FCP Delta extensions of rings
aa r X i v : . [ m a t h . A C ] A p r FCP ∆ -EXTENSIONS OF RINGS GABRIEL PICAVET AND MARTINE PICAVET-L’HERMITTE
Abstract.
We consider ring extensions, whose set of all subex-tensions is stable under the formation of sums, the so-called ∆-extensions. An integrally closed extension has the ∆-property ifand only it is a Pr¨ufer extension. We then give characterizationsof FCP ∆-extensions, using the fact that for FCP extensions, itis enough to consider integral FCP extensions. We are able togive substantial results. In particular, our work can be appliedto extensions of number field orders because they have the FCPproperty. Introduction and Notation
In this paper, we work inside the category of commutative and unitalrings, whose epimorphisms will be involved. If R ⊆ S is a (ring)extension, [ R, S ] denote the set of all R -subalgebras of S . An extension R ⊆ S is said to have FCP (or is called an FCP extension) if the poset([ R, S ) , ⊆ ) is both Artinian and Noetherian, which is equivalent to eachchain in [ R, S ] is finite.The so-called ∆-extensions have been the subject of many papers.The seminal paper on the subject was authored by Gilmer and Huckaba[13]. A ring extension R ⊂ S is called a ∆ -extension if T + U ∈ [ R, S ] for each
T, U ∈ [ R, S ] ( i.e. T + U = T U ) [13, Definition, page414]. Although the notion of ∆-extensions originates in CommutativeAlgebra, the lattice properties of [
R, S ] associated to an extension R ⊆ S bring a new point of view to their theory. We will explain what weare aiming to show about them in some contexts that have not beenconsidered yet.We consider lattices of the following form. For an extension R ⊆ S ,the poset ([ R, S ] , ⊆ ) is a complete lattice, where the supremum of anynon-void subset is the compositum of its elements, which we call product Mathematics Subject Classification.
Primary:13B02, 13B21, 13B22, 06E05,06C05; Secondary: 13B30.
Key words and phrases.
FIP, FCP extension, ∆-extension, minimal extension,integral extension, support of a module, lattice, modular lattice, Boolean lattice,pointwise minimal extension. from now on and denote by Π when necessary, and the infimum of anynon-void subset is the intersection of its elements.As a general rule, an extension R ⊆ S is said to have some propertyof lattices if [ R, S ] has this property. We use lattice definitions andproperties described in [16].Any undefined material is explained at the end of the section or inthe next sections.A representative example of the use of lattices is given by the follow-ing result. A catenarian ( i.e. verifying the Jordan-H¨older condition)integral FCP extension R ⊂ S , with t -closure T , is a ∆-extension ifand only if R ⊆ T and T ⊆ S are ∆-extensions. Note also that aninfra-integral (integral, with isomorphic residual field extensions) FCPextension has the ∆-property if and only if it is modular.In case we are dealing with an integrally closed extension, a char-acterization is immediately given as follows. Such extensions are ∆-extensions if and only if they are Pr¨ufer extensions (defined by Knebushand Zhang [14]) or equivalently they are normal pairs. This result isoften reproved by authors working in some particular contexts.We mainly consider FCP ∆-extensions. The FCP condition allowsus to prove results by induction. FCP extensions are of the form R ⊆ R ⊆ S , where R is the integral closure of R in S and R ⊆ S is Pr¨ufer [22, Proposition 1.3]. We show that these extensions are∆-extensions if and only if R ⊆ R is a ∆-extension (Theorem 4.5).Therefore, we need only to consider integral FCP extensions. Inter-esting examples of integral FCP extensions are given by extensions ofnumber field orders. We exhibit examples of such extensions, show-ing that everything is possible. Note also that an extension R ⊂ R [ t ]where t is either idempotent or nilpotent of index 2, and such that R is a SPIR, is a ∆-extension.Section 2 is devoted to some recalls and results on ring extensionsand their lattice properties.The general properties of ∆-extensions are given in Section 3.In Section 4, the main result is Theorem 4.28, where we give a char-acterization of ∆-extensions using the canonical decomposition of anintegral extension R ⊆ S through the seminormalization and the t -closure of R in S . Actually the ∆-property of R ⊆ S is equivalent tothe ∆-property of all the paths of the canonical decomposition, plusan extra lattice condition relative to some B -subextensions. The caseof infra-integral extensions is specially considered, while length twoextensions are strongly involved. For example, an infra-integral FCPextension of length two is a ∆-extension. -EXTENSIONS 3 The paper ends in Section 5 with Examples of ∆-extensions. Inparticular, we consider Boolean extensions, pointwise minimal exten-sions, extensions of the form R ⊂ R n . These special cases allow tocharacterize more generally some ∆-extensions.We denote by ( R : S ) the conductor of R ⊆ S , and by R the integralclosure of R in S . We set ] R, S [:= [
R, S ] \ { R, S } (with a similardefinition for [ R, S [ or ]
R, S ]).The extension R ⊆ S is said to have FIP (for the “finitely manyintermediate algebras property”) or is an FIP extension if [ R, S ] isfinite. A chain of R -subalgebras of S is a set of elements of [ R, S ] thatare pairwise comparable with respect to inclusion. We will say that R ⊆ S is chained if [ R, S ] is a chain. We also say that the extension R ⊆ S has FCP (resp.; FMC) (or is an FCP (resp.; FMC) extension) ifeach chain in [ R, S ] is finite (resp.; there exists a maximal finite chain).Clearly, each extension that satisfies FIP must also satisfy FCP andeach extension that satisfies FCP must also satisfy FMC. Dobbs andthe authors characterized FCP and FIP extensions [6].Our principal tool are the minimal (ring) extensions, a concept thatwas introduced by Ferrand-Olivier [11]. In our context, minimal exten-sions coincide with lattice atoms. They are completely known (seeSection 2). Recall that an extension R ⊂ S is called minimal if[ R, S ] = { R, S } . The key connection between the above ideas is thatif R ⊆ S has FCP, then any maximal (necessarily finite) chain C of R -subalgebras of S , R = R ⊂ R ⊂ · · · ⊂ R n − ⊂ R n = S , with length ℓ ( C ) := n < ∞ , results from juxtaposing n minimal extensions R i ⊂ R i +1 , ≤ i ≤ n −
1. An FCP extension is finitely generated,and (module) finite if integral. For an FCP extension R ⊆ S , the length ℓ [ R, S ] of [
R, S ] is the supremum of the lengths of chains of R -subalgebras of S . Notice that this length is finite and there does exist some maximal chain of R -subalgebras of S with length ℓ [ R, S ] [7,Theorem 4.11].The characteristic of a field k is denoted by c( k ). Finally, | X | is thecardinality of a set X , ⊂ denotes proper inclusion and, for a positiveinteger n , we set N n := { , . . . , n } .2. Recalls and results on ring extensions
This section is devoted to two types of recalls: commutative ringsand lattices.2.1.
Rings and ring extensions. A local ring is here what is calledelsewhere a quasi-local ring. As usual, Spec( R ) and Max( R ) are theset of prime and maximal ideals of a ring R . For an extension R ⊆ S G. PICAVET AND M. PICAVET and an ideal I of R , we write V S ( I ) := { P ∈ Spec( S ) | I ⊆ P } . Thesupport of an R -module E is Supp R ( E ) := { P ∈ Spec( R ) | E P = 0 } ,and MSupp R ( E ) := Supp R ( E ) ∩ Max( R ). Note that if R ⊆ S is anFMC (or FCP) extension, then | Supp R ( S/R ) | < ∞ [6, Corollary 3.2].If E is an R -module, L R ( E ) (also denoted L( E )) is its length as amodule.If R ⊆ S is a ring extension and P ∈ Spec( R ), then S P is both thelocalization S R \ P as a ring and the localization at P of the R -module S . We denote by κ R ( P ) the residual field R P /P R P at P .The following notions and results are deeply involved in the sequel. Definition 2.1. [3, Definition 2.10] An extension R ⊂ S is called M -crucial if Supp( S/R ) = { M } . Such M is called the crucial (maximal)ideal C ( R, S ) of R ⊂ S . Theorem 2.2. [11, Th´eor`eme 2.2]
A minimal extension R ⊂ S is ei-ther integral (module-finite) or a flat epimorphism and | Supp(
S/R ) | =1 . Moreover, if Supp(
S/R ) = { M } , then M is the crucial (maximal)ideal of R ⊂ S (such that R P = S P for all P ∈ Spec( R ) \ { M } ). Recall that an extension R ⊆ S is called Pr¨ufer if R ⊆ T is a flatepimorphism for each T ∈ [ R, S ] (or equivalently, if R ⊆ S is a normalpair) [14, Theorem 5.2, page 47]. In [22], we called an extension which isa minimal flat epimorphism, a Pr¨ufer minimal extension. Three typesof minimal integral extensions exist, characterized in the next theorem,(a consequence of the fundamental lemma of Ferrand-Olivier), so thatthere are four types of minimal extensions, mutually exclusive.
Theorem 2.3. [6, Theorems 2.2 and 2.3]
Let R ⊂ T be an extensionand M := ( R : T ) . Then R ⊂ T is minimal and finite if and only if M ∈ Max( R ) and one of the following three conditions holds: inert case : M ∈ Max( T ) and R/M → T /M is a minimal field exten-sion. decomposed case : There exist M , M ∈ Max( T ) such that M = M ∩ M and the natural maps R/M → T /M and R/M → T /M areboth isomorphisms, or equivalently, there exists q ∈ T \ R such that T = R [ q ] , q − q ∈ M and M q ⊆ M . ramified case : There exists M ′ ∈ Max( T ) such that M ′ ⊆ M ⊂ M ′ , [ T /M : R/M ] = 2 , and the natural map
R/M → T /M ′ is anisomorphism, or equivalently, there exists q ∈ T \ R such that T = R [ q ] , q ∈ M and M q ⊆ M .In each of the above cases, M = C ( R, T ) . The crucial ideals of minimal subextensions of an FCP extension givemany useful properties as we can see in the following. -EXTENSIONS 5
Lemma 2.4. [23, Lemma 1.5]
Let R ⊂ S be an extension and T, U ∈ [ R, S ] such that R ⊂ T is finite minimal and R ⊂ U is Pr¨ufer minimal.Then, C ( R, T ) = C ( R, U ) , so that R is not a local ring. Lemma 2.5. (Crosswise exchange) [6, Lemma 2.7]
Let R ⊂ S and S ⊂ T be minimal extensions, M := C ( R, S ) , N := C ( S, T ) and P := N ∩ R be such that P M . Then there is S ′ ∈ [ R, T ] such that R ⊂ S ′ is minimal of the same type as S ⊂ T and P = C ( R, S ′ ) ; and S ′ ⊂ T is minimal of the same type as R ⊂ S with M S ′ = C ( S ′ , T ) .Moreover, [ R, T ] = { R, S, S ′ , T } and R Q = S Q = S ′ Q = T Q for Q ∈ Max( R ) \ { M, P } . Some special ring extensions.
Let R ⊂ S be an extension and C := { T i } i ∈ N n ⊂ ] R, S [ , n ≥ R ⊂ S is pinched at C if [ R, S ] = ∪ ni =0 [ T i , T i +1 ], where T := R and T n +1 := S ,which means that any element of [ R, S ] is comparable to the T i ’s.If R ⊂ S is an extension, we say that R is unbranched in S if R islocal. We also say that R ⊂ S is unbranched. If R ⊂ S is unbranchedand FCP, then each T ∈ [ R, S ] is a local ring [25, Lemma 3.29]. Anextension R ⊂ S is said locally unbranched if R M ⊂ S M is unbranchedfor all M ∈ MSupp(
S/R ). In particular, for an FCP extension R ⊂ S ,this is equivalent to Spec( R ) → Spec( R ) is bijective. An extensionis said branched if it is not unbranched. An extension R ⊂ S is said almost unbranched if each T ∈ [ R, R [ is a local ring. Then unbranchedimplies almost unbranched.
Remark 2.6.
Let R ⊂ S be a ring extension and T ∈ [ R, S ]. Then,it is easily seen that T ∩ R (resp.; T R ) is the integral closure of R in T (resp.; of T in S ). Similar relations exist for the t-closure and theseninormalization. We warn the reader that these properties will beoften used in this paper. Corollary 2.7.
An FCP almost unbranched extension is pinched at R .Proof. Assume that R ⊂ S is not pinched at R , so that there exists T ∈ [ R, S ] \ [ R, R ] ∪ [ R, S ]. Set U := T ∩ R ∈ [ R, R ]. Since R ⊂ S isalmost unbranched and FCP, then U is a local ring when U = R , thatis T [ R, S ], which is satisfied. Because T [ R, R ], it follows that U = T . Then, there exist V ∈ [ U, T ] and W ∈ [ U, R ] such that U ⊂ V is minimal Pr¨ufer and U ⊂ W is minimal integral, a contradiction byLemma 2.4. (cid:3) Proposition 2.8.
Let R ⊂ S be an FCP extension such that R = R, S .Then, R ⊂ S is pinched at R if and only if, for any U ∈ [ R, S ] suchthat U ⊂ R is minimal, then MSupp R ( S/R ) ⊆ V R (( U : R )) . G. PICAVET AND M. PICAVET
Proof.
