Picavet Ascending the divided and going-down properties by absolute flatness

This paper aims to show that the “going-down ring” and the “divided ring” properties ascend along flat morphisms whose co-diagonal morphisms are flat, the so-called absolutely flat morphisms introduced by Olivier. But unibranchedness hypotheses are necessary as any henselization morphism shows. As a by-product, we get that the “unibranched divided ring” property is preserved by the formation of factor domains and by localization with respect to prime ideals. Moreover, we exhibit some ascent results for the “going-down ring” and “divided ring” properties along integral extensions. Mathematics Subject Classification 13B24 · 13G05 · 13C15 · 14A05 · 13B40 · 13A15

going-down morphism of rings D → C [7,8,12]. Then an arbitrary ring is called (universally) going-down if each of its factor domains is (universally) going-down. Following Badawi, a prime ideal P of a ring R is called a divided prime ideal if P is comparable under inclusion to each (principal) ideal of R [1]. Then a ring R is called divided if each of its prime ideals is divided. A divided ring is local and irreducible. Hence the reduced ring R red of a divided ring R is an integral domain. This last property is crucial in the sequel because we will consider (geometrically) unibranched rings whose definitions follow.
A local ring R with integral closure R is unibranched if R red is an integral domain and R is a local ring. Clearly, a unibranched local ring is irreducible. Moreover, a unibranched local ring R is geometrically unibranched if the residual extension at the maximal ideals of R and R is purely inseparable (see [36,Chapter IX]). Then a local ring R is unibranched (geometrically unibranched), amounts to saying that its henselization R h (strict henselization R sh ) is irreducible [36,Definitions 1 & 2,p. 100].
We present now our main results. Our motivation comes from some papers by Cinquegrani [4][5][6]. She mainly considers ascent by henselization of the "divided domain" property and, as a corollary, by integral extensions. A recent paper by Badawi and Dobbs gives criteria that characterize a going-down ring R in terms of some divided extension S of R, within the universe of either primary rings or reduced rings [2]. Moreover, we established that the "universally going-down ring" property is ascended by étale morphisms within the context of weak Baer rings [35]. Primary rings and weak Baer rings share a common property; they have a zero-dimensional total quotient ring and their classes contain integral domains. Thus our first task is the generalization of Cinquegrani's results to absolutely flat morphisms within the above contexts. Our second aim consists in giving ascent results for the "going-down ring" property by using the results of Badawi and Dobbs. As a by-product, we get new results about the ascent of the "going-down ring" property by integral extensions. But all our results are established under either a unibranchedness hypothesis or at least, an irreducibility condition. That the consideration of unibranchedness is necessary is clear, because for instance, if R h is divided, then R is unibranched and R → R h is absolutely flat.
In Sect. 1, we develop some material about absolutely flat morphisms. Ascent of the primary property by absolutely flat morphisms is considered.
Section 2 begins with a crucial result that is used in the rest of the paper. If R → S is an injective ring morphism, Spec(S) → Spec(R) is a homeomorphism and S is (locally) primary, then S is (locally) divided if R is. It follows that an absolutely flat ring morphism, whose spectral map is a homeomorphism, ascends the "weak Baer locally divided ring" property.
We establish that the "divided unibranched ring" property is sufficiently strong to be preserved by the formation of factor domains and localization with respect to prime ideals. These two results are essential to proving ascent of the "divided ring" property. Our first ascent result is as follows: If R → S is a local absolutely flat ring morphism and S is unibranched, then R is a primary divided ring if and only if S is a primary divided ring. In this case, Spec(S) → Spec(R) is a homeomorphism. This result applies to (strict) henselizations and is globalized. Moreover, in the above result, we can replace the unibranched hypothesis on S by R is geometrically unibranched. As a matter of fact, this property is preserved by absolute flatness. An immediate consequence is that the "locally divided ring" property ascends along absolutely flat morphisms whose domains are normal.
These results combine to yield that if R is a divided unibranched primary ring, then so is its integral closure. Since a weak Baer ring is locally primary, there naturally exists a version for weak Baer rings. We call a ring morphism a min morphism if minimal prime ideals of the target contract to minimal prime ideals. Then one of our main results is the following. Let R be a divided unibranched ring and R → S a min integral extension where S is unibranched. If S is either primary or Nil(S) is divided, then S is a divided ring.
Then Sect. 2 ends with some transfer results for properties like the PVD property and the "i-domain" property.
Section 3 is devoted to the ascent of the "(universally) going-down ring" property along absolutely flat morphisms. We prove two versions in the "weak Baer ring" context. Let R be a universally going-down weak Baer ring and an absolutely flat ring morphism f : R → S. Then S is a universally going-down weak Baer ring. McAdam proved a result (see [23]) that gives in our context an ascent criterion: if R → S is an absolutely flat ring morphism between weak Baer rings and R is a going-down ring, then S is a going-down ring if and only if S is treed. We deduce from this criterion the next result. Let R → S be an absolutely flat ring morphism between weak Baer rings. If R is going-down and S is locally unibranched (for instance, if R is locally geometrically unibranched), then S is going-down.
Next we look at the primary context and get a stronger result than Badawi and Dobbs obtained because we add a unibranchedness hypothesis [2,Theorem 3.3]. Let R be a local primary ring, such that Nil(R) is a divided prime ideal. Then R is a unibranched going-down ring if and only if its integral closure R is a divided ring. We deduce from this last theorem the following ascent result: Let R be a local unibranched going-down primary ring, such that Nil(R) is a divided prime ideal. If R → S is a local absolutely flat ring morphism, such that S is unibranched, then S is a going-down primary ring, such that Nil(S) is a divided prime ideal.
We now describe notational conventions. For a ring R, we denote, respectively, by Z(R), Tot(R), R red , R , Nil(R), Min(R) and Spec(R) the set of all its zero-divisors, its total quotient ring, its reduced ring, its integral closure in Tot(R), its nilradical, its set of all minimal prime ideals and its set of all prime ideals endowed with the Zariski topology. The residue field R P /P R P Tot(R/P) of R at P ∈ Spec(R) is denoted by k(P). Let f : R → S be a ring morphism with spectral map a f : Spec(S) → Spec(R), then k(P) R S is the fiber ring at P of f and its spectrum is homeomorphic to a f −1 ({P}).
We now introduce some useful material.
To begin with, the reader is referred to [26,27] which state the following results that are used in this paper. Absolute flatness is universal (preserved in any base change), stable under composition and preserved by direct limits. Moreover, f : R → S is absolutely flat if and only if R P → S Q is absolutely flat for each Q ∈ Spec(S) and P := f −1 (Q).
If R is a local ring, a local-étale R-algebra is a local ring morphism R → S P where R → S is étale and P a prime ideal of S. A direct limit of local-étale R-algebras with local transition morphisms is called a local ind-étale algebra. For instance, R h and R sh are local ind-étales over R (see [36,Chapter VIII]).
(Local) ind-étale morphisms, flat epimorphisms are absolutely flat. Absolutely flat morphisms and indétale morphisms are closely related as the next lemma shows. Some results already known are recalled for the reader's convenience.

