prp. 2.1. Suppose A is locally formally equidimensional. Then so is A0. Note that Frac(A0) = Frac(A). Denote by 0 mVi the maximal ideal of Vi. Set pi := mVi ∩ A . By Cohen’s dimension inequality (see [SH06, Thm. B.2.5])

tr. degκ(pi)κ(mVi ) ≤ ht(pi) − 1. 0 Because pi = qi ∩ A , then by [Ran20, Thm. 1.1 (iii)] we get ht(pi) = 1. Therefore, tr. degκ(pi)κ(mVi ) = ht(pi) − 1 = 0. Hence each Vi is a divisorial valuation ring in Frac(A). Remark 2.4. As it’s well-known, an integrally closed domain equals the intersection of all valuation rings in its field of fractions that contain it. From here one derives set-theoretically that A = ∩V∩B where the intersection is taken over all valuation rings in Frac(A) that contain the integral closure of A in Frac(A). Because A0 and A have the same integral closure and A0 is Noetherian (see the proof of Prp. 2.1), then in the intersection we can take only DVRs. Thus, the real contribution of Thm. 1.1 is that under the additional hypothesis that B is a finitely generated A-algebra, one can take finitely many uniquely determined DVRs each of which is a localization of A at a height one prime ideal.

Proof of Cor. 1.2 and Cor. 1.3

Suppose AssA(B/A) 6= {(0)} and A 6= B. Because AV = AV and A ⊂ AV we get A ⊂ AV . Thus for each i = 1, . . . , r A ⊂ AVi ⊂ Vi. ∞ r But AVi = ⊕j=0AjVi. Also, by Thm. 1.1 A = ∩i=1Vi ∩ B. Thus, set-theoretically An = r ∩i=1AnVi ∩ Bn. Set K := Frac(R). Suppose Vi 6= K for each i = 1, . . . , r. Showing that the decomposition is minimal and unique is done in the same way as in Thm. 1.1. Here we will 0 0 show just the uniqueness of the valuations. Suppose there exist DVRs V1,...,Vs in K such that s 0 (4) A = ∩j=1AVj ∩ B 0 is minimal. As in the proof of Thm. 1.1 we can assume that Frac(B) = Frac(A). If Vj = K for 0 some j, then because (4) is minimal we get s = 1 and V1 = K. So A = AK ∩ B. Localizing at q1 we obtain that V1 = (AK)q1 . But K ⊂ (AK)q1 . Thus V1 = K, a contradiction. 0 Therefore, Vj 6= K for each j. Again, by localizing (4) at qi, we get that there is a j such that A VALUATION THEOREM FOR NOETHERIAN RINGS 5

0 0 0 Vi = (AVj )qi . But Vi = Vi ∩ Frac(R) and Vj ∈ (AVj )qi ∩ Frac(R). Because Vi 6= Frac(R), 0 0 then Vi = Vj . Thus r = s by minimality and after possible renumbering Vi = Vi for each i = 1, . . . , r. Consider Cor. 1.3. Apply Cor. 1.2 with A := R[It] and B := R[t]. In the introduction we proved that R[It]V = R[It]V for each valuation V in K. What remains to be shown is that Vi 6= K for each i. Indeed, the prime ideals in R[It] are contractions of extensions of prime ideals of R to R[t]. Thus ht(q ∩ R) ≥ 1 and so V 6= K for each i otherwise q R[It] is a unit i i i qi ideal which is impossible. A version of Cor. 1.2 for Rees algebras of modules is proved by Rees in [R87, Thm. 1.7]. Proof of Theorem 1.4

First, we show that mi ∈ S(ht(mi) + e − 1). Recall that the maximal ideal of Vi contracts to qi in A. Set pi := qi ∩ A. Then by [Ran20, Thm. 1.1 (iii)] ht(pi) = 1. Consider the map

Proj(Ami ) → Spec(Rmi ). It’s closed, surjective and of ﬁnite type. By the dimension formula