Assume that R ⊂ S is pinched at R , so that [ R, S ] = [
R, R ] ∪ [ R, S ]. Let U ∈ [ R, S ] be such that U ⊂ R is minimal and set P :=( U : R ) ∈ Max( U ). Let M ∈ MSupp R ( S/R ). According to [23, Lemma1.8], there exists V ∈ [ R, S ] such that R ⊂ V is minimal Pr¨ufer with M = C ( R, V ). If M ∩ U P , by the Crosswise Exchange, thereexists T ∈ [ R, S ] such that U ⊂ T is minimal Pr¨ufer, so that T [ R, R ]. Then, T ∈ ] R, S ], a contradiction with U ⊂ R ⊂ T and U ⊂ T minimal. Then, M ∩ U ⊆ P , and, more precisely, M ∩ U = P because M ∈ Max( R ) and U ⊂ R is integral. To conclude, P ⊆ M , that is M ∈ V R (( U : R )).Conversely, assume that MSupp R ( S/R ) ⊆ V R (( U : R )) for any U ∈ [ R, S ] such that U ⊂ R is minimal. Supppose that [ R, S ] = [ R, R ] ∪ [ R, S ] and let T ∈ [ R, S ] \ ([ R, R ] ∪ [ R, S ]). Set U := T ∩ R ⊂ R because U = R implies T ∈ [ R, S ]. We also have U = T because U = T implies T ∈ [ R, R ]. Then, there exist U ∈ [ U, R ] and T ∈ [ U, T ] such that U ⊂ U is minimal integral and U ⊂ T is minimal Pr¨ufer. By Lemma2.4, we have C ( U, U ) = C ( U, T ). It follows from [9, Proposition 7.10]that U ⊂ U T is minimal Pr¨ufer and T ⊂ T U is minimal integralwith C ( U , U T ) MSupp U ( R/U ). Of course, U T [ R, R ] because T ∈ [ U, U T ].If T U ∈ [ R, S ], then U ⊆ R ⊂ T U implies U = R , so that U ⊂ R is minimal.If T U [ R, S ], then T ∈ [ R, S ] \ ([ R, R ] ∪ [ R, S ]). It follows thatwe get U = U T ∩ R , because U is the integral closure of U ⊆ U T .Since ℓ [ U , R ] < ℓ [ U, R ], an easy induction shows that there exists amaximal finite chain { U i } ni =0 such that U = U, U n = R with T U i [ R, R ] ∪ [ R, S ] for each i ∈ { , . . . , n − } . We have the followingcommutative diagram: T −→ S ↑ ↑ T → T U → T U n − → T R ↑ ↑ ↑ ↑ U → U → U n − → R Then, in both cases, we get that U n − ⊂ R is minimal integral, with U n − ⊂ T U n − minimal Pr¨ufer as R ⊂ T R , so that T R ∈ [ R, S ]. Westill have C ( U n − , U n − T ) = C ( U n − , R ) ( ∗ ) with C ( R, RT ) ∩ U n − = C ( U n − , U n − T ) = C ( U n − , R ) = ( U n − : R ). But M := C ( R, RT ) ∈ MSupp R ( S/R ) gives by assumption that ( U n − : R ) ⊆ M , so that M ∩ U n − = ( U n − : R ) because ( U n − : R ) ∈ Max( U n − ), a contradiction -EXTENSIONS 7 with ( ∗ ), which is M ∩ U n − = ( U n − : R ). Hence, [ R, S ] = [
R, R ] ∪ [ R, S ]. (cid:3) The following definitions are needed for our study.
Definition 2.9.
An integral extension R ⊆ S is called infra-integral [19] (resp.; subintegral [29]) if all its residual extensions κ R ( P ) → κ S ( Q ), (with Q ∈ Spec( S ) and P := Q ∩ R ) are isomorphisms (resp . ;and the natural map Spec( S ) → Spec( R ) is bijective). An exten-sion R ⊆ S is called t-closed (cf. [19]) if the relations b ∈ S, r ∈ R, b − rb ∈ R, b − rb ∈ R imply b ∈ R . The t - closure tS R of R in S is the smallest element B ∈ [ R, S ] such that B ⊆ S is t-closed andthe greatest element B ′ ∈ [ R, S ] such that R ⊆ B ′ is infra-integral.An extension R ⊆ S is called seminormal (cf. [29]) if the relations b ∈ S, b ∈ R, b ∈ R imply b ∈ R . The seminormalization + S R of R in S is the smallest element B ∈ [ R, S ] such that B ⊆ S is seminormaland the greatest element B ′ ∈ [ R, S ] such that R ⊆ B ′ is subintegral.The canonical decomposition of an arbitrary ring extension R ⊂ S is R ⊆ + S R ⊆ tS R ⊆ R ⊆ S . Proposition 2.10. [23, Proposition, 4.5] and [20, Lemma 3.1]
Letthere be an integral extension R ⊂ S admitting a maximal chain C of R -subextensions of S , defined by R = R ⊂ · · · ⊂ R i ⊂ · · · ⊂ R n = S ,where each R i ⊂ R i +1 is minimal. The following statements hold: (1) R ⊂ S is subintegral if and only if each R i ⊂ R i +1 is ramified. (2) R ⊂ S is infra-integral if and only if each R i ⊂ R i +1 is eitherramified or decomposed. (3) R ⊂ S is seminormal and infra-integral if and only if each R i ⊂ R i +1 is decomposed. (4) R ⊂ S is t-closed if and only if each R i ⊂ R i +1 is inert.Moreover, Spec( S ) → Spec( R ) is bijective if and only if each R i ⊂ R i +1 is either ramified or inert.Proof. The last result comes from Theorem 2.3 where it is shown thatdecomposed minimal extensions are the only extensions whose spectralmaps are not bijective. (cid:3)
Lattice Properties.
Let R ⊆ S be an FCP extension, then [ R, S ]is a complete Noetherian Artinian lattice, R being the least elementand S the largest. In the context of the lattice [ R, S ], some definitionsand properties of lattices have the following formulations. (see [16])(1) An element T ∈ [ R, S ] is an atom if and only if R ⊂ T is aminimal extension. We denote by A the set of atoms of [ R, S ]. G. PICAVET AND M. PICAVET (2) R ⊆ S is called catenarian , or graded by some authors working inthe lattices context, if R ⊂ S has FCP and all maximal chains betweentwo comparable elements have the same length.(3) R ⊆ S is called semimodular if, for each T , T ∈ [ R, S ] such that T ∩ T ⊂ T i is minimal for i = 1 ,
2, then T i ⊂ T T is minimal for i = 1 , R ⊆ S is called modular if T ∩ ( T T ) = T ( T ∩ T ) for each T , T , T ∈ [ R, S ] such that T ⊆ T .(5) R ⊆ S is called distributive if intersection and product are eachdistributive with respect to the other. Actually, each distributivityimplies the other [16, Exercise 5, page 33].Moreover, R ⊆ S distributive ⇒ R ⊆ S modular ⇒ R ⊆ S semi-modular ⇒ R ⊆ S catenarian.(6) Let T ∈ [ R, S ]. Then, T ′ ∈ [ R, S ] is called a complement of T if T ∩ T ′ = R and T T ′ = S .(7) R ⊆ S is called Boolean if ([
R, S ] , ∩ , · ) is a distributive latticesuch that each T ∈ [ R, S ] has a (necessarily unique) complement.(8) R ⊂ S is called a B - extension if R ⊂ S is a Boolean extensionof length 2 (which is equivalent to ℓ [ R, S ] = 2 and | [ R, S ] | = 4). See[15, Fig. 1]. In particular, if R ⊂ S ⊂ T is an extension satisfying theCrosswise Exchange, then R ⊂ T is a B -extension.(9) R ⊆ S is called simple if there exists x ∈ S \ R such that S = R [ x ].(10) R ⊆ S is called arithmetic if R P ⊆ S P is chained for each P ∈ Spec( R ). Proposition 2.11. [16, Corollary 1.3.4, p. 172]
A modular lattice offinite length is catenarian (the Jordan-H¨older chain condition holds).
Proposition 2.12. [26, Proposition 4.7]
An infra-integral FCP exten-sion is catenarian.
According to [4, Exercise 3.2, p. 37], a finite length lattice is modularif and only if it satisfies both covering conditions , which means that fora ring extension R ⊂ S , for each T, U ∈ [ R, S ] such that T ∩ U ⊂ T is minimal, then U ⊂ T U is minimal (upper covering condition), andfor each
T, U ∈ [ R, S ] such that U ⊂ T U is minimal, then T ∩ U ⊂ T is minimal (lower covering condition). In particular, we have thefollowing: Lemma 2.13.
Let R ⊂ S be an FCP modular extension. Then, ℓ [ T, U V ] = 2 for
T, U, V ∈ [ R, S ] , U = V such that T ⊂ U and T ⊂ V are minimal.Proof. Assume that R ⊂ S is modular and let T, U, V ∈ [ R, S ] , U = V such that T ⊂ U and T ⊂ V are minimal. Then, T = U ∩ V implies -EXTENSIONS 9 that U ⊂ U V is minimal. Since any maximal chains of R ⊂ S havethe same length, ℓ [ T, U V ] = ℓ [ T, U ] + ℓ [ U, U V ] = 2. (cid:3)
Proposition 2.14.
A length 2 extension is modular, whence catenar-ian.Proof.
Let R ⊂ S be a length 2 extension with T , T , T ∈ [ R, S ] suchthat T ⊆ T . It T = T , then obviously T ∩ ( T T ) = T ( T ∩ T ). If T = T , then either T = R or T = S , and T ∩ ( T T ) = T ( T ∩ T )also holds. (cid:3) The following Proposition summarizes [9, Propositions 7.1, 7.4, 7.6and 7.10]. Different cases occur when considering two minimal integralextensions with the same domain. The discussion is organized withrespect to their crucial maximal ideals.
Proposition 2.15.
Let R ⊂ T and R ⊂ U be two distinct minimalintegral extensions, whose compositum S := T U exists. Set M := C ( R, T ) and N := C ( R, U ) . The following statements hold: (1) Assume that M = N . Then [ R, S ] = { R, T, U, S } . (2) Assume that M = N, R ⊂ T is inert and R ⊂ U is not inert.Then R ⊂ S is not catenarian. (3) Assume that M = N, R ⊂ T, R ⊂ U are both non-inert and P Q ⊆ M for some P ∈ Max( T ) and some Q ∈ Max( U ) bothlying above M . Then R ⊂ S is catenarian infra-integral and ℓ [ R, S ] = 2 . (4) Assume that M = N, R ⊂ T, R ⊂ U are both non-inert and P Q M for any P ∈ Max( T ) and any Q ∈ Max( U ) bothlying above M . Then R ⊂ S is catenarian infra-integral and ℓ [ R, S ] = 3 .Proof.
We only prove that [
R, S ] = { R, T, U, S } in (1). Indeed, [9,Proposition 7.10] says that T ⊂ S is a minimal extension such that C ( T, S ) lies over N in R . In particular, N = R ∩ C ( T, S ) M shows that | [ R, S ] | = 4 by the Crosswise Exchange (Lemma 2.5). Since { R, T, U, S } ⊆ [ R, S ], the result follows. (cid:3) General properties of ∆ -extensions We begin by recalling a Gilmer-Huckaba’s result about ∆-extensions.
Proposition 3.1. [13, Proposition 1]
Let R ⊂ S be a ring extension.The following conditions are equivalent: (1) R ⊂ S is a ∆ -extension; (2) R [ s, t ] = R [ s ] + R [ t ] for all s, t ∈ S ; (3) R [ s , . . . , s n ] = P ni =1 R [ s i ] for all integer n and s , . . . , s n ∈ S . Corollary 3.2.
A ring extension R ⊂ S is a ∆ -extension if and onlyif T ⊂ U is a ∆ -extension for each T, U ∈ [ R, S ] .Proof. Obvious. (cid:3)
Lemma 2.13 and Proposition 2.14 show that modular extensions andextensions of length 2 are linked. All along the paper, we will see thatextensions of length 2 play a significant role. In particular, we have thefollowing Corollary:
Corollary 3.3.
A ring extension R ⊂ S is not a ∆ -extension if thereexist T, U ∈ [ R, S ] , T ⊂ U , such that ℓ [ T, U ] = 2 and T ⊂ U is not a ∆ -extension.Proof. Obvious according to Corollary 3.2. (cid:3)
Proposition 3.4.
A ring extension R ⊂ S is a ∆ -extension if andonly if T + U = T U for each
T, U ∈ [ R, S ] .Proof. Let R ⊂ S be a ∆-extension and let T, U ∈ [ R, S ]. Since T + U is an R -algebra, we get that T, U ⊆ T + U ⊆ T U , so that T + U = T U .Conversely, T + U = T U shows that T + U ∈ [ R, S ]. (cid:3) Corollary 3.5.
Let R ⊂ S be a ring extension and C := { T i } i ∈ N n ⊂ ] R, S [ , n ≥ be a finite chain such that R ⊂ S is pinched at C . Set T := R and T n +1 := S . Then, R ⊂ S is a ∆ -extension if and only if T i ⊂ T i +1 is a ∆ -extension for each i ∈ { , . . . , n } .Proof. One implication results from Corollary 3.2. Conversely, assumethat T i ⊂ T i +1 is a ∆-extension for each i ∈ { , . . . , n } Let
U, V ∈ [ R, S ] = ∪ ni =0 [ T i , T i +1 ]. If U, V are both in some [ T i , T i +1 ], then U + V = U V by Proposition 3.4 applied to [ T i , T i +1 ]. Assume that U ∈ [ T i , T i +1 ]and V ∈ [ T j , T j +1 ] with i = j . Let, for instance, i < j , so that i + 1 ≤ j .Then, U ⊆ T i +1 ⊆ T j ⊆ V leads to U + V = V = U V . Anotherapplication of Proposition 3.4 shows that R ⊂ S is a ∆-extension. (cid:3) Proposition 3.6.