Lemma 1.1 Let R → S be an absolutely flat ring morphism.
(1) If R is reduced (respectively, absolutely flat), so is S. (2) Let Q ⊆ Q be two prime ideals of S both lying over P in R. The fiber k(P) R S is absolutely flat because so is k(P) and moreover, k(P) → k(P) R S is absolutely flat. Therefore, each prime ideal of the fiber is maximal. As a f −1 (P) is homeomorphic to Spec(k(P) R S), we get Q = Q . Then Nil(S) = Nil(R)S is a consequence of (1) applied to the absolutely flat morphism R/Nil(R) → S/Nil(R)S. To complete the proof, first observe that a ring A is zero-dimensional if and only if A red is absolutely flat and then use (1).
We prove (3). Observe that R/M → S/M S is absolutely flat. An appeal to (1) shows that S/M S is absolutely flat. As this ring is local, it is a field and M S is a maximal ideal. Statement (4) is a consequence of [26, Proposition 3.1 (iii)] because each residual extension of R → S identifies with a residual extension of k(P) → k(P) R S for some P ∈ Spec(R).
(5) Assume that R → S is local where S is henselian and R and S have respective maximal ideals M and N . There is a factorization R → R h → S into local morphisms. In view of (3), S/M S is a field and N = M S. Thus R/M → S/M S is a separable algebraic extension of fields by (4 Badawi and Dobbs observed that divided rings R verifying the condition Z(R) = Nil(R) have the same behavior as divided domains [2]. This condition is clearly equivalent to (0) is a primary ideal of R. For this reason, rings verifying this condition are called primary by Bowman, O'Carroll and Qureshi in [3,24] where it is proved that the tensor product of finitely many fields over a common subfield is locally primary. The reader is warned that this terminology we adopt is different from the usual. A primary ring R is irreducible and if Min(R) = {m}, then Tot(R) = R m , a zero-dimensional ring. Moreover, there is a factorization R → R P → Tot(R) into injective morphisms for each P ∈ Spec(R). An overring of a primary ring is primary [2, Lemma 2.6]. Clearly, injective ring morphisms descend the primary property.
Then a primary ring R is divided if and only if P R P = P for each P ∈ Spec(R). This statement makes sense since we can consider that R P ⊆ Tot(R). Besides, P is divided if and only if R is the pullback P(P) defined by R P → k(P) and R/P → k(P) [2, Proposition 2.5 (c)].
In this paper, we consider another class of rings whose total quotient rings are zero-dimensional. A ring is called a weak Baer ring if the annihilator of each of its elements is generated by an idempotent. Then R is a weak Baer ring amounts to saying that its total quotient ring is absolutely flat and its local rings are integral domains (see for instance, [35,Definition 1.4 ff ]). An overring S of a weak Baer ring R is a weak Baer ring. To see this, let x = a/s and y = b/t be elements of S where s and t are regular elements of R and such that x y = 0. If e is the idempotent of R such that 0 : R a = Re, then b = be so that y ∈ Se. It follows that 0 : S x = Se.
Let R be a ring such that Tot(R) is absolutely flat and let R → S be an absolutely flat morphism. Then Tot(S) is absolutely flat and identifies to Tot(R) R S [33,Lemme 2.5]. This result can be generalized to rings whose total quotient rings are zero-dimensional and thus applies to primary rings and weak Baer rings. We first look at the ascent of the "primary ring" property by absolutely flat morphisms. Recall that an attached prime ideal P of an R-module M is a prime ideal P of R such that for each finitely generated ideal I ⊆ P there is some x ∈ M such that I ⊆ 0 : R x ⊆ P. We denote by Att(R) the set of all attached prime ideals of the R-module R. Then we have Z(R) = ∪ [P | P ∈ Att(R)] while Nil(R) = ∩ [P | P ∈ Att(R)]. For these definitions and further results, see [14,20,32]. Hence, a ring R is primary if and only if Att(R) has only one element m. In this case, m is the unique minimal prime ideal of R. [14, p.407]. It follows that the "primary ring" property localizes (see also [2, Lemma 2.6]).
Proof Since Tot(R) is zero-dimensional, we can use Proposition 1.2 from which it follows that Tot(S) = S m . Now an element Q of Att(S) verifies f −1 (Q) ⊆ m since it can be lifted up to Tot(S). We thus get f −1 (Q) = m. But n is contained in Q and f −1 (n) = m by flatness of f . We deduce from Lemma 1.1(2) that Q = n. It follows then that Z(S) = n = Nil(S) and S m = S n . Since Nil(S) = Nil(R)S by Lemma 1.1(2), we get that n = mS.