([Stks, Tag 02JX]) dim Proj(Ami ) = ht(mi) + e. But Ami is a local formally equidimensional ring. Because ht(piAmi ) = 1, by [SH06, Lem. B.4.2] dim Proj(A ⊗ k(mi)) = ht(mi) + e − 1. Thus mi ∈ S(ht(mi) + e − 1). Next, suppose there exists a prime ni in R with ni ⊂ mi and ni ∈ S(ht(mi) + e − 1). Then dim Proj(A ⊗ k(ni)) ≥ ht(mi) + e − 1. Note that ni 6= (0) for otherwise ni ∈ S(e) which forces ht(mi) = 1, a contradiction. Because dim Proj(Ani ) = ht(ni) + e then dim Proj(A ⊗ k(ni)) ≤ ht(ni) + e − 1. Therefore, ht(mi) + e − 1 ≤ ht(ni) + e − 1. But ni ⊂ mi. Thus ni = mi and mi is a minimal prime in S(ht(mi) + e − 1). This completes the proof of Thm. 1.4. Thm. 1.4 generalizes [Ran18, Thm. 7.8], which is a result for Rees algebras of modules. To see that Thm. 1.4 is sharp, let (R, m) be a Noetherian regular local ring of dimension at least 2, and let h ∈ m be an irreducible element. Let B be the polynomial ring R[y1, . . . , ye+1] for some e ≥ 0. Set A := R[hy1, . . . , hye+1]. Thus A is a polynomial subring of B. It is normal because R is regular. In the setup of Cor. 1.2 there is only one V1 = Aq1 where q1 = hA. Note that S(k) = S(e) for all k ≥ 0 because A is a polynomial ring over R generated by e + 1 elements. Thus the only minimal prime in S(e) is (0), whereas mVi = (h) is a height one prime ideal in R. Zariski’s Main Theorem Let A ⊂ B be Noetherian rings. Suppose B is a ﬁnitely generated A-algebra. Denote by A the integral closure of A in B. Denote by IB/A the intersection of all elements in AssA(B/A). In all remaining results with the exception of Thm. 2.9 we assume that A and B are integral domains. The following result characterizes the support of B/A. Proposition 2.5. Let A ⊂ B be integral domains. Suppose A is Noetherian and B is a ﬁnitely generated A-algebra. Then V(IB/A) = SuppA(B/A).

Proof. If Frac(A) 6= Frac(B), then (0) ∈ AssA(B/A) and trivially V(IB/A) = SuppA(B/A) =

Spec(A). Suppose Frac(A) = Frac(B). Then by [Ran20, Thm. 1.1 (i)] IB/A = q1 ∩...∩qs with ht(q ) = 1 for each i. If q ∈ (I ), then q ⊂ q for some j and thus q A ∈ Ass (B /A ). i V B/A j j q Aq q q Hence V(IB/A) ⊂ SuppA(B/A). Suppose q ⊂ SuppA(B/A). Then there is x/y ∈ B with x, y ∈ A, such that its image in Bq is not Aq. In other words, if J := ((x): A y), then J ⊂ q. 6 ANTONI RANGACHEV

But by Prp. 2.3 the minimal primes of J are among the qis. Thus there exists qj such that qj ⊂ q, i.e. SuppA(B/A) ⊂ V(IB/A). In [EGAIII, Cor. 4.4.9] Grothendieck derives the following result as a consequence of Zariski’s main theorem (ZMT): if g : X → Y is a birational, proper morphism of noetherian integral schemes with Y normal and g−1(y) finite for each y ∈ Y , then g is an isomorphism. Below we show that in the affine case we can reach the same conclusion assuming just that g is surjective. To do this we do not have to appeal to ZMT. In fact, our result proves ZMT in codimension one as shown below. Theorem 2.6. Let A ⊂ B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra. Assume Frac(A) = Frac(B) and Spec(B) → Spec(A) is surjective. Then B = A.