Let R ⊆ S be a ring extension. The following state-ments are equivalent: (1) R ⊆ S is a ∆ -extension; (2) R M ⊆ S M is a ∆ -extension for each M ∈ MSupp(
S/R ) ; (3) R P ⊆ S P is a ∆ -extension for each P ∈ Supp(
S/R ) ; (4) R/I ⊆ S/I is a ∆ -extension for some ideal I shared by R and S .Proof. Obvious with the help of Proposition 3.4. (cid:3) -EXTENSIONS 11
Remark 3.7.
According to the equivalences of Proposition 3.6, andas we mostly consider in the rest of the paper FCP extensions, thereis no harm to replace FCP integral extensions by locally FCP integralextensions. Let R ⊂ S be an integral extension satisfying the follow-ing property: any T ∈ [ R, S ] such that R ⊂ T is finite implies that | MSupp(
T /R ) | < ∞ . Then R ⊂ S is locally FCP if and only if R ⊂ S is locally finite, and for any T ∈ [ R, S ] such that R ⊂ T is finite, then R ⊂ T has FMC.First assume that R ⊂ S is locally finite and for any T ∈ [ R, S ]such that R ⊂ T is finite, then R ⊂ T has FMC. Let M ∈ Max( R ),so that there exist x , . . . , x n ∈ R, s ∈ R \ M such that S M = R M [ x /s, . . . , x n /s ]. Set T := R [ x , . . . , x n ], which is a finite exten-sion of R . In particular, R ⊂ T has FMC and so has R M ⊆ T M = S M .It follows that R M ⊂ S M has FCP by [6, Theorem 4.2].Conversely, assume that R ⊂ S is locally FCP. The previous refer-ence shows that R M ⊂ S M is locally finite. Let T ∈ [ R, S ] be such that R ⊆ T is finite. In particular, R M ⊆ T M is finite for each M ∈ Max( R )and ( R M : T M ) = ( R : T ) M , so that R M / ( R M : T M ) = R M / ( R : T ) M = ( R/ ( R : T )) M . Since R M ⊂ S M has FCP, so has R M ⊆ T M .But, | MSupp(
T /R ) | < ∞ implies that R ⊂ T has FCP according to[6, Proposition 3.7], and then has FMC. Proposition 3.8.
Let R ⊂ S be a ring extension, f : R → R ′ a ringmorphism and S ′ := R ′ ⊗ R S . (1) If f : R → R ′ is a faithfully flat ring morphism and if R ′ ⊂ S ′ is a ∆ -extension, then so is R ⊂ S . (2) If f : R → R ′ is a flat ring epimorphism (for example, a lo-calization with respect to a multiplicatively closed subset) and R ⊂ S is a ∆ -extension, then so is R ′ ⊂ S ′ .Proof. (1) The ring morphism ϕ : S → S ′ defines a map ψ : [ R, S ] → [ R ′ , S ′ ] by ψ ( T ) = R ′ ⊗ R T and θ : [ R ′ , S ′ ] → [ R, S ] by θ ( T ′ ) = T ′ ∩ S such that θ ◦ ψ is the identity of [ R, S ] by [2, Proposition 10, p.52] (itis enough to take F = S and to observe that if M is an R -submoduleof S , then with the notation of the above reference, R ′ M identifies to R ′ ⊗ R M ). The same reference shows that ψ ( T + U ) = ψ ( T ) + ψ ( U )and ψ ( T ∩ U ) = ψ ( T ) ∩ ψ ( U ) for U, T ∈ [ R, S ]. It is easy to show that ψ ( T U ) = ψ ( T ) ψ ( U ). Then the result follows.(2) The proof is a consequence of the following facts. Let f : R → R ′ be a flat epimorphism and Q ∈ Spec( R ′ ), lying over P in R , then R P → R ′ Q is an isomorphism by [22, Scholium A]. Moreover, we have( R ′ ⊗ R S ) Q ∼ = R ′ Q ⊗ R P S P , so that R P → S P identifies to R ′ Q → ( R ′ ⊗ R S ) Q . (cid:3) Remark 3.9.
We deduce from the above statement (1), that the ∆-property is local on the spectrum. This means that for a ring extension R ⊂ S and any finite set { r , . . . , r n } of elements of R , such that R = Rr + · · · + Rr n , then R ⊂ S is a ∆-extension if and only if all theextensions R r i ⊂ S r i have the ∆-property.Given a ring R , recall that its Nagata ring R ( X ) is the localization R ( X ) = T − R [ X ] of the ring of polynomials R [ X ] with respect to themultiplicatively closed subset T of all polynomials with content R . In[8, Theorem 32], Dobbs and the authors proved that when R ⊂ S is an extension, whose Nagata extension R ( X ) ⊂ S ( X ) has FIP, themap ϕ : [ R, S ] → [ R ( X ) , S ( X )] defined by ϕ ( T ) = T ( X ) is an order-isomorphism. We look at the transfer of the ∆-property between R ⊂ S and R ( X ) ⊂ S ( X ). We recall the following [20, Corollary 4.3]: If R ⊂ S has FIP, then R ( X ) ⊂ S ( X ) has FIP if and only if R ⊂ + S R isarithmetic. Proposition 3.10.
An FCP extension R ⊂ S is a ∆ -extension if R ( X ) ⊂ S ( X ) is a ∆ -extension. The converse hold if R ( X ) ⊂ S ( X ) has FIP.Proof. Assume that R ( X ) ⊂ S ( X ) is a ∆-extension. By [7, Corollary3.5], we have S ( X ) = R ( X ) ⊗ R S . Since R ⊂ R ( X ) is faithfully flat,an application of Proposition 3.8 gives the result.Conversely, assume that R ⊂ S is a ∆-extension and R ( X ) ⊂ S ( X )has FIP. According to the previous remark, the map ϕ : [ R, S ] → [ R ( X ) , S ( X )] defined by ϕ ( T ) = T ( X ) is an order-isomorphism, andmore precisely, a lattice isomorphism. Then, two elements of [ R ( X ) , S ( X )]are of the form T ( X ) , U ( X ) for some T, U ∈ [ R, S ]. In particular, T ( X ) + U ( X ) = ( T + U )( X ) = ( T U )( X ) = T ( X ) U ( X ). Then, Propo-sition 3.4 shows that R ( X ) ⊂ S ( X ) is a ∆-extension. (cid:3) Proposition 3.11.
An arithmetic extension is a ∆ -extension.Proof. [20, Proposition 5.16]. (cid:3) Corollary 3.12.
Let R ⊂ S be a ring extension and T, U ∈ [ R, S ] suchthat R ⊂ T and R ⊂ U are minimal. Assume that either C ( R, T ) = C ( R, U ) , or R ⊂ T and R ⊂ U are minimal of different types with ℓ [ R, T U ] = 2 . Then, R ⊂ T U is a B -extension and a ∆ -extension.Proof. Set M := C ( R, T ) and N := C ( R, U ) with M = N . Obviously,Supp( T U/R ) = MSupp(
T U/R ) = { M, N } . Since R M = U M and R N = T N , it follows that R ⊂ T U is locally minimal, then arithmeticand a ∆-extension by Proposition 3.11. Moreover, [6, Theorem 3.6]shows that the map ϕ : [ R, T U ] → [ R M , T M U M ] × [ R N , T N U N ] defined -EXTENSIONS 13 by V ( V M , V N ) for any V ∈ [ R, T U ], is bijective, with T M U M = T M and T N U N = U N , so that | [ R, T U ] | = 4, with ℓ [ R, T U ] = 2. Therefore, R ⊂ T U is a B -extension.Assume now that M := C ( R, T ) = C ( R, U ) with R ⊂ T and R ⊂ U minimal of different types and ℓ [ R, T U ] = 2. In this case, neither R ⊂ T nor R ⊂ U is minimal Pr¨ufer by Lemma 2.4. In particular, R ⊂ T U is catenarian by Proposition 2.14 and integral, so that nei-ther R ⊂ T nor R ⊂ U is minimal inert according to Proposition2.15 (2). It follows that R ⊂ T U is infra-integral with + T U R = R, T U .Assume that R ⊂ T is minimal ramified and R ⊂ U is minimal de-composed. Since T ⊂ T U is minimal, it is necessarily decomposed, sothat T = + T U R . We now show that [ R, T U ] = { R, T, U, T U } . Assumethat there exists some W ∈ [ R, T U ] \ { R, T, U, T U } , so that R ⊂ W and W ⊂ T U are minimal. Because R ⊂ + T U R is minimal, we cannothave R ⊂ W minimal ramified. But, R ⊂ U and R ⊂ W both mini-mal decomposed implies R ⊂ U W seminormal infra-integral accordingto [9, Proposition 7.6], with
U W = S , a contradiction. To conclude,[ R, T U ] = { R, T, U, T U } . Then, R ⊂ T U is a B -extension and a∆-extension. (cid:3) Proposition 3.13. A ∆ -extension is modular, and hence is catenarianwhen it has FCP.Proof. Let R ⊂ S be a ∆-extension and let T, U, V ∈ [ R, S ] be suchthat U ⊆ T . We always have U ( T ∩ V ) ⊆ T ∩ ( U V ). Since T ∩ ( U V ) = T ∩ ( U + V ) ⊆ U + ( T ∩ V ) = U ( T ∩ V ), we get that T ∩ U V = U ( T ∩ V )and R ⊂ S is a modular extension. In particular, an FCP ∆-extensionis catenarian by Proposition 2.11. (cid:3) Proposition 3.14. [13, Theorem 1]
Let k ⊂ L be a field extension.Then k ⊂ L is a ∆ -extension if and only if k ⊂ L is chained. In particular, the previous Proposition is satisfied when k ⊂ L iseither an FIP purely inseparable extension [24, Lemma 4.1], or a cyclicextension whose degree is a power of a prime integer.In [26, before Proposition 2.7], we define the following notion: Aproperty ( T ) of ring extensions R ⊂ S is called convenient if the fol-lowing conditions are equivalent.(1) R ⊂ S satisfies ( T ).(2) R M ⊂ S M satisfies ( T ) for any M ∈ MSupp(
S/R ).(3)
R/I ⊂ S/I satisfies ( T ) for any ideal I shared by R and S . Proposition 3.15.
Let R ⊆ S be an FCP extension and ( T ) a con-venient property. Assume that R = Q ni =1 R i is a product of rings. For each i ∈ N n , there exist FCP ring extensions R i ⊆ S i such that S ∼ = Q ni =1 S i . Moreover R ⊆ S satisfies property ( T ) if and only ifso does R i ⊆ S i for each i ∈ N n . Since the ∆ -property is convenient, R ⊆ S is a ∆ -extension if and only if so is R i ⊆ S i for each i ∈ N n .Proof. The first part of the statement is [5, Lemma III.3], except forthe FCP property of the R i ⊆ S i . But this property is obvious sinceany T ∈ [ R, S ] is of the form Q ni =1 T i , where T i ∈ [ R i , S i ] for each i ∈ N n .We recall the following of [21, the last paragraph of Section 1]: If R , . . . , R n are finitely many rings, the ring R ×· · ·× R n localized at theprime ideal P × R ×· · ·× R n is isomorphic to ( R ) P for P ∈ Spec( R ).This rule works for any prime ideal of the product. Since a maximalideal M ∈ Max( R ) is of the form R × · · · × M i × · · · × R n for some i ∈ N n and M i ∈ Max( R i ), we get that R M ∼ = ( R i ) M i and S M ∼ = ( S i ) M i .In particular, R M ⊆ S M can be identified with ( R i ) M i ⊆ ( S i ) M i . Itfollows that M ∈ MSupp(
S/R ) ⇔ R M = S M ⇔ ( R i ) M i = ( S i ) M i ⇔ M i ∈ MSupp( S i /R i ). Since property ( T ) holds for R ⊆ S if and only ifit holds for R M ⊆ S M for each M ∈ MSupp(
S/R ) because convenient,the previous isomorphisms give the last result. (cid:3) Characterization of ∆ -extensions First characterizations of ∆ -extensions. Our results mainlyhold for FCP extensions.
Proposition 4.1.
Let R ⊂ S be a ring extension. The following state-ments hold: (1) R ⊂ S is a Pr¨ufer extension if and only if R ⊂ S is an integrallyclosed ∆ -extension. (2) If R ⊂ S is integrally closed and | Supp(
S/R ) | < ∞ , the follow-ing conditions are equivalent: (a) R ⊂ S is a ∆ -extension; (b) R ⊂ S is an FCP extension; (c) R ⊂ S is an FIP extension.Proof. (1) A Pr¨ufer extension is integrally closed by [22, Scholium B],then arithmetic by [20, Theorem 5.17] and a ∆-extension by Proposi-tion 3.11.Conversely, if R ⊂ S is an integrally closed ∆-extension, then R ⊂ S is a Pr¨ufer extension by [14, Theorem 1.7, page 88].(2) Using (1), [22, Proposition 1.3] gives the equivalence of (a) and(b) because an integrally closed extension R ⊂ S has FCP if and onlyif R ⊂ S is Pr¨ufer with | Supp(
S/R ) | < ∞ . The equivalence of (b) and -EXTENSIONS 15 (c) follows from [6, Theorem 6.3] because an integrally closed extensionhas FIP if and only if it has FCP. (cid:3) Lemma 4.2.