Then (2) is a consequence of Proposition 1.2.
Assume that m is divided. We can write nS n = nS m = mR m S = mS = n. Remark 1.4 As in Proposition 1.3, we will have to consider absolutely flat morphisms f : R → S where S is irreducible. It would have been pleasant to have a transfer result for irreducibility, at least when R is unibranched and f is local. But this is definitely wrong. It is enough to consider a unibranched ring R which is not geometrically unibranched because R → R sh is absolutely flat and R sh is not irreducible. Part (7) of these remarks provides a stronger argument. Here are some positive/negative results.
(1) When R is unibranched, R h is irreducible. Moreover, Lemma 1.1 (7) shows that S is irreducible when R is geometrically unibranched. Assume that f : R → S is a flat epimorphism and that R is irreducible with minimal prime ideal m. As f satisfies going-down, each minimal prime of S contracts to m. Since Spec(S) → Spec(R) is injective by [22, Proposition 1.9, p. 109], S is irreducible. (2) Suppose that R is irreducible and R → S is absolutely flat, then Nil(S) = mS where m is the minimal prime ideal of R. Then S is irreducible if and only if mS is a prime ideal. As R → S is going-down, the preceding condition amounts to the irreducibility of S m . To see this, use Nil(S) = mS and the fact that f −1 (n) = m for n ∈ Min(S). (3) Now consider an absolutely flat ring morphism R → S where R is an integral domain and S is irreducible; then S is an integral domain because S is reduced. (4) Let R be a normal ring that is, each localization at a prime ideal is an integrally closed domain. If R → S is absolutely flat, then S is normal whence locally irreducible (apply [27,Corollary,p. 57] to each local morphism R P → S Q ). (5) Let f : R → S be an absolutely flat morphism such that R is a weak Baer ring and S is locally irreducible; then S is a weak Baer ring. It is enough to remark that Tot(S) is absolutely flat and that S Q is an integral domain for each Q ∈ Spec(S). This last fact follows from the absolute flatness of R P → S Q where P := f −1 (Q) and (3). (6) Let R be a local ring. Then R is (geometrically) unibranched if and only if the integral domain R red is (geometrically ) unibranched. Denote by S either R h or R sh and consider the absolutely flat morphism R → S. We can read in Lemma 1. [17,Section 18]. (7) It does not hold in general that irreducibility ascends along absolutely flat morphisms. Assume the contrary and consider an absolutely flat integral extension R ⊂ S where R is an integral domain (for instance, an étale covering of R). In view of (3), S is an integral domain. Because S → S R S is an absolutely flat integral extension, S R S is also an integral domain. Since this integral extension is essential by [ It is easy to show that if R → S is submersive with an injective spectral map, then Spec(S) → Spec(R) is actually a homeomorphism. This will be used in next sections for either injective integral morphisms or going-down morphisms whose spectral maps are surjective (for instance, faithfully flat morphisms). Indeed, such morphisms are submersive (see for instance [31]).
The following remark will also be useful. Let f : R → S be a ring morphism. If a f : Spec(S) → Spec(R) is a homeomorphism, then Q = √ P S for each Q ∈ Spec(S) and P := f −1 (Q). Indeed, the relations a f −1 (V(P)) = V(P S) and a f (V(Q)) = V(P) hold because a f is a continuous closed map (see [18, I, Section 1.2]).

Ascent of the divided property
Our goal in this section is the extension of some Cinquegrani's results to rings which may not be integral domains and absolutely flat morphisms. Moreover, we will exhibit new results. In this paper a ring morphism R → S is called an i-morphism (respectively, unibranched, h-morphism) if the map Spec(S) → Spec(R) is injective (respectively, bijective, a homeomorphism). Recall that an h-morphism R → S induces an order isomorphism Spec(S) → Spec(R). Also, an overring B of a ring A is called a unibranched overring if A → B is unibranched (this does not mean that the overring is a unibranched ring!) We give here a useful result which is outside of the absolutely flat context.

