aa r X i v : . [ m a t h . A C ] N ov A VALUATION THEOREM FOR NOETHERIAN RINGS
ANTONI RANGACHEV
Abstract.
Let
A ⊂ B be integral domains. Suppose A is Noetherian and B is a finitelygenerated A -algebra. Denote by A the integral closure of A in B . We show that A isdetermined by finitely many unique discrete valuation rings. Our result generalizes Rees’classical valuation theorem for ideals. We also obtain a variant of Zariski’s main theorem. Introduction
Let
A ⊂ B be integral domains. Denote the integral closure of A in B by A . Suppose thereexist valuation rings V , . . . , V r in Frac( A ) such that(1) A = ∩ ri =1 V i ∩ 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 V i violates (1). Themain 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 Ass A ( B / A ) = { (0) } , or A = B , or there exist unique discrete valuation rings V , . . . , V r in Frac( A ) such that A = ∩ ri =1 V i ∩ B is minimal. Furthermore, if A is locally formally equidi-mensional, then each V i 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. 1.1rests upon three key observations. First, we show that A is generically Noetherian. Thenwe use this to prove that Ass A ( B / A ) is finite by results of [Ran20]. We set each V i to bethe localization of A at a prime in Ass A ( 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 whichis the annihilator of an element of B / A are in Ass A ( B / A ). As another application of theseobservations we obtain a variant of Zariski’s main theorem.Let R be a Noetherian domain. Suppose A = ⊕ ∞ i =0 A i ⊂ B = ⊕ ∞ i =0 B i is a homogeneousinclusion of graded Noetherian domains with A = B = R . Suppose B is a finitely generated A -algebra. For each n denote by A n the integral closure of A n in B n . It’s the R -moduleconsisting of all elements in B n that are integral over A . For the discrete valuations V i inThm. 1.1 set V i := V i ∩ Frac( R ). Define A n V i ∩ B n to be the set of elements in B n that mapto A n V i as a submodule of B n V i . The following is a corollary to our main result. Corollary 1.2.
Suppose A V = A V for each valuation V in Frac( R ). Then either A = B , orAss A ( B / A ) = { (0) } , or(2) A n = ∩ ri =1 A n V i ∩ B n Mathematics Subject Classification.
Key words and phrases.
Integral closure of rings, discrete valuation rings, Rees valuations, Chevalley’sconstructability result, Zariski’s main theorem. for each n . Furthermore, if V i = Frac( R ) for i = 1 , . . . , r , then (2) is minimal and the V i s 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 ⊕ I t ⊕ · · · is called the Rees algebra of I . It’s contained in the polynomialring R [ t ] := R ⊕ Rt ⊕ Rt ⊕ · · · . For each n denote by I n the integral closure of I n 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 A V = V [ t ] or A V = V [ at ] where IV = ( a ) for some a ∈ I . Thus A V is integrally closed, andso A V = A V . 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 nonzeroideal in R . There exists unique discrete valuations V , . . . , V r in the field of fractions of R such that I n = ∩ ri =1 I n V i ∩ 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 V i s in R using Chevalley’s con-structability result as follows. Consider the structure map c : Proj( A ) → Spec( R ). For eachinteger l ≥ 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 m i the center of V i in R . Set e := dim Proj( A ⊗ R Frac( R )). Theorem 1.4.
Suppose R is locally formally equidimensional. If ht( m i ) > for some i , then m i is a minimal prime of S (ht( m i ) + e − . Acknowledgements.