Let R ⊂ S be a ring extension and T, U ∈ [ R, S ] . Then, T + U = T U if MSupp(
T /R ) ∩ MSupp(
U/R ) = ∅ .Proof. Let M ∈ Max( R ) \ [MSupp( T /R ) ∪ MSupp(
U/R )]. Then, R M = T M = U M yields ( T + U ) M = T M + U M = R M = T M U M = ( T U ) M .Let M ∈ MSupp(
T /R ), so that M MSupp(
U/R ). Then, R M = U M yields ( T + U ) M = T M + U M = T M = T M U M = ( T U ) M .Let M ∈ MSupp(
U/R ), so that M MSupp(
T /R ). Then, R M = T M yields ( T + U ) M = T M + U M = U M = T M U M = ( T U ) M .Since for each M ∈ Max( R ), it holds that ( T + U ) M = ( T U ) M , wehave T + U = T U . (cid:3) In [22, Definition 4.1], we call an extension R ⊂ S almost-Pr¨ufer ifit can be factored R ⊆ U ⊆ S , where R ⊆ U is Pr¨ufer and U ⊆ S isintegral. Actually, U is the Pr¨ufer hull e R of the extension, that is thegreatest T ∈ [ R, S ] such that R ⊆ T is Pr¨ufer. Lemma 4.3.
Let R ⊂ S be an FCP extension. Then, R ⊆ R is a ∆ -extension if and only if T ⊆ T U is a ∆ -extension for any subextension T ⊂ U of R ⊂ S .Proof. One implication is obvious. Now, assume that R ⊆ R is a ∆-extension and consider the tower R ⊆ T ⊂ U ⊆ S . Set R := R T and S := T U . We get the following commutative diagram, where i , (resp. p ) indicates an integral (resp. Pr¨ufer) extension: T i → S p → Up ↑ p ↑ ↓ R i → R i → R U → S The definition of R and S implies that R ⊆ T and R U ⊆ S = T U are integrally closed, and then Pr¨ufer extensions by [22, Proposition 1.3(2)], while T ⊂ S and R ⊂ R U are integral. Then, R ⊆ S is almost-Pr¨ufer, with T = f R S , the Pr¨ufer hull of the extension R ⊆ S and R U = R S . It follows that MSupp( T /R ) ∩ MSupp( R U /R ) = ∅ andMSupp( S /R ) = MSupp( T /R ) ∪ MSupp( R U /R ) by [22, Proposition4.18].Since R U ⊆ R , we deduce from Corollary 3.2 that R ⊆ R U is a∆-extension, and so is ( R ) M ⊆ ( R U ) M for each M ∈ Max( R ) byProposition 3.6. Let V, W ∈ [ T, S ] and M ∈ Max( R ). If M MSupp( R U /R ), we get that ( R U ) M = ( R ) M , which implies that( R ) M ⊆ ( S ) M is Pr¨ufer, so that T M = ( S ) M = V M = W M . Then,( V + W ) M = V M + W M = V M = V M W M = ( V W ) M . If M ∈ MSupp( R U /R ), we get that M MSupp(
T /R ), so that T M = ( R ) M ,which in turn implies that ( R ) M ⊆ ( S ) M is integral and ( S ) M =( R U ) M . This shows that T M = ( R ) M ⊆ ( S ) M = ( R U ) M is a ∆-extension, so that ( V + W ) M = V M + W M = V M W M = ( V W ) M . Toconclude, we have ( V + W ) M = ( V W ) M for each M ∈ Max( R ), andthen V + W = V W , showing that T ⊂ S = T U is a ∆-extension. (cid:3) Proposition 4.4.
Let R ⊂ S be an FCP extension and T, U ∈ [ R, S ] such that R ⊆ T is integral and R ⊆ U is Pr¨ufer. Then U + T = U T .Proof.
Since R ⊆ T is integral, we have T ∈ [ R, R ] and since R ⊆ U isPr¨ufer, we have U ∈ [ R, ˜ R ] by definition of the Pr¨ufer hull. Then,MSupp( T /R ) ⊆ MSupp(
R/R ) and MSupp(
U/R ) ⊆ MSupp( ˜
R/R ).But MSupp(
R/R ) ∩ MSupp( ˜
R/R ) = ∅ by [22, Proposition 4.18]. Itfollows that MSupp( T /R ) ∩ MSupp(
U/R ) = ∅ and Lemma 4.2 showsthat U + T = U T . (cid:3) Theorem 4.5.
An FCP extension R ⊂ S is a ∆ -extension if and onlyif R ⊆ R is a ∆ -extension.Proof. Corollary 3.2 gives one implication. Conversely, assume that R ⊂ S is an FCP extension such that R ⊆ R is a ∆-extension. Let V, W ∈ [ R, S ] and set T := V ∩ W, U := V W . We denote by U (resp. V , W ) the integral closure of T in U (resp. V, W ). According toLemma 4.3, T ⊆ U is a ∆-extension. In particular, V + W = V W (1)since
V , W ⊆ U .We get the following commutative diagram, where i , (resp. p ) indi-cates an integral (resp. Pr¨ufer) extension: V p → V i → V Wi ր i ց p ր p ց T V W Ui ց i ր p ց p ր W p → W i → V W
Since V ⊆ V is Pr¨ufer and V ⊆ V W is integral, Proposition 4.4leads to V + V W = V V W = V W (2). For the same reason, we get W + V W = V W (3).Moreover, V ⊆ V W and W ⊆ V W are almost-Pr¨ufer with V (resp. W ) as Pr¨ufer hull of V ⊆ V W (resp. W ⊆ V W ). According to [22, -EXTENSIONS 17
Proposition 4.16], MSupp(
V /V ) ∩ MSupp(
V W /V ) = ∅ . We claim that, V W ⊆ V W and
V W ⊆ V W are Pr¨ufer. Indeed, MSupp(
V W /V ) =MSupp(
V /V ) ∪ MSupp(
V W /V ). Let M ∈ MSupp(
V W /V ). If M ∈ MSupp(
V /V ), then V M = V M W M , so that V M W M ⊆ V M = V M W M isPr¨ufer. If M ∈ MSupp(
V W /V ), then V M = V M leads to V M W M = V M W M . It follows that V W ⊂ V W is Pr¨ufer. The same holds for
V W ⊂ V W . It follows that
V W ⊆ ( V W )( V W ) =
V W is Pr¨ufer since
V W and
V W are both contained in the Pr¨ufer hull of
V W ⊂ U . Then, V W ⊆ V W is a ∆-extension by Proposition 4.1, so that
V W + V W = V W V W = V W (4).Adding (2) and (3) and using (1) and (4), this leads to V + V W + W + V W = V + V + W + W + V + W = V + W = V W + V W = V W .To conclude, we obtain V + W = V W for any
V, W ∈ [ R, S ] and R ⊂ S is a ∆-extension. (cid:3) Remark 4.6.
When R ⊂ S is an almost unbranched FCP extensionthe result of Theorem 4.5 is gotten immediately. Indeed, Corollary 2.7shows that R ⊂ S is pinched at R . Since R ⊆ S is a ∆-extension byProposition 4.1, then Corollary 3.5 gives the result of Theorem 4.5.Theorem 4.5 shows that it is enough to characterize integral ∆-extensions that are FCP, which we are aiming to do by using the pathsof the canonical decomposition. Proposition 4.7.
A t-closed FCP extension R ⊂ S is a ∆ -extensionif and only if R ⊂ S is arithmetic.Proof. One implication is Proposition 3.11. Conversely, assume that R ⊂ S is a ∆-extension. By Proposition 3.6, we can assume that( R, M ) is a local ring, so that ( R : S ) = M because ( S, M ) is a localring by [7, Lemma 3.17]. Moreover,
R/M ⊂ S/M is a ∆-extensionagain by Proposition 3.6 and a field extension. Then,
R/M ⊆ S/M ischained by Proposition 3.14 and so is [
R, S ]. (cid:3) Lemma 4.8. If R ⊂ S is a catenarian FCP extension and T, U ∈ ] R, S ] are such that R ⊂ T is infra-integral and R ⊂ U is t-closed, then, T + U = T U and
MSupp(
T /R ) ∩ MSupp(
U/R ) = ∅ .Proof. We claim that MSupp(
T /R ) ∩ MSupp(
U/R ) = ∅ . Deny andlet M ∈ MSupp(
T /R ) ∩ MSupp(
U/R ). According to [23, Lemma 1.8],there exist V ∈ [ R, T ] and W ∈ [ R, U ] such that R ⊂ V is minimalinfra-integral (that is either ramified or decomposed) and R ⊂ W isminimal inert, while both extensions have M as conductor. It followsthat there exist two maximal chains from R to V W whose lengths aredifferent by Proposition 2.15, contradicting that R ⊂ S is catenarian. Moreover, MSupp(
T U/R ) = MSupp(
T /R ) ∪ MSupp(
U/R ). Now let M ∈ MSupp(
T U/R ).If M ∈ MSupp(
T /R ), then M MSupp(
U/R ), so that R M = U M ,giving ( T + U ) M = T M + U M = T M = T M U M = ( T U ) M . If M ∈ MSupp(
U/R ), then, M MSupp(
T /R ), so that R M = T M , giving( T + U ) M = T M + U M = U M = T M U M = ( T U ) M .If finally M MSupp(
T U/R ), then R M = T M = U M , giving ( T + U ) M = T M + U M = R M = T M U M = ( T U ) M .Since ( T + U ) M = ( T U ) M for any M ∈ Max( R ), we get T + U = T U . (cid:3) Lemma 4.9.
Let R ⊂ S be a catenarian integral FCP extension. Then, tS R ⊆ S is arithmetic if and only if tU T ⊆ U is arithmetic for anysubextension T ⊂ U of R ⊂ S .Proof. One implication is obvious. Now, assume that tS R ⊆ S is arith-metic. Consider the tower R ⊆ T ⊂ U ⊆ S and set R := tU R, R := tS R and T := tU T . We get the following commutative diagram, where i , (resp. t ) indicates an infra-integral (resp. t-closed) extension: T i → T t → U ↑ t ↑ ց R i → R i → R t → S Since R is the t-closure of R in U and R ⊆ T ⊆ U ⊆ S , we get that R ⊆ T and R ⊆ R . The definition of R , R and T implies that R ⊆ T , T ⊆ U and R ⊆ S are t-closed, while R ⊆ R , T ⊆ T and R ⊆ R are infra-integral.By Lemma 4.8 and because R ⊂ S is catenarian, MSupp R ( U/R ) ∩ MSupp R ( R /R ) = ∅ . Moreover, MSupp R ( U/R ) ∪ MSupp R ( R /R )= MSupp R ( U R /R ). Let N ∈ MSupp T ( U/T ) and set M := N ∩ R , so that M ∈ MSupp R ( U/R ). According to [7, Lemma 3.17], M ( R ) M = (( R ) M : ( T ) M ) = (( R ) M : U M ) and is the maximal idealof the local rings ( T ) M and U M because ( R ) M ⊂ U M is t-closed asis ( R ) M ⊂ ( T ) M . In particular, N is the only maximal ideal of T lying above M and we get ( T ) M = ( T ) N and U M = U N . Moreover, M MSupp( R /R ) which yields ( R ) M = ( R ) M . To conclude, we getthe tower ( R ) M = ( R ) M ⊆ ( T ) M ⊂ U M ⊆ S M , where ( R ) M ⊆ S M is chained, and so is ( T ) M = ( T ) N ⊂ U M = U N . Then, tU T ⊆ U isarithmetic. (cid:3) Lemma 4.10.
Let R ⊂ S be a catenarian integral FCP extension.Then, R ⊆ tS R is a ∆ -extension if and only if T ⊆ tU T is a ∆ -extensionfor any subextension T ⊂ U of R ⊂ S . -EXTENSIONS 19 Proof.
One implication is obvious. Now, assume that R ⊆ tS R is a ∆-extension. Consider the tower R ⊆ T ⊂ U ⊆ S . Set R := tU R, R := tS R, T := tT R and T := tU T . There is no harm to assume that T = T .We get the following commutative diagram, where i , (resp. t ) indicatesan infra-integral (resp. t-closed) extension: T t → T i → T t → Ui ↑ i ց t ր ↓ R i → R i → R t → S Since R is the t-closure of R in U and R ⊆ T ⊆ U ⊆ S , we getthat T ⊆ R ⊆ T and R ⊆ R . Then, R is also the t-closureof T in T . It follows from the definitions of R , R , T and T that T ⊆ T, R ⊆ T , T ⊆ U and R ⊆ S are t-closed, while R ⊆ T , T ⊆ R , R ⊆ R , T ⊆ T and R ⊆ R are infra-integral. Since R ⊆ R is a ∆-extension, so is T ⊆ R . Let N ∈ MSupp T ( T /T )and set M := N ∩ T ∈ Max( T ). Then M ∈ MSupp T ( T /T ) and( T ) M ⊂ ( T ) M is not t-closed, whence ( T ) M = ( R ) M , from whchwe deduce that M ∈ MSupp T ( R /T ). Now Lemma 4.8 shows that M MSupp T ( T /T ). In particular, N is the only maximal ideal of T lying above M . Then, we have ( T ) M = T M = T N by [6, Lemma 2.4].For the same reason, ( T ) M = ( T ) N = ( R ) M , where the last equalityis consequence of the following facts: ( T ) M ⊂ ( T ) M is infra-integraland R ⊂ T is t-closed with R ∈ [ T , T ]. Then, ( T ) M = T M = T N ⊂ ( T ) N = ( T ) M = ( R ) M is a ∆-extension because so is R ⊆ R .Since this property holds for any N ∈ MSupp T ( T /T ), we get that T ⊆ T = tU T is a ∆-extension by Proposition 3.6. (cid:3) The proof of the next theorem mimics the proof of Theorem 4.5.
Theorem 4.11.