Proposition 2.1 Let f : R → S be an injective h-morphism of rings (for instance, if R → S is unibranched and either has the going-down property or is integral).
(1) S Q = S P for each Q ∈ Spec(S) and P := Q ∩ R so that R P → S Q is injective.
(2) If S is (locally) primary and R is (locally) divided, then S is (locally) divided.
(3) If S is a weak Baer ring and R is locally divided, then S is locally divided.
Proof We prove (1). As Spec(S) → Spec(R) is a homeomorphism, we get that Q = √ P S by Remark 1.5. Consider the multiplicative subset f (R \ P) and its saturated associated multiplicative subset T . Then S \ T is the union of all prime ideals q of S such that Assume that R is divided and S is primary. Let Q ∈ Spec(S) lying over P ∈ Spec(R). It follows that P R P = P because R is primary. Let x ∈ QS Q . From S Q = S P and Q = √ P S, we deduce that there is some integer n such that x n ∈ P R P S = P S so that x ∈ Q. Therefore, QS Q = Q for each Q implies that S is divided. Now the local statement follows easily because R P → S Q is injective. To get (3), we can reduce to the local case of (2) because S Q is an integral domain for Q ∈ Spec(S).

Corollary 2.2 Let f : R → S be an h-morphism. If R is a weak Baer locally divided ring and f is absolutely flat, then S is a weak Baer locally divided ring.
Proof From Remark 1.4(5), we deduce that S is a weak Baer ring because the homeomorphism Spec(S) → Spec(R) forces S to be locally irreducible. Then we can use Proposition 2.1(2), because each local morphism R P → S Q , associated with Q ∈ Spec(S) and P := f −1 (Q), is faithfully flat whence injective and S Q is an integral domain.
We next outline a generalization to arbitrary rings of [6, Proposizione 2.1] because it exhibits an interesting morphism and gives Proposition 2.3 when S = R h . Actually this result is crucial to proving [5, Proposizione 1.1] in the integral domain case used in Proposition 2.4. Moreover, this result has its own interest.

Proposition 2.3
If R is a divided (geometrically) unibranched ring, then so is each factor domain of R. Moreover, for each P ∈ Spec(R), there is a factorization of the natural map R/P → (R/P) → R /P R into injective integral morphisms.
Proof We mimic with suitable changes [6, Proposizione 2.1], using Badawi's definition of divided rings. Let R be a unibranched divided ring. The integral closure R of R is a local ring and so is R /P R for P ∈ Spec(R). Then we can exhibit an injective ring morphism ϕ : (R/P) → R /P R by setting ϕ(r /s) = r/s, wherer /s is an element of the quotient field k(P) of R/P and r/s is the class of r/s in R /P R . To build the ring morphism ϕ, copy the proof of [6, Proposizione 2.1], taking into account that if an element s ∈ R does not belong to P then P ⊆ Rs n for each integer n. Then ϕ is injective because R → R is injective integral from which it follows that P R ∩ R = P since R → R has lying-over. Next observe that R/P → (R/P) → R /P R (R/P) R R is the natural map and is therefore integral and injective. It follows that (R/P) → R /P R is integral and hence (R/P) is a local integral domain. Thus R/P is divided and unibranched. Assume that R is geometrically unibranched and denote by M and M the respective maximal ideals of R and R . Then M /P R is above the unique maximal ideal M of (R/P) lying over M/P. As R/M → R /M is purely inseparable, so is (R/P)/(M/P) → (R/P) /M, showing that R/P is geometrically unibranched.

Proposition 2.4
Let f : R → S be a local absolutely flat ring morphism where R is a divided ring and S is a henselian irreducible ring. Then R/P is unibranched for each P ∈ Spec(R) and the following statements hold: (1) P S is a prime ideal of S for each P ∈ Spec(R).
(2) R → S is an h-morphism whose set-theoretic inverse is given by P → P S for P ∈ Spec(R). In particular, Spec(S) is linearly ordered. Proof There is a factorization R → R h → S where the second morphism is faithfully flat by Lemma 1.1 (5). It follows that R h is irreducible and R is therefore unibranched. We can assume that R → S is local ind-étale by Lemma 1.1 (5). Let P ∈ Spec(R), then R/P is unibranched by Proposition 2.3. Let m be the minimal prime ideal of R. In view of Lemma 1.1(2), mS is the nilradical of S so that mS = n, the minimal prime ideal of S. Then A := R/m is a divided domain and B := S/mS is local ind-étale over A and an integral domain. As S is henselian, so is B because n is the nilradical of S [36, Remarque p. 5]. We are now in position to apply a result of Cinquegrani: S/P S is an integral domain for each P ∈ Spec(R) [5, Proposizione 1.1]. Then (1) follows easily.
Next suppose that Q, Q ∈ Spec(S) both contract to P in R; then by incomparability we get that Q = P S = Q . Hence Spec(S) → Spec(R) is injective and surjective, because R → S is faithfully flat. Now R → S has the going-down property. It follows from Remark 1.5 that Spec(R) → Spec(S) is a homeomorphism because R → S is a submersive morphism.
Next We are now ready for our first transfer result.