I thank Madhav Nori and Bernard Teissier for stimulating andhelpful 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 isa finitely generated A -algebra. Then there exists f ∈ A such that A f is Noetherian.Proof. Denote by E the algebraic closure of Frac( A ) in Frac( B ). By Zariski’s lemma E is afinite field extension of Frac( A ). Because E = Frac( A ), there exist f , . . . , f k ∈ A such that E = Frac( A )( f , . . . , f k ). Set A ′ := A [ f , . . . , f k ]. Then A ′ is Noetherian and Frac( A ′ ) =Frac( A ). By [Ran20, Prp. 2.1] Ass A ′ ( B / A ′ ) is finite. But Ass A ′ ( A / A ′ ) ⊂ Ass A ′ ( B / A ′ ). SoAss A ′ ( A / A ′ ) is finite, too. Select f ∈ A ′ from the intersection of all minimal primes inAss A ′ ( A / A ′ ). Then A ′ f = A f ; hence A f is Noetherian. (cid:3) 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 isa finitely generated A -algebra. Then Ass A ( B / A ) and Ass A ( B / A ) are finite.Proof. By Prp. 2.1 there exists f ∈ A such that A f is Noetherian. Let q ∈ Ass A ( B / A ). If f q , then q ∈ Ass A f ( B f / A f ). The last set is finite by [Ran20, Prp. 2.1]. Suppose f ∈ q . Asbefore, denote by E the algebraic closure of Frac( A ) in Frac( B ). It’s a finite field extensionof Frac( A ). Denote by L the integral closure of A in E . By the Mori–Nagata Theorem L isa Krull domain ([Bour75, Prp. 12, pg. 209] and [SH06, Ex. 4.5]). But L is also the integral VALUATION THEOREM FOR NOETHERIAN RINGS 3 closure of A in its field of fractions. Let q ′ be a prime in L that contracts to q . We have A q ⊂ L q ′ . By Thm. 1.1 (i) A q is a DVR. As A and L have the same field of fractions, A q = L q ′ . Thus ht( q ′ ) = 1. Because L is a Krull domain, there are finitely many height oneprime ideals in L containing f . Thus there are finitely many q ∈ Ass A ( B / A ) containing f .This proves the finiteness of Ass A ( B / A ). Alternatively, apply directly [Ran20, Thm. 1.1 (ii)]for A ′ and B noting that A and A ′ have the same integral closure in B .Let p ∈ Ass A ( B / A ). If f p , then p is a contraction from a prime in Ass A f ( B f / A f ) whichis finite by [Ran20, Prp. 2.1]. If f ∈ p , then the proof of [Ran20, Thm. 1.1 (ii)] shows that p ∈ Ass A ( A /f A ) which is finite because A is Noetherian. The proof is now complete. (cid:3) Proposition 2.3.
Suppose
A ⊂ B are integral domains. Suppose A is Noetherian and B isa 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 Ass A ( B / A ) .Proof. If J = (0), then clearly J ∈ Ass A ( B / A ). Suppose J = (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 integralover A . So by incomparability each minimal prime of ( h ) is a contraction of a prime of heightone in L which has to be a minimal prime of hL . Therefore, ( h ) has finitely many minimalprimes, and so does J .Denote by q , . . . , q l the minimal primes of J . First, we want to show that for each 1 ≤ i ≤ l there exists a positive integer s i such that q s i i ⊂ J A q i . We proceed as in the proof of [Ran20,Thm. 1.1 (i)]. Set p i := q i ∩ A . We can assume that A is local at p i . Let b A be the completionof A with respect to p i . Set A ′ := A ⊗ A b A and B ′ := B ⊗ A b A . Replace b A , A ′ and B ′ bytheir reduced structures. Because b A is a reduced complete local ring and B ′ is a finitelygenerated b A -algebra, then by [Stks, Tag 03GH] A ′ is module-finite over b A . In particular, A ′ is Noetherian. Clearly, J A ′ is primary to q i A ′ . Thus there exists s i such that q s i i A ′ ⊂ J A ′ .Hence q s i i b ∈ A ′ . But q s i i b ∈ B . Thus by [Ran20, Prp. 2.2] q s i i b ∈ A p i , and so q s i i b ∈ A q i . Thisimplies q s i i ⊂ J A q i by [AtM69, Prp. 3.14] applied for ( A , b ) / A .Assume that the s i defined above are the minimal possible. Fix 1 ≤ j ≤ l . For each i = j by prime avoidance we can select c i ∈ q s i i and c i q j . Let c j ∈ q s j − j with c j J A q j . Set c := c · · · c l . Then q j = ( A : A cb ) and thus q j ∈ Ass A ( B / A ). (cid:3) Proof of Theorem 1.1
We can proceed with the proof of Thm. 1.1. Suppose Ass A ( B / A ) = { (0) } and A 6 = B .Then by Prp. 2.2 Ass A ( B / A ) contains finitely many nonzero prime ideals which we denoteby q , . . . , q r . By [Ran20, Thm. 1.1 (i)] V i := A q i is a DVR for each i = 1 , . . . , r . Obviously, A ⊆ ∩ ri =1 V i ∩ B . Let b = x/y ∈ ∩ ri =1 V i ∩ B . Set J := ( y A : A x ). We have J V i = V i for each i .Thus J * q i for each i . But J b ∈ A and J = (0). So by Prp. 2.3 if J is a nonunit ideal, itsminimal primes are among q , . . . , q r 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 V j in (1) forsome j . Then localizing both sides of (1) at q j we obtain that A q j = B q j which contradictswith q j ∈ Supp( B / A ). Thus (1) is minimal. We are left with proving the uniqueness of the ANTONI RANGACHEV V i s. Suppose(3) A = ∩ sj =1 V ′ j ∩ B is a minimal discrete valuation decomposition. Set B ′ := Frac( A ) ∩ B . We have A q i = B ′ q i for each i = 1 , . . . , r . As the intersection in (3) takes place in B ′ we can replace in it B by B ′ . Localizing both sides of (3) at q we obtain that there exists a valuation V ′ l such that( V ′ l ) q = Frac( A ). But V = A q ⊂ ( V ′ l ) q . Thus V = ( V ′ l ) q . Also, V ′ l ⊂ ( V ′ l ) q , and so V ′ l = ( V ′ l ) q . Therefore, V = V ′ l . Continuing this process we obtain that each V i appears in(3). As (3) is minimal, we obtain that s = r and after possibly renumbering we get V i = V ′ i for i = 1 , . . . , r .Let A ′ be the module-finite A -algebra defined in the proof of of Prp. 2.1. Suppose A islocally formally equidimensional. Then so is A ′ . Note that Frac( A ′ ) = Frac( A ). Denote by m V i the maximal ideal of V i . Set p i := m V i ∩ A ′ . By Cohen’s dimension inequality (see [SH06,Thm. B.2.5]) tr . deg κ ( p i ) κ ( m V i ) ≤ ht( p i ) − . Because p i = q i ∩ A ′ , then by [Ran20, Thm. 1.1 (iii)] we get ht( p i ) = 1. Therefore,tr . deg κ ( p i ) κ ( m V i ) = ht( p i ) − V i is a divisorial valuation ring in Frac( A ). (cid:3) Remark 2.4.