A catenarian integral FCP extension R ⊂ S is a ∆ -extension if and only if R ⊆ tS R and tS R ⊆ S are ∆ -extensions. Thelast condition is equivalent to tS R ⊆ S is arithmetic.Proof. One implication is obvious because of Proposition 4.7 and Corol-lary 3.2. Conversely, assume that R ⊂ S is a catenarian FCP integralextension such that R ⊆ tS R is a ∆-extension and tS R ⊆ S is arithmetic.Let V, W ∈ [ R, S ] and set T := V ∩ W, U := V W and U := tU T (resp. V := tV T, W := tW T ).According to Lemma 4.10, we get that T ⊆ U is a ∆-extension. Inparticular, V + W = V W (1) since V , W ∈ [ T, U ].We get the following commutative diagram, where i , (resp. t ) indi-cates an infra-integral (resp. t-closed) extension: V t → V i → V W i ր i ց t ր t ց T V W U = V Wi ց i ր t ց t ր W t → W i → V W Since V ⊆ V is t-closed and V ⊆ V W is infra-integral because T ⊆ V , W ⊆ U implies T ⊆ V W ⊆ U , the t-closure of T in U , so that T ⊆ U is infra-integral, Lemma 4.8 gives V + V W = V V W = V W (2). For the same reason, we get W + V W = V W (3). Now, V W ⊆ V W and V W ⊆ V W are t-closed. In-deed, Lemma 4.8 yields MSupp V ( V /V ) ∩ MSupp V ( V W /V ) = ∅ and MSupp V ( V /V ) ∪ MSupp V ( V W /V ) = MSupp V ( V W /V ). Let M ∈ MSupp V ( V W /V ).If M ∈ MSupp V ( V /V ), then ( V ) M = ( V ) M ( W ) M entails that( V ) M = ( V ) M ( W ) M ⊂ V M ( W ) M = ( V ) M V M ( W ) M = ( V ) M V M = V M is t-closed. If M ∈ MSupp V ( V W /V ), then ( V ) M = V M , so that( V ) M ( W ) M = V M ( W ) M . It follows that V W ⊂ V W is t-closed.For the same reason, V W ⊂ V W is t-closed.Now, V W ⊂ V W is also t-closed: assume that the contrary holds.Let U ′ be the t-closure of V W in V W and M ∈ MSupp( U ′ /V W ).Lemma 4.8 implies that M MSupp(
V W /V W ) ∪ MSupp( V W/V W )because R ⊂ S is catenarian. Then, ( V W ) M = V M ( W ) M = ( V W ) M = ( V ) M ( W ) M = ( V W ) M = ( V ) M W M , which leads to ( V W ) M =( V W ) M , a contradiction because M ∈ MSupp( U ′ /V W ) and U ′ ∈ [ V W , V W ].Then, V W ⊂ V W is a ∆-extension by Proposition 4.7 and Lemma4.9, so that
V W + V W = V W V W = V W (4).Now (2) and (3), using (1) and (4), combine to yield that
V W = V W + V W = V + V W + W + V W = V + V + W + W + V + W = V + W .To conclude, we obtained V + W = V W for any
V, W ∈ [ R, S ] and R ⊂ S is a ∆-extension. (cid:3) Corollary 4.12.
An unbranched integral FCP extension R ⊂ S is a ∆ -extension if and only if R ⊂ S is pinched at tS R, R ⊆ tS R is a ∆ -extension and tS R ⊆ S is chained. In particular, if these conditionshold, R ⊂ S is catenarian.Proof. Since R ⊂ S is integral unbranched, then S is local. Let N bethe maximal ideal of S . Then N = ( tS R : S ) and is also the maximalideal of tS R because tS R ⊆ S is t-closed [6, Proposition 4.10]. Moreover, -EXTENSIONS 21 + S R = tS R because R ⊂ S is spectrally bijective by Proposition 2.10, sothat any T ∈ [ R, S ] is local [25, Lemma 3.29] and R ⊆ tS R is subintegral.Assume that R ⊂ S is a ∆-extension, and then is catenarian byProposition 3.13. Then R ⊆ tS R is a ∆-extension by Corollary 3.2 and tS R ⊆ S is chained by Proposition 4.7. Moreover, R ⊂ S is pinched at tS R according to [26, Theorem 4.13].Conversely, assume that R ⊂ S is pinched at tS R, R ⊆ tS R is a ∆-extension and tS R ⊆ S is chained. Let C := { R i } ni =0 be a maximalchain of [ R, S ]. Since any R i ∈ [ R, tS R ] ∪ [ tS R, S ] and R i − ⊂ R i isminimal for each i ∈ N n , there exists some k ∈ { , . . . , n } such that R k = tS R . If k = 0, then R ⊂ S is t-closed and chained, then a ∆-extension. If k = n , then R ⊂ S is a ∆-extension by assumption. Now,assume that k ∈ N n − . It follows that { R i } ki =0 is a maximal chain of[ R, tS R ] and { R i } ni = k is a maximal chain of [ tS R, S ]. But, ℓ ( C ) = n = k + ( n − k ) = ℓ [ R, tS R ] + ℓ [ tS R, S ] because R ⊆ tS R is a ∆-extension andthen catenarian by Proposition 3.13 and tS R ⊆ S is chained. Hence,all the maximal chains of [ R, S ] have the same length. Then R ⊂ S iscatenarian and is a ∆-extension by Theorem 4.11. (cid:3) In case of a t-closed extension, we recover Proposition 4.7, but thisProposition was necessary to prove Theorem 4.11 and Corollary 4.12.
Example 4.13.
Here is an example of an integral FCP ∆-extension k ⊂ S such that k is a local ring but S is not a local ring, so thatthe extension is branched. As a corollary we get that k ⊂ k is a∆-extension.Let k be a field, k ⊂ k the ring extension defined as the diagonalmap, { e , e , e } the canonical basis of k , whose elements are idempo-tents of the ring k and k ⊂ K a minimal field extension. We knowthat k ⊂ k has FIP by [21, Proposition 2.1]. According to [7, Proposi-tion 4.15], [ k, k ] = { k, R , R , R , k } , where R i := ke i + k (1 − e i ), for i = 1 , ,
3. Set R := Ke + k (1 − e ) and S := Ke + ke + ke = Rk . Inparticular, N := ( k : S ) = ke + ke , is a maximal ideal of k and S . Itis easily seen that we have the following commutative diagram, where d , (resp. i ) indicates a minimal decomposed (resp. inert) extension.( R and R do not appear to make the situation clearer): Ri ր d ց k d → R Sd ց i ր k In particular, k ⊂ S is seminormal because so are k ⊂ k and k ⊂ S .Then k = + S k . Since k = tS k , obviously, R [ k, tS k ] ∪ [ tS k, S ]. We claimthat [ k, S ] = { k, R , R , R , k , R, S } . We first show that there does notexist some L ∈ [ k, S ] \ { R i } i =1 such that k ⊂ L is minimal. Supposethat the contrary holds. We proved above that k ⊂ L can be neitherdecomposed, nor ramified since L [ k, k ]. Then, k ⊂ L is inert, anda minimal field extension with some x ∈ L \ k . But L ⊂ S implies that x = ( y, z, t ), with y ∈ K and z, t ∈ k . Let P ( X ) ∈ k [ X ] be the (monic)minimal polynomial of x over k . It follows that P ( X ) is irreducible in k [ X ] and satisfies P ( y ) = P ( z ) = P ( t ) = 0, a contradiction since x k .Hence, there does not exists such L . We have just also proved that theonly minimal extensions of R are R ⊂ k and R ⊂ R . Indeed, R ⊂ R is the only minimal inert subextension of R ⊂ S startingfrom R and R ⊂ k is the only minimal decomposed subextension of R ⊂ S starting from R .Let i = 1, for instance i = 2 (the proof would be the same for i = 3).Then, R ⊂ k is the only minimal decomposed extension of R since k = tS k . There does not exist any U ∈ [ R , S ] such that R ⊂ U is minimal ramified for the same reason. Assume that there is some U ∈ [ R , S ] such that R ⊂ U is minimal inert. Moreover, R has twomaximal ideals N := ke and N := k (1 − e ) = k ( e + e ) with ( R : k ) = N . We claim that ( R : U ) = N . Indeed, by [9, Proposition7.1], ( R : U ) = N leads to a contradiction: k ⊂ S is not minimal.Then, N ∈ Max( U ). But N S = ( ke )( Ke + ke + ke ) = ke = N shows that N = ( R : S ) is also an ideal of U , a contradiction since N + N = R . It follows that k ⊂ k and k ⊂ R are the only extensionsof k of length 2. Since k ⊂ S and R ⊂ S are minimal, we get that[ k, S ] = { k, R , R , R , k , R, S } = [ k, k ] ∪ { R, S } . It remains to showthat U + V = U V for any
U, V ∈ [ k, S ]. The result is obvious if either U, V ∈ [ k, k ] or U ∈ [ k, k ] and V = S , or U, V ∈ {
R, S } . The two lastcases are when either U ∈ [ k, k ] and V = R , or U = R and V ∈ [ k, k ],which are also obvious. Then, k ⊂ S is a ∆-extension. In particular, k ⊂ k is a ∆-extension, by Corollary 3.2 because k is a subextensionof S = Rk .According to Theorem 4.11, we can reduce our study to infra-integralFCP extensions. These extensions are catenarian by Proposition 2.12.We will see in the next subsection that for such extensions, the ∆-property is linked to extensions of length 2.4.2. Properties linked to extensions of length 2.
We begin withthe following lemmas. -EXTENSIONS 23
Lemma 4.14.
Let R ⊂ S be an FCP extension and T, U ∈ [ R, S ] , T = U be such that ℓ [ T ∩ U, T ] + ℓ [ T ∩ U, U ] = ℓ [ T ∩ U, T U ] . If R ⊂ T and R ⊂ U are minimal, so are T ⊂ T U and U ⊂ T U . In particular, R = T ∩ U .Proof. Since we have the chain R ⊆ T ∩ U ⊆ T with R ⊂ T minimal,it follows that either T ∩ U = R or T ∩ U = T . In the last case, weshould have R ⊂ T ⊂ U , with R ⊂ U minimal, a contradiction. Then, R = T ∩ U and ℓ [ R, T ] + ℓ [ R, U ] = 2 = ℓ [ R, T U ], with
T, U ⊂ T U bothminimal. (cid:3)
Lemma 4.15.
Let R ⊂ S be an FCP extension and T, U ∈ [ R, S ] , T = U be such that R ⊂ T U is infra-integral with R ⊂ T and R ⊂ U minimal. Then, R ⊂ T U is a ∆ -extension if and only if ℓ [ R, T U ] = 2 .Proof.
Assume first that R ⊂ T U is a ∆-extension. Then, [
R, T U ] ismodular by Proposition 3.13 and ℓ [ R, T U ] = 2 by Lemma 2.13.Conversely, assume that ℓ [ R, T U ] = 2 and let
V, W ∈ [ R, T U ]. Weare aiming to show that V + W = V W . The result is obvious if either V = W , or V ∈ { R, T U } , or W ∈ { R, T U } . In the other cases, R ⊂ V, W and
V, W ⊂ T U are minimal infra-integral extensions. Inparticular,
V W = T U . Set M := C ( R, V ) and N := C ( R, W ).Assume first that M = N . Then, Corollary 3.12 shows that R ⊂ T U is a ∆-extension.Assume now that M = N . In particular, M = ( R : T U ). UsingProposition 3.6, working with the extension
R/M ⊂ ( T U ) /M andsetting k := R/M , we get that dim k ( V /M ) = dim k ( W/M ) = 2 (byminimality of the infra-integral extensions R ⊂ V and R ⊂ W andTheorem 2.3). This leads to dim k (( V + W ) /M ) = 3. Since k ⊂ V /M and k ⊂ W/M are still minimal infra-integral extensions, there exist x ∈ ( V /M ) \ k and y ∈ ( W/M ) \ k such that V /M = k [ x ] = k + kx and W/M = k [ y ] = k + ky . Moreover, we may choose x (resp. y ) generatingthe (or one of the two) maximal ideal(s) of V /M (resp.
W/M )). Sincewe still have ℓ [ k, ( T U ) /M ] = ℓ [ k, ( V W ) /M ] = 2, it follows from [9,Proposition 7.6] that xy = 0 so that ( V W ) /M = k [ x, y ] = k + kx + ky ,giving dim k (( V W ) /M ) = 3. To end, ( V + W ) /M ⊆ ( V W ) /M leads to V + W = V W and R ⊂ T U is a ∆-extension. (cid:3)
Theorem 4.16.
An infra-integral FCP extension of length two is a ∆ -extension.Proof. Let R ⊂ S be an infra-integral FCP extension of length two.Let T, U ∈ [ R, S ] , T = U be such that R ⊂ T and R ⊂ U are minimal.Since T ⊂ T U ⊆ S , it follows that T U = S , giving ℓ [ R, T U ] = 2.Hence, R ⊂ S is a ∆-extension by Lemma 4.15. (cid:3) Lemma 4.17.
Let R ⊂ S be an infra-integral FCP extension. For any T, U, V ∈ [ R, S ] , U = V such that U ⊂ T and V ⊂ T are minimal,then ℓ [ U ∩ V, T ] = 2 .Proof.
Set W := U ∩ V, M := ( U : T ) and N := ( V : T ). Weuse several dual results of Proposition 2.15. We claim that M = N .Deny. Then, [9, Proposition 5.7] asserts that M ∈ Max( W ), so that M = ( W : U ), a contradiction because W ⊂ U is infra-integral andnot t-closed. Then, M = N .If either M and N are incomparable or not, at least one of W ⊂ U and W ⊂ V is minimal infra-integral by [9, Proposition 6.6]. It followsthat ℓ [ W, T ] = 2 because W ⊂ T is infra-integral, and then catenarianaccording to Proposition 2.12. (cid:3) Theorem 4.18.