Theorem 2.6 Let f : R → S be an absolutely flat local ring morphism where S is unibranched. Then R is a primary divided ring if and only if S is a primary divided ring. In this case, R → S is an h-morphism.
Proof A faithfully flat ring morphism f : A → B descends the "divided ring" property because f −1 (I B) = I for each ideal I of A. Moreover, absolute flatness is stable under composition. Thus to establish the proof, we can as well assume that S is an irreducible henselian ring by considering R → S → S h . The "primary ring" property is also descended by faithful flatness (actually, by injectivity). First assume that R is divided and primary. Then S is primary by Lemma 2.5. In order to check that S is divided, it is enough to show that S is the pullback P(Q) for each Q ∈ Spec(S). Now set P := f −1 (Q); we know that S Q = S P by Proposition 2.4(3) while S/Q = S/P S by Proposition 2.4 (2). Then tensor the pullback P(P) over R by S. As R → S is flat, we get a new pullback defined by the ring morphisms S P = S Q → k(P) R S and S/P S = S/Q → k(P) R S. Since R → S is absolutely flat, so is k(P) → k(P) R S. It follows that k(P) R S is an absolutely flat ring by [27,Theorem 2.2]. But k(P) R S is the fiber ring at P and is therefore a field because Spec(S) → Spec(R) is injective. There is a factorization S → k(P) R S → k(Q) by definition of a pushout. The composite being an epimorphism of rings so is the last morphism. As its range is a field, k(P) R S → k(Q) is actually bijective [22, Corollaire 1.3 p.109]. We have thus proved that S is the pullback P(Q) so that Q is divided.
The converse is straightforward because f : R → S is faithfully flat.
Remark We may also use Proposition 2.1 to prove the previous theorem once we know that Spec(S) → Spec(R) is a homeomorphism and S is primary.

Corollary 2.7 Let R → S be an absolutely flat ring morphism. If R is locally primary (for instance, if R is a weak Baer ring), S is locally unibranched and R is locally divided, then S is locally divided.
Proof Let Q be a prime ideal of S lying over P ∈ Spec(R). Then R P → S Q is absolutely flat and local.

Theorem 2.9 Let R → S be a local absolutely flat morphism. Then R is divided, geometrically unibranched and primary if and only if S is divided, geometrically unibranched and primary. In this case R → S is an h-morphism.
Proof First assume that the conditions hold for R and let R → T be a strict henselization of R. There is a factorization R → S → T where the last morphism is local and faithfully flat by Lemma 1.1 (6). Then T and S are geometrically unibranched by [26,Corollaire 2.8]. Moreover, T is divided and primary thanks to Theorem 2.6. These two properties are descended by local faithfully flat morphisms. This has been shown at the beginning of the proof of Theorem 2.6. Hence, the conditions hold for S. Now R → T is unibranched by Theorem 2.6. As S → T has lying-over, R → S is unibranched. Conversely, assume that the conditions hold for S; then R is primary and divided by faithful flatness of R → S. It follows from the proof of a theorem by Paxia [ A ring is called normal if each of its localizations at a prime ideal is an integrally closed domain. Such a ring is evidently locally geometrically unibranched.

Proposition 2.10 Let R → S be an absolutely flat morphism where R is a normal ring. If R is locally divided, so is S.
Proof We can assume that R → S is local. In this case, R is geometrically unibranched and primary and then Theorem 2.9 is available.
Our previous results enable us to get ascent results for integral morphisms. As a first result, we show that seminormalization ascends the locally divided property, at least in the case where the base ring R is a weak Baer ring (see Sect. 1). As the total quotient ring of a weak Baer ring R is absolutely flat, its seminormalization R s is the seminormalization of R in its total quotient ring (see [37]). Let P be a prime ideal of R; then Proposition 1.2 ensures that Tot(R P ) = Tot(R) P so that (R P ) s = (R s ) P . Moreover, R s is a weak Baer ring because an overring of R.

Proposition 2.11 Let R be a weak Baer ring. If R is locally divided, so is R s .
Proof It is well-known that Spec(R s ) → Spec(R) is a homeomorphism. Then we can use Proposition 2.1(3) since R s is a weak Baer ring.
We now consider ascent of the divided property through integral extensions. We need a property of henselian pullbacks.

Proposition 2.12
Let R be a divided ring and P ∈ Spec(R). The following statements are equivalent: (1) R is henselian.
(2) R P and R/P are henselian.
Proof We may mimic the proof of Cinquegrani [4,Teorema 2.6]. We can also use the fact that a local ring is henselian if and only if its reduced ring is henselian [36,Remarque,p. 5]. As R is divided, this ring has only one minimal prime ideal n and the reduced rings of R, R P , R/P are, respectively, R/n, (R/n) P/n and (R/n)/(P/n). Moreover, R/n is a divided domain. Therefore, we can assume that R is an integral domain and then apply the above-quoted result.

Lemma 2.13 Let R be a unibranched divided primary ring and let P ∈ Spec(R). Then (R P ) h is isomorphic to (R h ) P . It follows that R P is a divided unibranched primary ring.
Proof The first remark is that R h is a divided ring and that R → R h is an h-morphism by Corollary 2.8. In view of Proposition 2.4, (R h ) P = (R h ) Q where Q is the unique prime ideal of R h such that P := R ∩ Q. Therefore, (R h ) P is henselian by Proposition 2.12. To complete the proof, it is enough to use the universal properties of henselization and localization (details can be seen in the proof of [5,Lemma 2.2]).
In view of Corollary 2.8, R h is a divided primary ring. Hence, R h is irreducible and so is (R h ) P for each P ∈ Spec(R). From the first part of the proof, we infer that (R P ) h has a unique minimal prime ideal. It follows that R P is unibranched.