As it’s well-known, an integrally closed domain equals the intersection of allvaluation rings in its field of fractions that contain it. From here one derives set-theoreticallythat A = ∩V ∩B where the intersection is taken over all valuation rings in Frac( A ) that containthe integral closure of A in Frac( A ). Because A ′ and A have the same integral closure and A ′ 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 afinitely generated A -algebra, one can take finitely many uniquely determined DVRs each ofwhich is a localization of A at a height one prime ideal. Proof of Cor. 1.2 and Cor. 1.3
Suppose Ass A ( B / A ) = { (0) } and A 6 = B . Because A V = A V and A ⊂ A V we get A ⊂ A V .Thus for each i = 1 , . . . , r A ⊂ A V i ⊂ V i . But A V i = ⊕ ∞ j =0 A j V i . Also, by Thm. 1.1 A = ∩ ri =1 V i ∩ B . Thus, set-theoretically A n = ∩ ri =1 A n V i ∩ B n . Set K := Frac( R ). Suppose V i = K for each i = 1 , . . . , r . Showing that thedecomposition is minimal and unique is done in the same way as in Thm. 1.1. Here we willshow just the uniqueness of the valuations. Suppose there exist DVRs V ′ , . . . , V ′ s in K suchthat(4) A = ∩ sj =1 A V ′ j ∩ B is minimal. As in the proof of Thm. 1.1 we can assume that Frac( B ) = Frac( A ). If V ′ j = K forsome j , then because (4) is minimal we get s = 1 and V ′ = K . So A = A K ∩ B . Localizingat q we obtain that V = ( A K ) q . But K ⊂ ( A K ) q . Thus V = K , a contradiction.Therefore, V ′ j = K for each j . Again, by localizing (4) at q i , we get that there is a j such that V i = ( A V ′ j ) q i . But V i = V i ∩ Frac( R ) and V ′ j ∈ ( A V ′ j ) q i ∩ Frac( R ). Because V i = Frac( R ),then V i = V ′ j . Thus r = s by minimality and after possible renumbering V i = V ′ i for each i = 1 , . . . , r .Consider Cor. 1.3. Apply Cor. 1.2 with A := R [ It ] and B := R [ t ]. In the introduction weproved that R [ It ] V = R [ It ] V for each valuation V in K . What remains to be shown is that VALUATION THEOREM FOR NOETHERIAN RINGS 5 V i = K for each i . Indeed, the prime ideals in R [ It ] are contractions of extensions of primeideals of R to R [ t ]. Thus ht( q i ∩ R ) ≥ V i = K for each i otherwise q i R [ It ] q i is a unitideal which is impossible. (cid:3) 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 m i ∈ S (ht( m i ) + e − V i contractsto q i in A . Set p i := q i ∩ A . Then by [Ran20, Thm. 1.1 (iii)] ht( p i ) = 1. Consider the mapProj( A m i ) → Spec( R m i ). It’s closed, surjective and of finite type. By the dimension formula([Stks, Tag 02JX]) dim Proj( A m i ) = ht( m i ) + e . But A m i is a local formally equidimensionalring. Because ht( p i A m i ) = 1, by [SH06, Lem. B.4.2] dim Proj( A ⊗ k ( m i )) = ht( m i ) + e − m i ∈ S (ht( m i ) + e − n i in R with n i ⊂ m i and n i ∈ S (ht( m i ) + e − A ⊗ k ( n i )) ≥ ht( m i ) + e −
1. Note that n i = (0) for otherwise n i ∈ S ( e ) which forcesht( m i ) = 1, a contradiction. Because dim Proj( A n i ) = ht( n i ) + e then dim Proj( A ⊗ k ( n i )) ≤ ht( n i ) + e −
1. Therefore, ht( m i ) + e − ≤ ht( n i ) + e − . But n i ⊂ m i . Thus n i = m i and m i is a minimal prime in S (ht( m i ) + e − (cid:3) Thm. 1.4 generalizes [Ran18, Thm. 7.8], which is a result for Rees algebras of modules. Tosee that Thm. 1.4 is sharp, let ( R, m ) be a Noetherian regular local ring of dimension at least2, and let h ∈ m be an irreducible element. Let B be the polynomial ring R [ y , . . . , y e +1 ] forsome e ≥
0. Set A := R [ hy , . . . , hy e +1 ]. Thus A is a polynomial subring of B . It is normalbecause R is regular. In the setup of Cor. 1.2 there is only one V = A q where q = h A .Note that S ( k ) = S ( e ) for all k ≥ A is a polynomial ring over R generated by e + 1elements. Thus the only minimal prime in S ( e ) is (0), whereas m V i = ( h ) is a height oneprime ideal in R . A Variant of 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 I B / A the intersection of all elements in Ass A ( B / A ).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 afinitely generated A -algebra. Then V ( I B / A ) = Supp A ( B / A ) . Proof.