Let R ⊂ S be an infra-integral FCP extension. Thefollowing conditions are equivalent: (1) R ⊂ S is a ∆ -extension, (2) R ⊂ S is modular, (3) ℓ [ T ∩ U, T ] + ℓ [ T ∩ U, U ] = ℓ [ T ∩ U, T U ] for any T, U ∈ [ R, S ] , (4) R ⊂ S is semi-modular, (5) ℓ [ T, U V ] = 2 for any
T, U, V ∈ [ R, S ] such that T ⊂ U and T ⊂ V are minimal with U = V .Proof. (1) ⇒ (2) is Proposition 3.13.(2) ⇔ (3) by [28, Theorem 4.15].(3) ⇒ (1) Assume that ℓ [ T ∩ U, T ] + ℓ [ T ∩ U, U ] = ℓ [ T ∩ U, T U ] forany
T, U ∈ [ R, S ]. Let
V, W ∈ [ R, S ]. We claim that V + W = V W .Choose two maximal chains R := V ⊂ V ⊂ . . . ⊂ V i ⊂ . . . ⊂ V n := V and R := W ⊂ W ⊂ . . . ⊂ W j ⊂ . . . ⊂ W m := W such that V i ⊂ V i +1 and W j ⊂ W j +1 are minimal infra-integral for each i ∈ { , . . . , n − } and each j ∈ { , . . . , m − } . We are going to show by inductionon k ∈ { , . . . , m + n } that V i − W j ⊂ V i W j and V i W j − ⊂ V i W j areminimal and that V i + W j = V i W j for each i, j ≥ i + j ≤ k, i ≤ n, j ≤ m .Let k = 2, so that i = j = 1. Since R ⊂ V and R ⊂ W are minimal,so are V = V W ⊂ V W and W = V W ⊂ V W by Lemma 4.14.Moreover, ℓ [ R, V W ] = 2, so that V + W = V W by Lemma 4.15.Then, the induction hypothesis holds for k = 2.Assume that the induction hypothesis holds for k < m + n , so thatfor any i, j such that i + j ≤ k , we have V i − W j , V i W j − ⊂ V i W j minimal and V i + W j = V i W j . Let α, β be such that α + β = k + 1.Then, ( α −
1) + β = α + ( β −
1) = k and the induction hypothesis holds -EXTENSIONS 25 for k . Consider the following commutative diagram: V α − → V α → V α W β − ց ր ց V α − W β − V α W β ր ց ր W β − → W β → V α − W β It follows that V α − W β − ⊂ V α − W β , V α W β − are minimal and so are V α − W β , V α W β − ⊂ V α W β by Lemma 4.14. Now, ℓ [ V α − W β − , V α W β ] =2, so that V α − W β + V α W β − = ( V α − W β )( V α W β − ) = V α W β ( ∗ ) byLemma 4.15. But V α − W β = V α − + W β and V α W β − = V α + W β − bythe induction hypothesis for k . Then, ( ∗ ) yields V α W β = V α − W β + V α W β − = V α − + W β + V α + W β − = V α + W β . Then, the inductionhypothesis holds for k + 1. As it holds for any k , in particular, for k = n + m , we get V n + W m = V + W = V n W m = V W . Then, R ⊂ S is a ∆-extension.(2) ⇒ (4) See the properties at the beginning of Subsection 2.3.(4) ⇒ (5) Assume that T ⊂ U and T ⊂ V are minimal with U = V ,so that T = U ∩ V . (4) implies that U ⊂ U V and V ⊂ U V are minimalby definition of a semi-modular extension. It follows that ℓ [ T, U V ] = 2,giving (5).(5) ⇒ (2) In order to prove that R ⊂ S is modular, we are goingto show that both covering conditions are satisfied, that is, for each T, U ∈ [ R, S ] such that T ∩ U ⊂ T is minimal, then U ⊂ T U isminimal, and for each
T, U ∈ [ R, S ] such that U ⊂ T U is minimal,then T ∩ U ⊂ T is minimal (see [28, Definition page 105 and Theorem4.15]). First, let T, U ∈ [ R, S ] be such that T ∩ U ⊂ T is minimal.We prove that U ⊂ T U is minimal by induction on l := ℓ [ T ∩ U, U ].If l = 1, then T ∩ U ⊂ U is minimal and U ⊂ T U is minimal by (5)since ℓ [ T ∩ U, T U ] = 2. Assume that the induction hypothesis holdsfor l − U ′ ∈ [ T ∩ U, U ] be such that U ′ ⊂ U is minimal, sothat T ∩ U = T ∩ U ′ and ℓ [ T ∩ U ′ , U ′ ] = l −
1. From the inductionhypothesis, we deduce that U ′ ⊂ T U ′ is minimal. Using (5) for theminimal extensions U ′ ⊂ T U ′ and U ′ ⊂ U , we get that ℓ [ U ′ , T U ] = 2,so that U ⊂ T U is minimal.In the same way, let
T, U ∈ [ R, S ] be such that U ⊂ T U is minimal.We prove that T ∩ U ⊂ T is minimal by induction on r := ℓ [ T, T U ]. If r = 1, the result holds by Lemma 4.17 since T, U ⊂ T U are minimal.Assume that the induction hypothesis holds for r − T ′ ∈ [ T, T U ] be such that T ⊂ T ′ is minimal, so that T U = T ′ U and ℓ [ T ′ , T ′ U ] = r −
1. From the induction hypothesis, we deduce that T ′ ∩ U ⊂ T ′ is minimal. Using again Lemma 4.17 for the minimal extensions T ⊂ T ′ and T ′ ∩ U ⊂ T ′ , we get that T ∩ U ⊂ T is minimal.To conclude, R ⊂ S is modular. (cid:3) We may remark that (1) ⇔ (2) in the above theorem re-proves The-orem 4.16 by Proposition 2.14. Proposition 4.19.
Let k be a field and n a positive integer, n > .Then, k ⊂ k n is an FIP ∆ -extension if and only if n ≤ .Proof. According to [21, Proposition 2.1], for any positive integer n, k ⊂ k n is an FIP extension. For n = 2, k ⊂ k is a minimal extension by[11, Lemme 1.2] and then a ∆-extension by Proposition 3.11, sincearithmetic. Example 4.13 asserts that the result is valid for n = 3.Let n ≥ B := { e , . . . , e n } be the canonical basis of the k -algebra k n . Set x := e + e and y := e + e . Then, xy = e ∈ k [ x, y ]and xy k [ x ] + k [ y ] = k [ e + e ] + k [ e + e ] = k + k ( e + e ) + k ( e + e ).Using Proposition 3.1, we get that k ⊂ k n is not a ∆-extension. (cid:3) Corollary 4.20.
Let R ⊂ S be a seminormal infra-integral FCP ex-tension. The following conditions are equivalent: (1) R ⊂ S is a ∆ -extension; (2) | V S ( M S ) | ≤ for any M ∈ MSupp(
S/R ) ; (3) ℓ [ R M , S M ] ≤ for any M ∈ MSupp(
S/R ) .Proof. (1) ⇔ (2) According to Proposition 3.6, R ⊂ S is a ∆-extensionif and only if R M ⊂ S M is a ∆-extension for any M ∈ MSupp(
S/R ).Moreover, | V S ( M S ) | ≤ M ∈ MSupp(
S/R ) if and only if | Max( S M ) | ≤ M ∈ MSupp(
S/R ). Indeed, Max( S M ) = { N S M | N ∈ Max( S ) , M ⊆ N } = { N S M | N ∈ Max( S ) , M ∈ V S ( M S ) } . Therefore, we can assume that ( R, M ) is a local ring. Since R ⊂ S is a seminormal FCP extension, we deduce from [6, Proposition4.9] that ( R : S ) = M is an intersection of finitely many maximal idealsof S . Moreover, R ⊂ S being infra-integral, S/M ∼ = ( R/M ) n , where n := | Max( S ) | by the Chinese Remainder Theorem. Moreover, R ⊂ S is a ∆-extension if and only if R/M ⊂ S/M is a ∆-extension by Propo-sition 3.6. Since
R/M is a field,
R/M ⊂ ( R/M ) n is a ∆-extension if andonly if n ≤ R ⊂ S is a ∆-extensionif and only if | V S ( M S ) | ≤ M ∈ MSupp(
S/R ).(2) ⇔ (3) by [6, Lemma 5.4] since ℓ [ R M , S M ] = | Max( S M ) | − | V S ( M S ) | − (cid:3) Applications of extensions of length 2 to ∆ -extensions. Before giving a more striking characterization of FCP infra-integral∆-extensions, we need the following technical lemmas. -EXTENSIONS 27
Lemma 4.21.
Let R ⊂ S be an infra-integral FCP extension over thelocal ring ( R, M ) , such that R ⊂ + S R is a ∆ -extension and + S R ⊂ S is minimal decomposed. Let T, U ∈ [ R, S ] , T [ R, + S R ] be such that T ⊂ U is subintegral. Then the following hold: (1) | Max( S ) | = | Max( T ) | = | Max( U ) | = 2 . (2) Let V ∈ [ T, U ] and set W := V ∩ + S R = + V R . Then, ℓ [ T, V ] = ℓ [ + T R, W ] . (3) For any V , V ∈ [ T, U ] such that T ⊂ V , V are minimal, then V + V = V V and V , V ⊂ V V are minimal (ramified). (4) T ⊂ U is a ∆ -extension.Proof. (1) Since ( R, M ) is a local ring, so is + S R and | Max( S ) | = 2since + S R ⊂ S is minimal decomposed. We also have | Max( T ) | = 2,because T [ R, + S R ] implies that R ⊂ T is not subintegral, whichgives that | Max( R ) | < | Max( T ) | ≤ | Max( S ) | = 2. It follows that | Max( U ) | = | Max( T ) | = 2 because T ⊂ U is subintegral.(2) Let V ∈ [ T, U ] and W := V ∩ + S R = + V R . Since T ⊂ U issubintegral, so is T ⊂ V , which implies that | Max( V ) | = 2. But, + T R ⊆ T and W ⊆ V are seminormal infra-integral. Therefore, + T R ⊆ T and W ⊆ V are minimal decomposed and ℓ [ + T R, T ] = ℓ [ W, V ] = 1.Now, ℓ [ + T R, V ] = ℓ [ + T R, T ]+ ℓ [ T, V ] = ℓ [ + T R, W ]+ ℓ [ W, V ] because R ⊂ S is infra-integral, whence catenarian by Proposition 2.12. So, we get ℓ [ T, V ] = ℓ [ + T R, W ].(3) Let V , V ∈ [ T, U ] be such that T ⊂ V , V are minimal (rami-fied). As in (2), set W i := + S R ∩ V i = + V i R for i = 1 ,
2, which are localrings. Since T ⊂ V V is subintegral, because V V ∈ [ T, U ], there is noharm to assume that U = V V . As ( T : V ) , ( T : V ) ∈ Max( T ), weconsider two cases.(a) ( T : V ) = ( T : V ).Propositions 2.15 (1) and 3.4 and Lemma 4.15 say that V + V = V V . Moreover, Proposition 2.15 shows that V , V ⊂ V V are minimal(ramified).(b) ( T : V ) = ( T : V ).Set N := ( T : V ) = ( T : V ). For each i ∈ { , } , there is a unique N i ∈ Max( V i ) lying above N since Max( U ) → Max( T ) is bijective.Then, M i := N i ∩ W i is the maximal ideal of W i . From (2), we deducethat + T R ⊂ W i is minimal ramified with conductor M ′ := N ∩ + T R , themaximal ideal of + T R . In particular, W , W ∈ [ + T R, + S R ], where + T R ⊆ + S R is a ∆-extension, because so is R ⊆ + S R . Then, ℓ [ + T R, W W ] = 2 byLemma 4.15. An application of Proposition 2.15 shows that M M ⊆ M ′ ( ∗ ). Since | Max( T ) | = | Max( V i ) | = 2, let N ′ (resp. N ′ i ) be the othermaximal ideal of T (resp. of V i ). Then, M ′ = N N ′ because + T R ⊂ T is minimal decomposed. For the same reason, we have M i = N i N ′ i byTheorem 2.3. Then ( ∗ ) yields N N ′ N N ′ = M M ⊆ M ′ = N N ′ ( ∗∗ ).Since N ′ MSupp( V i /T ), we have T N ′ = ( V i ) N ′ = N N ′ = ( N i ) N ′ .Moreover, N ′ N = T N and ( N ′ i ) N = ( V i ) N . Localizing ( ∗∗ ) at the twomaximal ideals N, N ′ of T , we get ( N N ′ N N ′ ) N = ( N N ) N ⊆ N N and ( N N ) N ′ = T N ′ = N N ′ . It follows that N N ⊆ N , so that ℓ [ T, V V ] = 2 by Proposition 2.15 and V + V = V V since T ⊆ V V isa ∆-extension by Lemma 4.15. In particular, V ⊂ V V and V ⊂ V V are minimal (ramified) extensions.(4) Let V, V ′ ∈ [ T, U ]. We want to show that V + V ′ = V V ′ . Since R ⊂ S has FCP, there exists a maximal chain V := T ⊂ V ⊂ . . . ⊂ V i ⊂ · · · ⊂ V n := V and a maximal chain V ′ := T ⊂ V ′ ⊂ · · · ⊂ V ′ j ⊂ · · · ⊂ V ′ m := V ′ such that V i ⊂ V i +1 and V ′ j ⊂ V ′ j +1 are minimalextensions, for each i = 0 , . . . , n − , j = 0 , . . . , m −
1. We mimic theproof of Theorem 4.18 although the assumptions are not the same.We are going to show by induction on k ∈ { , . . . , m + n } that V i − V ′ j ⊂ V i V ′ j and V i V ′ j − ⊂ V i V ′ j are minimal and V i + V ′ j = V i V ′ j foreach i, j ≥ i + j ≤ k, i ≤ n, j ≤ m .If k = 2, then i = j = 1. Since T ⊂ V and T ⊂ V ′ are minimal, V + V ′ = V V ′ by (3), and the induction hypothesis holds for k = 2.Assume that the induction hypothesis holds for k . Hence, for any i, j such that i + j ≤ k , we have that V i − V ′ j , V i V ′ j − ⊂ V i V ′ j are minimaland V i + V ′ j = V i V ′ j . Let α, β be such that α + β = k + 1 and considerthe following commutative diagram: V α − → V α → V α V ′ β − ց ր ց V α − V ′ β − V α V ′ β ր ց ր V ′ β − → V ′ β → V α − V ′ β Then, ( α −
1) + β = α + ( β −
1) = k . Since the induction hypothesisholds for k , it follows that V α − V ′ β − ⊂ V α − V ′ β , V α V ′ β − are minimal.Moreover, V α − V ′ β − , V α V ′ β ∈ [ T, U ], implies that V α − V ′ β − ⊂ V α V ′ β issubintegral. Applying (3) to this extension, we get that V α − V ′ β + V α V ′ β − = ( V α − V ′ β )( V α V ′ β − ) = V α V ′ β ( ∗ ). But V α − V ′ β = V α − + V ′ β and V α V ′ β − = V α + V ′ β − by the induction hypothesis for k . Then, ( ∗ )yields V α V ′ β = V α − V ′ β + V α V ′ β − = V α − + V ′ β + V α + V ′ β − = V α + V ′ β .Moreover, V α − V ′ β , V α V ′ β − ⊂ V α V ′ β are minimal by (3). Then, theinduction hypothesis holds for k +1. As it holds for any k , in particular,for k = n + m , we get V n + V ′ m = V + V ′ = V n V ′ m = V V ′ . Then, T ⊂ U is a ∆-extension. (cid:3) -EXTENSIONS 29 Corollary 4.22.