Proposition 2.14 Let R be a unibranched divided ring. Then the rings R P are unibranched and divided for each P ∈ Spec(R). If in addition Tot(R) is zero-dimensional, then R → R is an h-morphism.
Proof Let P be a prime ideal of R, then Nil(R P ) is the unique minimal prime ideal of R P and contracts to the unique minimal prime ideal m of R. We have therefore Nil(R P ) = m P . Now observe that R/m is a divided unibranched integral domain by Proposition 2.3 if R is divided and unibranched. Then Lemma 2.13 asserts that T := (R/m) (P/m) is divided and unibranched. But T is isomorphic to R P /m P = (R P ) red . Then observe that for a local ring A, we have that (A h ) red = (A red ) h so that A is unibranched when A red is. It follows that R P is unibranched and evidently, divided.
Assume that Tot(R) is zero-dimensional. Then (R ) P = (R P ) is a consequence of Proposition 1.2. Because R P is unibranched, R P is a local ring. It follows easily that R → R is unibranched whence an h-morphism.

Theorem 2.15 Let R be a unibranched divided primary ring. Then its integral closure R is a divided ring and R → R is an h-morphism.
Proof Since R is primary, so is its overring R . We deduce from Proposition 2.1 that R is divided because R → R is an h-morphism by Proposition 2.14.

Theorem 2.16 Let R be a locally unibranched and divided weak Baer ring. Then its integral closure R is a locally divided ring.
Proof We know that R is a weak Baer ring. Let Q ∈ Spec(R ) and P := Q ∩ R. Since R P is primary, we get that (R P ) is a divided ring and that R P → (R P ) is an h-morphism by Theorem 2.15. As Tot(R) is zero-dimensional, we have (R P ) = (R ) P . These facts combine to yield that (R ) P is local and in fact is equal to R Q . Therefore, R is locally divided.
We studied 1-split rings in [35]. A ring morphism f is called essential if for any ring morphism g such that g • f exists and is injective, then g is injective. Then a ring R is called 1-split if each integral essential extension of R is unibranched.
Recall that a ring morphism f : R → S is called a min morphism if a f (Min(S)) ⊆ Min(R) [13]. If R → S is an essential injective morphism where Tot(R) is absolutely flat, then R → S is a min morphism. It is enough to reconsider the proof of [35, Lemma 1.6].

Lemma 2.17 Let R be a divided henselian ring with minimal prime ideal m.
(1) R/m is a 1-split ring.
(2) Let R → S be a min integral extension where S is irreducible, then R → S is an h-morphism.
Proof Proposition 2.12 states that the divided domain R/m is locally henselian. To conclude for (1), apply [35,Theorem 4.4]. Now let R → S be a min integral extension such that Min(S) = {n}, the unique minimal prime ideal of S. Then n contracts to m in R and R/m → S/n is an integral extension between two integral domains. Such an extension is essential by [35,Proposition 1.1]. Since R/m is 1-split [35,Theorem 4.4], R/m → S/n is unibranched so that Spec(S) → Spec(R) is a homeomorphism.

Theorem 2.18 Let R be a divided unibranched ring and R → S a min integral extension where S is unibranched. If S is either primary or Nil(S) is divided, then R → S is an h-morphism and S is a divided ring.
Proof First assume that S is primary so that R is also primary. Consider a min integral extension R → S where S is unibranched. Set A := R h and B := A R S. The natural map A → B is an integral extension because R → A is flat. This extension is also a min morphism. Indeed, a minimal prime ideal n of B contracts to m, the unique minimal prime ideal of R because S → B is flat and R → S is a min morphism. As R → A has incomparability, n contracts to an element of Min(A). Then observe that B is the henselization of S [17, Proposition 18.6.8]. It follows that B has a unique minimal prime ideal. Moreover, A is divided by Corollary 2.8. Therefore, R → S is unibranched thanks to Lemma 2.17(2) because injectivity of spectral maps is descended by surjective spectral maps. To conclude, use Proposition 2.1 since S is primary. Now in the general case, R red and S red are integral domains since R and S are unibranched. Moreover, Nil(R) and Nil(S) are the unique minimal prime ideals of R and S so that Nil(S) contracts to Nil(R). To conclude, it is enough to use the primary case since R/Nil(R) is a divided unibranched integral domain by Proposition 1.3 and S/Nil(S) is unibranched. Now we intend to establish transfer results for some classes of (locally) divided domains. They concern (global) pseudo-valuation integral domains. We will get other transfer results which are an introduction for the next section about going-down rings.
Next results are consequences of two ascent properties. A ring is called semi-hereditary if each of its finitely generated ideals is projective. Then the "semi-hereditary ring" property is ascended by absolutely flat morphisms [27, Definition, p. 54 ff ]. It follows that the transfer of the "valuation domain" property holds for local absolutely flat morphisms [27, Consequence, p. 55; 29, Teorema 1].