If Frac( A ) = Frac( B ), then (0) ∈ Ass A ( B / A ) and trivially V ( I B / A ) = Supp A ( B / A ) =Spec( A ). Suppose Frac( A ) = Frac( B ). Then by [Ran20, Thm. 1.1 (i)] I B / A = q ∩ . . . ∩ q s withht( q i ) = 1 for each i . If q ∈ V ( I B / A ), then q j ⊂ q for some j and thus q j A q ∈ Ass A q ( B q / A q ).Hence V ( I B / A ) ⊂ Supp A ( B / A ). Suppose q ⊂ Supp A ( B / A ). Then there is x/y ∈ B with x, y ∈ A , such that its image in B q is not A q . In other words, if J := (( x ) : A y ), then J ⊂ q .But by Prp. 2.3 the minimal primes of J are among the q i s. Thus there exists q j such that q j ⊂ q , i.e. Supp A ( B / A ) ⊂ V ( I B / A ). (cid:3) In [EGAIII, Cor. 4.4.9] Grothendieck derives the following result as a consequence ofZariski’s main theorem (ZMT): if g : X → Y is a birational, proper morphism of noetherianintegral schemes with Y normal and g − ( y ) finite for each y ∈ Y , then g is an isomorphism. ANTONI RANGACHEV
Below we show that in the affine case we can reach the same conclusion assuming that g issurjective in codimension one. To do this we do not have to appeal to ZMT. In fact, our resultproves ZMT in codimension one or in the special case when A is a UFD as shown below.Denote by g : Spec( B ) → Spec( A ) the induced map on ring spectra. Denote by V ( I g ) theZariski closure of Im( g ) ∩ Supp A ( B / A ). Theorem 2.6.
Let
A ⊂ B be integral domains. Suppose A is Noetherian and B is a finitelygenerated A -algebra. Assume Frac( A ) = Frac( B ) . (i) If g is surjective, then B = A . (ii) If B 6 = A , then ht( I g ) ≥ .Proof. Consider (i). Suppose there exists q ∈ Ass A ( B / A ). Because q = (0), by [Ran20, Thm.1.1 (i)] A q is a DVR. But A q = B q . Thus B q = Frac( B ). This contradicts the assumption thatthere exists a prime in B that contracts to q . Thus Ass A ( B / A ) is empty, and so by Prp. 2.5 B = A . Consider (ii). If q is a minimal prime in Supp A ( B / A ), then ht( q ) = 1 and B ⊗ A κ ( q )is empty as shown above. Thus the minimal primes of I g are of height at least 2. (cid:3) 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( B Q / p B Q ) = 0. We say that Spec( B ) → Spec( A ) is quasi-finite if it isquasi-finite at each prime in Spec( B ).The following two corollaries of Thm. 2.6 are special cases of ZMT. Corollary 2.8.
Let
A ⊂ B be integral domains. Suppose A is Noetherian and B is a finitelygenerated A -algebra. Assume Spec( B ) → Spec( A ) is quasi-finite at Q and ht( q ) = 1. Thenthere exists f ∈ I B / A with f q such that B f = A f . Proof.
Because ht( q ) = 1 then A q is a universally catenary Noetherian ring. Applying thedimension formula for A q and B q we get(5) ht( Q ) + tr . deg κ ( q ) κ ( Q ) = ht( q ) + tr . deg A B Because q B Q is Q B Q -primary we have ht( q ) ≥ ht( Q ). But κ ( Q ) is a finite field extension of κ ( q ). So tr . deg κ ( q ) κ ( Q ) = 0. Thus ht( Q ) = ht( q ) and Frac( A ) = Frac( B ). Applying Thm.2.6 to A q and B q we get A q = B q . Because Frac( A ) = Frac( B ), by Prp. 2.3 for each b ∈ B there exists a positive integer k b such that I k b B / A b ∈ A . Thus for each f ∈ I B / A we have A f = B f . By Prp. 2.5 I B / A q . So we can select f ∈ I B / A with f q . (cid:3) Corollary 2.9.