Let R ⊂ S be an infra-integral FCP extension overthe local ring ( R, M ) , such that R ⊂ + S R is a ∆ -extension and + S R ⊂ S is minimal decomposed. Let T, U ∈ [ R, S ] be such that T ⊂ U issubintegral. Then T ⊂ U is a ∆ -extension.Proof. According to Lemma 4.21, it is enough to verify only the casewhere T ∈ [ R, + S R ], which is obvious since U ∈ [ R, + S R ] and R ⊂ + S R isa ∆-extension (use Corollary 3.2). (cid:3) Lemma 4.23.
Let R ⊂ S be an infra-integral FCP extension suchthat ( R, M ) is a local ring, R ⊂ + S R and + S R ⊂ S are ∆ -extensions. Let T, U ∈ [ R, S ] , T ([ R, + S R ] ∪ [ + S R, S ]) be such that T ⊂ U is subintegral.Then, T ⊂ U is a ∆ -extension.Proof. Since + S R ⊂ S is a ∆-extension, | Max( U ) | ≤ | Max( S ) | ≤ + S R = S , we have | Max( S ) | ≥ ℓ [ + S R, S ] ≥
1, and there is at least one minimal decomposed extension + S R ⊂ V with V ∈ [ + S R, S ]. Then, use Theorem 2.3. We are done if | Max( S ) | = 2 by Lemma 4.21, since in this case + S R ⊂ S is minimaldecomposed. So, assume that | Max( S ) | = 3. If | Max( U ) | = 2, we mayapply Lemma 4.21 to the extension R ⊂ U because + U R ⊆ + S R and + U R ⊂ U is minimal decomposed, in which case we are done. Assumenow that | Max( U ) | = | Max( T ) | = 3 since T ⊂ U is subintegral. Then, ℓ [ + T R, T ] = | Max( T ) | − + T R ⊂ T isseminormal infra-integral. It follows that there exists T ′ ∈ [ + T R, T ]such that + T R ⊂ T ′ and T ′ ⊂ T are minimal decomposed. In particular, | Max( T ′ ) | = 2. Set U ′ := T ′ + U R ∈ [ + U R, U ] , n := ℓ [ + T R, + U R ], R := + T R and T = T ′ . We are going to prove that there is some m ≤ n whichwill be exhibited below and by induction on i ∈ N m , such that for each i ∈ N m , there exist R i ∈ [ + T R, + U R ] and T i ∈ [ T ′ , U ′ ] where T i = T ′ R i , ℓ [ R i − , R i ] ∈ { , } , R i ⊂ T i is minimal decomposed and T ′ ⊂ T i is subintegral. Since R ⊂ S has FCP, there exists R ′ ∈ [ + T R, + U R ]such that R ⊂ R ′ is minimal ramified. Consider T ′ R ′ ∈ [ T ′ , U ′ ].Using Proposition 2.15, two cases occur according to the behavior ofthe maximal ideals of T ′ and R ′ :Case (a): T ′ ⊂ T ′ R ′ is minimal ramified and R ′ ⊂ T ′ R ′ is minimaldecomposed. Then, we set T := T ′ R ′ ∈ [ T ′ , U ′ ] and R := R ′ . It fol-lows that R ⊂ R is minimal ramified, R ⊂ T is minimal decomposedand T ′ ⊂ T is subintegral.Case (b): We use [9, Proposition 7.6]. There exists T ′ ∈ [ T ′ , T ′ R ′ ]such that T ′ ⊂ T ′ and T ′ ⊂ T ′ R are minimal ramified and R ′′ ∈ [ R ′ , T ′ R ′ ] such that R ′′ ⊂ T ′ R ′ is minimal decomposed and R ′ ⊂ R ′′ isminimal ramified. We have the following commutative diagram where the horizontal maps are ramified and the vertical maps are decomposed: T ′ = T → T ′ → T ′ R ′ ↑ ↑ + T R = R → R ′ → R ′′ Set T := T ′ R ′ ∈ [ T ′ , U ′ ] and R := R ′′ . It follows that ℓ [ R , R ] = 2, R ⊂ T is minimal decomposed, T ′ ⊂ T is subintegral and T = T ′ R .The induction hypothesis holds for i = 1.Assume that the induction hypothesis holds for i ∈ N n − , so thatthere exist R i ∈ [ + T R, + U R [ and T i ∈ [ T ′ , U ′ ] such that ℓ [ R i − , R i ] ∈{ , } , R i ⊂ T i is minimal decomposed, T ′ ⊂ T i is subintegral and T i = T ′ R i . We mimic the proof made for i = 1, and use again Proposition2.15 and [9, Proposition 7.6]. Since R i ∈ [ + T R, + U R [, there exists R ′ i +1 ∈ [ + T R, + U R ] such that R i ⊂ R ′ i +1 is minimal ramified, and two cases occuraccording to the behavior of the maximal ideals of T i and R ′ i +1 (similarto the case i = 1).In case (a): T i ⊂ T i R ′ i +1 is minimal ramified and R ′ i +1 ⊂ T i R ′ i +1 is minimal decomposed. Then, we set R i +1 := R ′ i +1 ∈ [ + T R, + U R [ and T i +1 := T i R i +1 ∈ [ T ′ , U ′ ]. It follows that R i ⊂ R i +1 is minimal ramified, R i +1 ⊂ T i +1 is minimal decomposed, T ′ ⊂ T i +1 is subintegral and T i +1 = T i R i +1 = T ′ R i R i +1 = T ′ R i +1 . Then the induction hypothesisholds for i + 1.In case (b): There exists T ′ i ∈ [ T i , T i R ′ i +1 ] ⊆ [ T ′ , U ′ ] such that T i ⊂ T ′ i and T ′ i ⊂ T i R ′ i +1 are minimal ramified and R ′′ i +1 ∈ [ R i , T i R ′ i +1 ] suchthat R ′′ i +1 ⊂ T i R ′ i +1 is minimal decomposed and R ′ i +1 ⊂ R ′′ i +1 is minimalramified. We do not draw a new diagram, it is enough to replace 0 with i and 1 with i + 1 in the above diagram. Then we set T i +1 := T i R ′ i +1 and R i +1 := R ′′ i +1 . It follows that ℓ [ R i , R i +1 ] = 2 , R i +1 ⊂ T i +1 isminimal decomposed, T ′ ⊂ T i +1 is subintegral and T i +1 = T i R ′ i +1 = T i R ′ i +1 R ′′ i +1 = T i R ′′ i +1 = T ′ R i R ′′ i +1 = T ′ R ′′ i +1 = T ′ R i +1 because R ′ i +1 ⊂ R i +1 ⊂ T i R ′ i +1 . The induction hypothesis holds for i + 1.Once R i +1 = + U R is gotten, we set m := i + 1, in which case T m = T ′ + U R = U ′ ∈ [ + U R, U ], with + U R ⊂ U ′ minimal decomposed and T ′ ⊂ U ′ subintegral. We can apply Lemma 4.21 to the extension R ⊂ U ′ , whichgives that T ′ ⊂ U ′ is a ∆-extension. We may remark that when weget that R i ⊂ + U R is minimal (ramified), we are necessarily in case (a)since R ′ i +1 = + U R .Set Max( T ′ ) := { M, M ′ } , so that ( T ′ M , M M ) is a local ring. We arefirst going to show that T M ⊆ U M and T M ′ ⊆ U M ′ are ∆-extensions.Observe that the context is as follows: T ′ M ⊂ U M is an infra-integralFCP extension where the local ring ( T ′ M , M M ) verifies U ′ M = + U M T ′ M , T ′ ⊂ U ′ is a ∆-extension and so is T ′ M ⊆ U ′ M by Proposition 3.6. Either -EXTENSIONS 31 T ′ M ⊆ T M is minimal decomposed or T ′ M = T M . In this last case, U M = U ′ M , so that T M ⊆ U M is a ∆-extension. If T ′ M ⊆ T M is mini-mal decomposed, the assumptions of Lemma 4.21 are satisfied for theextension T ′ M ⊆ U M . As T M ⊆ U M is a subintegral subextension of T ′ M ⊆ U M , we get that T M ⊆ U M is a ∆-extension. In the same way, T M ′ ⊆ U M ′ is a ∆-extension.We intend to show that T ⊆ U is a ∆-extension by applying twiceProposition 3.6. Indeed, | Max( T ) | = 3. Moreover, ( T ′ : T ) = M since T ′ M ⊆ T M is minimal decomposed. Let N, P ∈ Max( T ) be lying over M and N ′ ∈ Max( T ) be lying over M ′ . Then, T M ′ = T N ′ , U M ′ = U N ′ ,and T N , T P (resp. U N , U P ) are localizations of T M (resp. U M ), sothat T N ⊆ U N , T P ⊆ U P and T N ′ ⊆ U N ′ are ∆-extensions, and so is T ⊆ U . (cid:3) Corollary 4.24.
Let R ⊂ S be an infra-integral FCP extension overthe local ring ( R, M ) , such that R ⊂ + S R and + S R ⊂ S are ∆ -extensions.Then a subintegral subextension T ⊂ U of [ R, S ] is a ∆ -extension.Proof. According to Lemma 4.23, it is enough to assume that T ∈ [ R, + S R ] ∪ [ + S R, S ]. If T ∈ [ R, + S R ], so is U , then T ⊂ U is a ∆-extensionbecause R ⊂ + S R is a ∆-extension. If T ∈ [ + S R, S ], then T ⊂ U ⊆ S implies that U ∈ [ + S R, S ], a contradiction with T ⊂ U is subintegral.Then, in any case, T ⊂ U is a ∆-extension (cid:3) Proposition 4.25.
An infra-integral FCP extension R ⊂ S is a ∆ -extension if and only if the following statements hold: (1) R ⊂ + S R and + S R ⊂ S are ∆ -extensions. (2) For each
T, U, V ∈ [ R, S ] such that T ⊂ U is minimal ramifiedand T ⊂ V is minimal decomposed, T ⊂ U V is a B -extension(or, equivalently, a ∆ -extension, or, equivalently, ℓ [ T, U V ] = 2 ).Proof.
One implication is obvious in the light of Corollary 3.2 andLemma 4.15. Moreover, in (2), the three conditions are equivalent. In-deed, if T ⊂ U is minimal ramified and T ⊂ V is minimal decomposed,then T ⊂ U V is infra-integral by Proposition 2.10. Then, T ⊂ U V isa ∆-extension if and only if ℓ [ T, U V ] = 2 by Lemma 4.15. At last,[23,Theorem 6.1 (5)] gives that ℓ [ T, U V ] = 2 if and only if | [ T, U V ] | = 4.Conversely, assume that (1) and (2) hold. Since R ⊂ S is infra-integral, there is no minimal inert subextension of R ⊂ S , always byProposition 2.10. We can assume that ( R, M ) is a local ring, and sois + S R . It follows that | Max( S ) | ≤ ℓ [ T, U V ] = 2 for any
T, U, V ∈ [ R, S ] such that T ⊂ U and T ⊂ V are minimal with U = V .If they are minimal of different types, that is one is minimal ramified and the other minimal decomposed, ℓ [ T, U V ] = 2 by (2). Assumethat T ⊂ U and T ⊂ V are both minimal ramified, then T ⊂ U V is subintegral. If T ∈ [ R, + S R ], so are U and V , then T ⊂ U V is a∆-extension by (1), so that ℓ [ T, U V ] = 2 by Theorem 4.18. Moreover,we cannot have T ∈ [ + S R, S ] since T ⊂ U is minimal ramified. If T [ R, + S R ] ∪ [ + S R, S ], then T ⊂ U V is a ∆-extension by Lemma4.23, so that ℓ [ T, U V ] = 2 by Theorem 4.18. At last, assume that T ⊂ U and T ⊂ V are both minimal decomposed. By Proposition 2.15, ℓ [ T, U V ] ≤
3. But [9, Proposition 7.6 (b)] says that when ℓ [ T, U V ] = 3,there exists U ′ ∈ [ U, U V ] such that U ⊂ U ′ and U ′ ⊂ U V are bothminimal decomposed. It follows that | Max( T ) | + 3 = | Max(
U V ) | ≤| Max( S ) | ≤
3, a contradiction. Then, ℓ [ T, U V ] = 2. This equalityholding in any case, R ⊂ S is a ∆-extension by Theorem 4.18. (cid:3) Remark 4.26.