Proof. Suppose there exists q ∈ AssA(B/A). Because q 6= (0), by [Ran20, Thm. 1.1 (i)] Aq is a DVR. But Aq 6= Bq. Thus Bq = Frac(B). This contradicts the assumption that there exists a prime in B that contracts to q. Thus AssA(B/A) is empty, and so by Prp. 2.5 B = A. Definition 2.7. Let Q ∈ Spec(B). Set q := Q ∩ A and p := Q ∩ A. We say that Spec(B) → Spec(A) is quasi-finite at Q if Q is isolated in its fiber, i.e. if the field extension κ(p) ⊂ κ(Q) is finite and dim(BQ/pBQ) = 0. We say that Spec(B) → Spec(A) is quasi-finite if it is quasi-finite at each prime in Spec(B). The following corollary to Thm. 2.6 is a special case of ZMT. Corollary 2.8. Let A ⊂ B be integral domains. Suppose A is Noetherian and B is a finitely generated A-algebra. Assume Spec(B) → Spec(A) is quasi-finite at Q and ht(q) = 1. Then there exists f ∈ IB/A with f 6∈ q such that Bf = Af .

Proof. Because ht(q) = 1 then Aq is a universally catenary Noetherian ring. Applying the dimension formula for Aq and Bq we get

(5) ht(Q) + tr. degκ(q)κ(Q) = ht(q) + tr. degAB

Because qBQ is QBQ-primary we have ht(q) ≥ ht(Q). But κ(Q) is a ﬁnite ﬁeld extension of κ(q). So tr. degκ(q)κ(Q) = 0. Thus ht(Q) = ht(q) and Frac(A) = Frac(B). Applying Thm. 2.6 to Aq and Bq we get Aq = Bq. Because Frac(A) = Frac(B), by Prp. 2.3 for each b ∈ B there exists a positive integer k such that Ikb b ∈ A. Thus for each f ∈ I we have b B/A B/A

Af = Bf . By Prp. 2.5 IB/A 6⊂ q. So we can select f ∈ IB/A with f 6∈ q. Theorem 2.9 (Zariski’s Main Theorem). Let A ⊂ B be Noetherian rings. Suppose B is a ﬁnitely generated A-algebra. Assume Spec(B) → Spec(A) is quasi-ﬁnite at Q. Then there exists f 6∈ p such that Bf = Af .

Proof. We will prove that Aq = Bq. Assume A is local with maximal ideal p. Let’s show we can reduce to the case when A is a complete Noetherian local domain and B is a domain. Here we follow the proof of [Ran20, Thm. 1.1 (i)]. The main observation we use is a descent property of the integral closure under a faithfully flat base change. 0 0 0 Let (Ab, pˆ) be the completion of A with respect to p. Set B := B ⊗A Ab and Q := QB . By flatness and associativity of the tensor product Q0 is maximal in B0 (see the second paragraph in the proof of [Ran20, Thm. 1.1 (i)]). Furthermore, Q0 is isolated in its fiber over pˆ by faithful A VALUATION THEOREM FOR NOETHERIAN RINGS 7