Let
A ⊂ B be integral domains. Suppose A is Noetherian and B is afinitely generated A -algebra. Assume A is universally catenary and A is a UFD. SupposeSpec( B ) → Spec( A ) is quasi-finite at Q . Then there exists f ∈ I B / A with f q such that B f = A f . Proof.
An application of the dimension formula as in (5) for A , B , Q and p implies thatht( Q ) = ht( p ) and Frac( B ) = Frac( A ). By [SH06, Prp. 4.8.6] ht( q ) = ht( Q ). We proceed byinduction on ht( q ). For the height zero case simply set f to be the common denominator ofthe generators of B as an A -algebra. The case ht( q ) = 1 was handled in Cor. 2.8. Supposeht( q ) = n . Let q be a prime ideal in A of height one. Because A is a UFD, there exists x ∈ A such that q = ( x ). Because A is universally catenary and A is integral over A , thenby [Ratl69, Thm. 3.1] there exists a chain of prime ideals (0) ⊂ q ⊂ . . . ⊂ q n := q . Assume VALUATION THEOREM FOR NOETHERIAN RINGS 7 A is local at q . Let x , x , . . . , x n be a system of parameters for q . Consider ( x , . . . , x n − ) B .Then there exists a prime Q ′ ⊂ Q minimal over ( x , . . . , x n − ) B . Since ht( Q ) = n , then byKrull’s height theorem Q ′ = Q . Set q ′ := Q ′ ∩ A . Then q ′ = q because Q is isolated in itsfiber. Thus ht( q ′ ) = n − Q ′ is isolated in its fiber over Q ′ ∩ A . Also, q ⊂ q . By theinduction hypothesis A q ′ = B q ′ . In particular, q Ass A ( B / A ). Therefore, A q = B q by Prp.2.5. The rest follows as in the proof of Cor. 2.8. (cid:3) Cor. 2.9 and Thm. 2.6 (ii) generalize [Mum99, Prp. 1, pg. 210].
References [AtM69] Atiyah, M., Macdonald, I., “Introduction to commutative algebra.” Addison-Wesley Publishing Co.,1969.[Bour75] Bourbaki, N.,“ ´El´ements de Math´ematique. Alg´ebre Commutative. Chapitres 5 ´a 7.” Herman, Paris,1975.[EGAIV] Grothendieck, A., and Dieudonn´e, J., ”EGA IV. ´Etude globale locale des sch´emas et des morphismesde sch´emas,” Troisieme partie, Publications Math´ematiques de l’IH´ES, 28, (1966).[EGAIII] Grothendieck, A., and Dieudonn´e, J., ”EGA III. ´Etude cohomologique des faisceaux coh´erents,”Premi´ere partie, Publications Math´ematiques de l’IH´ES, 21, (1961).[Mum99] Mumford, D., “The Red Book of Varieties and Schemes.” Lecture Notes in Mathematics, Springer-Verlag Berlin, Heidelberg, 1999.[Ran20] Rangachev, A.,
Associated primes and integral closure of Noetherian rings,
Journal of Pure andApplied Algebra, vol. , Issue 5, (2021): https://arxiv.org/abs/1907.10754 .[Ran18] Rangachev, A.,
Associated points and integral closure of modules , Journal of Algebra, (2018),301–338.[Ratl69] Ratliff, L. J.
On quasi-unmixed local domains, the altitude formula, and the chain condition for primeideals.
I. Amer. J. Math. (1969), 508–528.[R87] Rees, D., Reduction of modules,
Math. Proc. Cambridge Philos. Soc. (1987), 431—449.[R56] Rees, D.,
Valuations associated with ideals II,
J. London Math. Soc. , (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 MathematicalSociety Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006., 2016.[SH06] Swanson, I., and Huneke, C., “Integral closure of ideals, rings, and modules.” London MathematicalSociety Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006.