Proposition 2.19 Let R → S be an absolutely flat ring morphism between integral domains. If R is a Prüfer domain, so is S.
Proof For an integral domain R, the following properties are identical: R is semi-hereditary, R is a Prüfer domain [21,Corollary 3]. To conclude, it is enough to use the ascent of the "semi-hereditary ring" property.
If we consider an absolutely flat morphism R → S such that R is a Prüfer domain and S is connected, then S is an integral domain because R is an integrally closed domain [27,Corollary,p. 57]. Hence, S is a Prüfer domain by the previous proposition.
Global pseudo-valuation domains (GPVD) were defined by Dobbs and Fontana [11]. An integral domain R is called a GPVD if R is a subring of a Prüfer domain T such that R → T is unibranched and there is a common nonzero radical ideal I of R and T, such that R/I and T /I are zero-dimensional. Hypotheses on I are equivalent to I is a common ideal of R and T and R/I and T /I are absolutely flat.
Recall that a ring morphism R → S is radiciel if Spec(S) → Spec(R) is injective and its residual extensions are purely inseparable, or equivalently, R → S is universally an i-morphism [18, I, Section 3.7]. Hence, in view of Lemma 1.1(4), the residual extensions of an absolutely flat radiciel ring morphism are isomorphic. In the two following results we have not been able to show that the radiciel hypothesis is superfluous, but we suspect that these results are not valid without a kindred hypothesis. Indeed, since the beginning, spectral injectivity under various forms is a necessary condition.

Theorem 2.20
Let R → S be an absolutely flat radiciel ring morphism (for instance, a flat epimorphism) between integral domains. If R is a GPVD, so is S. Proof Using the above notation for R, we set U := T R S. Then U is a Prüfer domain and S → U is injective by flatness of R → S. We also set J := I S. Then J is nonzero because a flat morphism whose domain is an integral domain is injective [29,Proposizione 3,p. 109]. If K is the quotient field of T , then L := U T K is the quotient field of U by absolute flatness of T → U (see Proposition 1.2). Then U can be identified with its image W := T S in L. It follows that J W = I ST = I T S = J . Hence J is a common ideal of S and U . As R/I → S/I S and T /I → U/I S are absolutely flat, S/J and W/J are absolutely flat rings. Now let Q 1 and Q 2 be two prime ideals of U both contracting to the same prime ideal in S. As Spec(T ) → Spec(R) is injective and Spec(U ) → Spec(T ) is injective, we get Q 1 = Q 2 and S → U is an i-morphism. Finally, because spectral surjectivity is universal, we get that Spec(U ) → Spec(S) is bijective. This completes the proof.
We use the following definition of a pseudo-valuation domain (PVD): a local domain (R, M) is a PVD if there is a (unique) valuation overring V of R whose maximal ideal is M [19,Theorem 2.7]. This implies that Spec(R) = Spec(V ). Next corollary can be considered as a companion result of [19,Theorem 1.7]: if R is a PVD and T is an overring such that R → T satisfies incomparability, then T is also a PVD. Indeed, absolutely flat morphisms have the incomparability property.

Corollary 2.21 Let R → S be an absolutely flat radiciel local ring morphism between integral domains. If R is a PVD, then S is a PVD.
Proof As observed in [11], a PVD is nothing but a local GPVD.

Remark 2.22
It follows easily that the locally pseudo-valuation domain (LPVD) property is transferred by absolutely flat radiciel morphisms between integral domains since for such a ring morphism R → S, we have that R P → S Q is absolutely flat and radiciel for each Q ∈ Spec(S) lying over P ∈ Spec(R).
Recall that an integral domain R is called quasi-Prüfer if for each P ∈ Spec(R) and each Q ∈ Spec(R[X ]) such that Q ⊆ P[X ], then Q = (Q ∩ R) [X ]. An equivalent condition is that the integral closure of R is a Prüfer domain [16,Corollary 6.5.14]. As integral closure is preserved by absolutely flat morphisms between integral domains (see Proposition 1.3), the "quasi-Prüfer domain" property is ascended by absolutely flat ring morphisms between integral domains.
In view of Corollary 2.21, it may be asked whether a geometrically unibranched divided domain is a PVD. The answer is negative since there exists an integrally closed divided domain which is not a PVD [9, Remark 4.10 (b)] An i-domain is an integral domain R such that R → T is an i-morphism for each of its overrings T [28]. Then R is an i-domain amounts to saying that its integral closure R is Prüfer and R → R is an i-morphism [28,Proposition 2.14].

Proposition 2.23 Let R → S be a local absolutely flat morphism between integral domains. If R is an i-domain and S is unibranched, then S is an i-domain.
Proof Since R is quasi-Prüfer, so is S. As S is unibranched, its integral closure S = S R R is local whence a valuation domain. Since Spec(S ) is linearly ordered and R → S is incomparable, Spec(S ) → Spec(S) is injective because so is Spec(R ) → Spec(R).