flatness. Indeed, suppose there is Q00 ⊂ Q0 such that Q00 ∩ Ab = pˆ. Then Q00 ∩ B contracts to p, and so Q00 ∩ B = Q, which by faithful flatness forces Q00 = Q0. If we show that B0 is integral over Ab, then that will force the integrality of B over A by 0 0 0 0 [Ran20, Prp. 2.2]. Let Qmin be a minimal prime of B . If we prove that B /Qmin is integral 0 0 0 over Ab/(Qmin ∩ Ab) for each minimal prime Qmin, then that will force B to be integral over 0 Ab (cf. [SH06, Prp. 2.1.6]). Note that Ab/(Qmin ∩ Ab) is a complete local Noetherian ring and 0 0 0 0 0 0 B /Qmin is a finitely generated Ab/(Qmin ∩ Ab)-algebra. The image of Q in B /Qmin is isolated 0 in its fiber over pˆ/Qmin. From now on assume that (A, p) is a complete local Noetherian domain and B is a domain. To show that Aq = Bq it’s enough to prove that BQ is integral over A. Consider the associated n n+1 n n+1 graded rings grp(A) and grpB(B). From p /p ⊗ B p B/p B one deduces that grpB(B) is a homomorphic image of grp(A) ⊗κ(p) B/pB. But B/pB is finite over κ(p) because pB is QBQ-primary and κ(Q) is a finite field extension of κ(p). Thus grpB(B) is module-finite over grp(A). But A is complete and BQ is separated in the p-adic topology. Thus by [AK18, Lem. 22.24] we conclude that BQ is module-finite over A. Finally, Aq = Bq. Let b1, . . . , bs be generators of B as an A-algebra. Write each fi in the Qs form ai/fi where ai ∈ A and fi ∈ A\q. Set f := l=1 fl. Then Af = Bf .

In the proof of Thm. 2.9 we showed that BQ is integral over A assuming that the latter ring is a complete local Noetherian domain. This may seem too strong to be true for even if B is integral over A and has at least two maximal ideals of different height, say Q is the one of smaller height, then BQ is not integral over A (see Nagata’s [SH06, Ex. 2.2.6]). It’s the completeness of A that guarantees that A is a local ring. Indeed, let L be the integral closure of A in Frac(A). The latter is a finite field extension of Frac(A). Thus by [SH06, Thm. 4.3.4] L is a complete Noetherian local domain. But L is the integral closure of A in Frac(A). Thus A is local as well. As an immediate corollary we get an explicit description of the quasi-finite locus in B in the domain case. Corollary 2.10. Let A ⊂ B be integral domains. Suppose A is Noetherian and B is a finitely −1 generated A-algebra. Set g : Spec(B) → Spec(A) and U := Spec(A) − V(IB/A). Then g U is the set of primes at which Spec(B) → Spec(A) is quasi-finite.

Proof. By Thm. 2.9 if Spec(B) → Spec(A) is quasi-finite at Q, then Aq = Bq where q = Q∩A. Applying Prp. 2.5 finishes the proof. In general, Cor. 2.10 is usually stated as an existence result (cf. [Stks, Tag 03GT] and [Ray70, Cor. 1, p.42]). The following corollary due to Grothendieck shows that we can replace A by a subalgebra which is module-finite over A. It’s what is used in algebraic geometry to show that a qusi-finite morphism factors as a composition of an open imersion and a finite morphism. Corollary 2.11. [EGAIII, Cor. 4.4.7] Let A ⊂ B be integral domains. Suppose A is a local Noetherian ring with maximal ideal p and B is a finitely generated A-algebra. Assume Spec(B) → Spec(A) is quasi-finite at Q and Q ∩ A = p. Then there exists a finite A-algebra 0 0 0 A with a maximal ideal m such that BQ = Am0 . Proof. Let A0 be the subalgebra of A defined in the proof of Prp. 2.1. By Thm. 2.9 Frac(A) = 0 Frac(B). Thus there exists a ∈ A such that Ba = Aa = Aa. Moreover, a ∈ IB/A. Set 0 0 0 m := Q ∩ A . By Thm. 2.9 a 6∈ Q. Thus BQ = Am0 . 8 ANTONI RANGACHEV

Remark 2.12. Zariski worked under the assumption that A and B have the same field of fractions. In [Z43, pg. 522–527] he a gave a lengthy proof of his main theorem using an inductive argument on the number of generators for B as an A-algebra and and intricate study of conductor ideals. His approach was generalized by Peskine in [Pes66] for ring maps A → B of finite type (cf. [Stks, Tag 00Q9]). In [Z49] Zariski gave a simple proof of his theorem using valuations but under the additional assumption that A is a local domain of finite type over a field K and A is integrally closed in its field of fractions. In this case A is analytically irreducible. Using a precursor to Cohen’s structure theorem, Zariski shows that Ab and the completion BcQ of BQ with respect to Q are module finite over K[[x1, . . . , xd]] where (x1, . . . , xd) is a system of parameters for A. In particular, BQ is module-finite over Ab. To derive the integrality of BQ over A, Zariski uses in an essential way that A is analytically irreducible: in this case, there is a bijective correspondence of divisorial valuations associated with the field of fractions of Ab and A, respectively, given by restriction (see [SH06, Prps. 9.3.5; 6.5.4]). Thus BQ is contained in each divisorial valuation with respect to A, and so BQ is integral over A. In [EGAIII, Thm. 4.4.3] (see also [EGAIV, Thm. 8.12.6]) Grothendieck proved a global version of Zariski’s main theorem using the formal function theorem. Our use of the associated graded and completions is implicit in Grothendieck’s proof of Zariski’s connectedness theorem. The use of both techniques in this context, however, predates Grothendieck’s treatment. The method of associated graded rings ([ZS60, pg. 248; Cor. 1 pg. 259]) is used to show that the completion of a Notherian ring is Noetherian and it appears often in valuation theory and in the theory of multiplicities. The use of passing to the completion in our setup goes back at least to the proof of the Mori–Nagata theorem [Nag55, Thm. 1].

References [AK18] Altman, A., and Kleiman, S., ”A term in commutative algebra,” Wordwide Center of Mathematics, 2018. [AtM69] Atiyah, M., Macdonald, I., “Introduction to commutative algebra.” Addison-Wesley Publishing Co., 1969. [Bour75] Bourbaki, N.,“ Eléments´ de Mathématique.AlgébreCommutative. Chapitres 5 á7.” Herman, Paris, 1975. [EGAIV] Grothendieck, A., and Dieudonné,J., ”EGA IV. Etude´ globale locale des schémaset des morphismes de schémas,”Troisieme partie, Publications Mathématiquesde l’IHES,´ 28, (1966). [EGAIII] Grothendieck, A., and Dieudonné, J., ”EGA III. Etude´ cohomologique des faisceaux cohérents,” Premiérepartie, Publications Mathématiques de l’IHES,´ 21, (1961). [Nag55] Nagata, M., On the derived normal rings of Noetherian integral domains, Memoirs of the College of Science, University of Kyoto. Series A: Mathematics, 29, (1955), 293–303. [Pes66] Peskine, C., Une généralisation du main theorem de Zariski, Bull. Sci. Math. (2), 90, (1966), 119–127. [Ran20] Rangachev, A., Associated primes and integral closure of Noetherian rings, Journal of Pure and Applied Algebra, vol. 225, Issue 5, (2021): https://arxiv.org/abs/1907.10754. [Ran18] Rangachev, A., Associated points and integral closure of modules, Journal of Algebra, 508 (2018), 301–338. [Ray70] Raynaud, M., “Anneaux locaux henséliens,”Lecture Notes in Mathematics, 169, Springer-Verlag, (1970). [R87] Rees, D., Reduction of modules, Math. Proc. Cambridge Philos. Soc. 101 (1987), 431-449. [R56] Rees, D., Valuations associated with ideals II, J. London Math. Soc. , 31 (1956), 221–228. [Stks] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2016. [SH06] Swanson, I., and Huneke, C., “Integral closure of ideals, rings, and modules.” London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. [ZS60] Zariski, O., Samuel, P., “Commutative algebra. Vol. II.”, The University Series in Higher Mathemat- ics. D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. A VALUATION THEOREM FOR NOETHERIAN RINGS 9

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Department of Mathematics, University of Chicago, Chicago, IL 60637, Institute of Math- ematics, and Informatics, Bulgarian Academy of Sciences, Akad. G. Bonchev 8, Sofia 1113, Bulgaria