Ascent of the going-down property
Ascent of the "going-down ring" property by absolutely flat morphisms has only been considered in our paper [35] where we proved that if R → S is an étale morphism and R is a universally going-down weak Baer ring, then S is a universally going-down weak Baer ring [35,Theorem 3.3]. Moreover, the local rings of a universally going-down weak Baer ring are geometrically unibranched [35,Theorem 3.4]. In this section we extend the above result to absolutely flat morphisms. Besides, it was natural to look at the ascent of the "going- In a joint paper with Dobbs, we studied weak Baer (universally) going-down rings [13]. If R is a weak Baer ring, then R is (universally) going-down amounts to saying that each overring S (in Tot(R)) defines a (universally) going-down ring morphism R → S. In the primary rings context this statement holds also for the going-down property [10, Corollary 2.6]. Hence, the "going-down ring" property admits in both contexts the same overrings characterization as in the integral domain context.
Our main tools are the two following results of Badawi and Dobbs: Note that the parenthetical statement (unibranched) in (1) is superfluous because the spectrum of a divided ring S is linearly ordered. Hence, an integral extension R → S is unibranched by incomparability.
We begin our study within the weak Baer rings context. ) s is absolutely flat. This morphism is also local because R P → (R P ) s and S Q → (S Q ) s are h-morphisms. Since R P is a treed going-down domain, (R P ) s is a divided integral domain by Proposition 3.1 (2). Observe that (R P ) s is geometrically unibranched because so is R P . It follows that (S Q ) s is geometrically unibranched and divided by Theorem 2.9 whence locally divided. Moreover, (R P ) s → (S Q ) s is an h-morphism. Using again Proposition 3.1(2), we find that S Q is going-down and R P → S Q is an h-morphism. It follows that S Q is irreducible, whence an integral domain by Remark 1.4(3).  [13,Remark 3.14 (b)]. Now consider Q ∈ Spec(S) and P := f −1 (Q). We get an absolutely flat local ring morphism R P → S Q and a factorization R P → S Q → (R P ) sh where the second morphism is faithfully flat by Lemma 1.1. Hence, S Q is a universally going-down integral domain for each Q ∈ Spec(S). Therefore, S is universally going-down by [13, Proposition 3.2].
Next we give a version of McAdam's result quoted at the beginning of the section.

Proposition 3.5 Let R → S be an absolutely flat ring morphism between weak Baer rings. If R is a going-down ring, then S is a going-down ring if and only if S is treed.
Proof We can reduce to the case of a local morphism between integral domains. Then it is enough to apply [23, Lemma A] to the incomparable injective morphism R → S.
Fontana proved that for an integral domain R, the statement "Spec(R) is linearly ordered" is logically equivalent to "there exists a unibranched (overring) extension R ⊆ S where S is a divided integral domain" [15,Théorème 2.14]. This allows us to give a slightly different version of Theorem 3.3.

Theorem 3.6 Let R → S be an absolutely flat ring morphism between weak Baer rings. If R is going-down and S is locally unibranched (for instance, if R is locally geometrically unibranched), then S is going-down.
Proof We can assume that R → S is an absolutely flat local extension between integral domains and that S is unibranched. We first show that S has a unibranched extension S → U where U is divided. There is a unibranched integral overring R → T where T is divided by Proposition 3.1(1). Then U := S R T is an integral overring of S by Proposition 1.3 and T → U is absolutely flat. Since S → U is integral, we have S = U so that U is unibranched. It follows that T → U is local. We infer from Theorem 2.6 that U is a divided domain and T → U is unibranched. Hence, R → S → U is unibranched. As S → U satisfies lying-over, S → U is unibranched. By the above-quoted Fontana's result, Spec(S) is linearly ordered. In short, S is treed and the proof is complete by Proposition 3.5.
The preceding results are deduced from the going-down criterion of Proposition 3.1(2) working in the universe of reduced rings. Our goal is now to use the criterion of Proposition 3.1(1) on primary rings. Theorem 3.7 Let R be a local going-down unibranched primary ring, such that Nil(R) is divided. Then R h is a local going-down primary ring, such that Nil(R h ) is divided.
Proof There exists a divided integral overring S of R by Proposition 3.1(1). As R and S have the same integral closure, S is unibranched. Moreover, S is primary by [2,Lemma 2.6]. From [17,Proposition 18.6.8], we deduce that S h = R h R S because R ⊆ S is an integral extension and S is local. Hence R h → S h is an integral extension. By Corollary 2.7, S h is divided and primary and R h → S h is unibranched. Now S h = S R R h is an overring of R h by Proposition 1.3(2) because S is an overring of R. Moreover, Z(R h ) = Nil(R h ) is a divided prime ideal by Proposition 1.3(3) since R h is irreducible. Another application of Proposition 3.1 (1) shows that R h is a going-down local ring. Theorem 3.8 Let R be a local primary ring, such that Nil(R) is a divided prime ideal. Then R is a unibranched going-down ring if and only if its integral closure R is a divided ring.
Proof Assume that R is a unibranched going-down primary ring. We deduce from Proposition 3.1(1) that R has a divided integral overring S such that R → S is unibranched. As R = S , we find that S is unibranched. Besides, S is divided and primary. Thus R = S is divided by Theorem 2.15. The converse is clear in view of Proposition 3.1(1). Theorem 3.9 Let R be a local unibranched going-down primary ring, such that Nil(R) is a divided prime ideal. If R → S is a local absolutely flat ring morphism such that S is unibranched, then S is a going-down primary ring, such that Nil(S) is a divided prime ideal.
Proof Thanks to Proposition 1.3 (1) and (3), S is primary and Nil(S) is a divided prime ideal since S is irreducible. In view of Proposition 1.3(2), the integral closure of S is S = R R S so that R → S is absolutely flat. Moreover, S is local and hence, R → S is local. Then Theorem 2.9 applies since R is geometrically unibranched, primary as an overring of a primary ring and is divided by Theorem 3.8. Hence S is divided and the conclusion follows from Theorem 3.8.

Corollary 3.10 Let R be a local geometrically unibranched going-down primary ring, such that Nil(R) is a divided prime ideal. If R → S is a local absolutely flat ring morphism, then S is a geometrically unibranched going-down primary ring, such that Nil(S) is a divided prime ideal.
Proof Use Theorem 3.9 and [26, Corollaire 2.8] which states that S is geometrically unibranched.
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