The condition (2) of Proposition 4.25 is necessary inorder to have a ∆-extension as it is shown in the following example.We use [27, Example 5 of 16.4] and its results. Let S be the ring ofintegers of Q ( √ , i ). Then S is a Z -module of basis B := { , √ , t, u } ,where t := ( √ i ) / u := (1 + i √ /
2. It is shown that 2 S = Q Q , where Q , Q are the maximal ideals of S lying above 2 Z . Set R := Z + 2 S and W := + S R . Then, 2 S = ( R : S ) is a maximalideal of R . It is easy to see that W = Z + Q Q , so that W ⊂ S isminimal decomposed, and then a ∆-extension. Hence, the second partof condition (1) of Proposition 4.25 is satisfied. Set U := Z + Q Q and V := Z + Q . A short calculation shows that R ⊂ U and V ⊂ S areboth minimal ramified, while U ⊂ V is minimal decomposed. Then, ℓ [ R, S ] = 3, which leads to ℓ [ R, W ] = 2. It follows that R ⊂ W isa ∆-extension by Theorem 4.16 and the first part of condition (1) ofProposition 4.25 is satisfied. We show that condition (2) of Proposition4.25 is not satisfied.Set T := R [ x ], where x := 1 + √
7. Since x = 8 + 2 √ ∈ S and2 xS ⊆ S , we get that R ⊂ T is minimal ramified by Theorem 2.3.Set T ′ := R [ u ]. Since u − u = − ∈ S and 2 uS ⊆ S , we getthat R ⊂ T ′ is minimal decomposed by Theorem 2.3. Now, T T ′ = R [ x, u ] = R [ √ , (1 + i √ / √ i √ / t + 3 i ∈ T T ′ . But, i = 2 t − √ ∈ T ⊂ T T ′ implies that t ∈ T T ′ . To conclude T T ′ = S and ℓ [ R, T T ′ ] > R ⊂ S is not a ∆-extension. We cancheck that T + T ′ is not a ring, because it does not contain √ u .We will see in Section 5 an example where conditions of Proposition4.25 hold in the context of number field orders. Lemma 4.27.
Let R ⊂ S be an integral FCP extension such that ℓ [ T, U V ] = 2 for each
T, U, V ∈ [ R, S ] such that T ⊂ U and T ⊂ V -EXTENSIONS 33 are minimal of different types. Then, for any maximal chain C ′ of [ R, S ] , there exists a maximal chain C of [ R, S ] containing tS R suchthat ℓ ( C ) = ℓ ( C ′ ) .Proof. Let C ′ be a maximal chain of [ R, S ]. According to [26, Propo-sition 4.11], there exists a maximal chain C of [ R, S ] containing tS R such that ℓ ( C ) ≥ ℓ ( C ′ ). Assume that ℓ ( C ) = ℓ ( C ′ ). The same ref-erence shows that there exist A, B, C ∈ C ′ such that A ⊂ B is in-ert, B ⊂ C is minimal non-inert, and ( A : B ) = ( B : C ). Set W := tC A = A, C because A ⊂ C is neither infra-integral nor t-closed,and M := ( A : B ) = ( B : C ) ∈ Max( A ). In particular, M = ( A : C ).Let W ∈ [ A, W ] be such that A ⊂ W is minimal non-inert. In par-ticular, M = ( A : W ). Then, Proposition 2.15 asserts that A ⊂ BW is not catenarian, a contradiction with ℓ [ A, BW ] = 2 by assumptionand [26, Proposition 3.4]. To conclude, ℓ ( C ) = ℓ ( C ′ ). (cid:3) Theorem 4.28.
An FCP extension R ⊂ S is a ∆ -extension if andonly if the following conditions hold: (1) R ⊂ + R R , + R R ⊂ tR R and tR R ⊆ R are ∆ -extensions (the lastcondition being equivalent to tR R ⊆ R is arithmetic). (2) For each
T, U, V ∈ [ R, R ] such that T ⊂ U and T ⊂ V areminimal of different types, then T ⊂ U V is a B -extension.Proof. Assume that R ⊂ S is a ∆-extension, and then catenarian, sothat (2) holds by Lemma 2.13 and Corollary 3.12. Moreover, (1) holdsbecause of Proposition 4.7 and Corollary 3.2.Conversely, assume that conditions (1) and (2) hold. By Proposition4.25, we get that R ⊆ tR R is a ∆-extension thanks to (1) and (2), andis catenarian since infra-integral.We now observe that (2) implies that R ⊆ R is catenarian. Indeed,any maximal chain from R to tR R has the same length ℓ [ R, tR R ] =: k ,and any maximal chain from tR R to R has the same length ℓ [ tR R, R ] := l because tR R ⊆ R is a ∆-extension by Proposition 3.13, whence catenar-ian. Now, let R := R ⊂ · · · ⊂ R i ⊂ · · · R n := R be a maximal chain C such that R i ⊂ R i +1 is a minimal extension for each i = 0 , . . . , n − m := k + l . If tR R ∈ C , then tR R = R k and ℓ ( C ) = n = m because R ⊆ tR R and tR R ⊆ R are catenarian. Now, assume that tR R C . Ac-cording to Lemma 4.27, (2) implies that there exists a maximal chain C ′ of [ R, S ] containing tR R such that ℓ ( C ) = ℓ ( C ′ ) = m . It follows thatany maximal chain of [ R, R ] have length m , and R ⊆ R is catenarian.Now, Theorem 4.11 shows that R ⊆ R is a ∆-extension, which im-plies at last that R ⊂ S is a ∆-extension by Theorem 4.5. (cid:3) Adding some assumptions in the statement of Theorem 4.28, we geta simpler characterization of ∆-extensions.
Corollary 4.29.
Let R ⊂ S be a catenarian FCP extension. Then, R ⊂ S is a ∆ -extension if and only if the following conditions hold: (1) R ⊂ + R R , + R R ⊂ tR R and tR R ⊆ R are ∆ -extensions (the lastcondition being equivalent to tR R ⊆ R is arithmetic). (2) For each
T, U, V ∈ [ R, R ] such that T ⊂ U and T ⊂ V areminimal non-inert of different types, then T ⊂ U V is a B -extension.Proof. One implication is Theorem 4.28. Conversely, in order to provethat R ⊂ S is a ∆-extension, it is enough to prove that (2) impliescondition (2) of Theorem 4.28. So, let T ⊂ U and T ⊂ V be minimalextensions of different types, with one of them inert. Since R ⊂ S isa catenarian extension, Proposition 2.14 says that C ( T, U ) = C ( T, V ),and [
T, U V ] = { T, U, V, U V } . Then, T ⊂ U V is a B -extension. There-fore, R ⊂ S is a ∆-extension. (cid:3) Corollary 4.30.
A modular FCP extension R ⊂ S is a ∆ -extensionif and only if tR R ⊆ R is arithmetic.Proof. If R ⊂ S is a ∆-extension, tR R ⊆ R is arithmetic by Corollary4.29. Conversely, assume that tR R ⊆ R is arithmetic. Then, Corollary4.29 (1) holds since R ⊂ + R R is modular, and then a ∆-extension byTheorem 4.18. Let T, U, V ∈ [ R, R ] be such that T ⊂ U and T ⊂ V are minimal non-inert of different types. Then, they are infra-integral,and so is T ⊂ U V , which is also modular. It follows that ℓ [ T, U V ] = 2by Theorem 4.18, so that [
T, U V ] = { T, U, V, U V } by [23, Theorem 6.1(5)]. Therefore, T ⊂ U V is a B -extension. To conclude, R ⊂ S is a∆-extension by Corollary 4.29. (cid:3) Examples
The preceding section shows that we are lacking of a characterizationof arbitrary subintegral ∆-extensions, except those of Theorem 4.18which says that an infra-integral FCP extension is a ∆-extension ifand only if it is modular, this last condition being unsatisfactory. Inthis section, we give examples of subintegral ∆-extensions with variousproperties. We also characterize some special types of FCP extensionsthat are ∆-extensions.
Proposition 5.1.
Let R ⊂ S be an FCP subintegral extension. As-sume that ( R : S ) M = M M for any M ∈ MSupp(
S/R ) . Then R ⊂ S is a ∆ -extension . -EXTENSIONS 35 Proof.
According to Proposition 3.6, we may assume first that (
R, M )is a local ring such that M = ( R : S ), and after, considering thefactor ring R/ ( R : S ), so that ( R : S ) = 0, with R which is a localArtinian ring by [6, Theorem 4.2], and not a field. First assume that R/M is infinite. Then [6, Proposition 5.15] asserts that R ⊂ S ischained, and then a ∆-extension by Proposition 3.11. Assume nowthat R/M is finite, and so is R because Artinian [6, Lemma 5.11].Set R ′ := R ( X ) , S ′ := S ( X ) and M ′ := M R ( X ). Then R ′ is alocal Artinian ring with maximal ideal M ′ which has the same indexof nilpotency as M , so that R ′ is not a field, R ′ /M ′ is infinite and( R ′ : S ′ ) = 0. Then, the first part of the proof says that R ′ ⊂ S ′ is a∆-extension, and so is R ⊂ S by Proposition 3.10. (cid:3) Since a ∆-extension R ⊂ S is modular (Proposition 3.13), a firstapproach consists in exhibiting conditions that imply the distributivityof a modular extension. For a distributive FCP extension R ⊂ S andits Loewy series { S i } ni =0 (see the definition before Corollary 5.3), [25,Proposition 3.8] says that S i ⊂ S i +1 is Boolean for each i ∈ { , . . . , n } .We begin with the characterization of Boolean ∆-extensions. In thelight of [24, Proposition 3.5], we first consider extensions R ⊂ S over alocal ring. Proposition 5.2.
Let R ⊂ S be a Boolean FCP extension, where ( R, M ) is a local ring. Then R ⊂ S is a ∆ -extension if and only ifeither R ⊂ S is minimal, or an infra-integral extension (in fact B ).Proof. Let R ⊂ S be a Boolean FCP extension. By [24, Theorem 3.30],one of the following conditions is satisfied:(1) R ⊂ S is a minimal extension.(2) There exist U, T ∈ [ R, S ] such that R ⊂ T is minimal ramified, R ⊂ U is minimal decomposed and [ R, S ] = { R, T, U, S } .(3) R ⊂ S is a Boolean t-closed extension.In case (1), R ⊂ S is a ∆-extension by Proposition 3.11.In case (2), S = T U , ℓ [ R, S ] = 2 and R ⊂ S is a B -extension,because T = U . In particular, R ⊂ S is infra-integral since S = T U = tS R . Then Theorem 4.16 implies that R ⊂ S is a ∆-extension. To sumup, a Boolean infra-integral extension is always a ∆-extension.In case (3), Proposition 4.7 shows that R ⊂ S is a ∆-extension ifand only if R ⊂ S is chained. But a chain is Boolean if and only if itis minimal by [24, Example 3.3 (1)]. Then, we recover case (1). (cid:3) In [25], we studied the Loewy series { S i } ni =0 associated to an FCPring extension R ⊆ S defined as follows in [25, Definition 3.1]: the socle of the extension R ⊂ S is S [ R, S ] := Q A ∈ A A , the product of the atomsof [ R, S ], and the
Loewy series of the extension R ⊂ S is the chain { S i } ni =0 defined by induction: S := R, S := S [ R, S ] and for each i ≥ S i = S , we set S i +1 := S [ S i , S ]. Of course, since R ⊂ S has FCP, there is some integer n such that S n − = S n = S n +1 = S .We said that R ⊂ S is a P -extension if R ⊂ S is pinched at the chain { S i } n − i =1 [25, Definition 3.19]. Corollary 5.3.
A distributive FCP P -extension R ⊂ S with Loewyseries { S i } ni =0 is a ∆ -extension if and only if S i ⊂ S i +1 is locally eitherminimal or an infra-integral B -extension for each ≤ i ≤ n − .Proof. By [25, Proposition 3.8], S i ⊂ S i +1 is Boolean for each 0 ≤ i ≤ n − R ⊂ S is distributive. Since R ⊂ S is a P -extension, R ⊂ S is pinched at the chain { S i } n − i =1 . Then, R ⊂ S is a ∆-extensionif and only if S i ⊂ S i +1 is a ∆-extension for each 0 ≤ i ≤ n − S i ⊂ S i +1 is a ∆-extension if and only if S i ⊂ S i +1 is locally a ∆-extension ifand only if S i ⊂ S i +1 is locally either minimal or an infra-integral B -extension. (cid:3) Proposition 5.4.
A locally unbranched distributive FCP extension R ⊂ S is a ∆ -extension if and only if it is arithmetic.Proof. If R ⊂ S is arithmetic, then R ⊂ S is a ∆-extension by Propo-sition 3.11.Conversely, assume that R ⊂ S is a ∆-extension. According toProposition 3.6, there is no harm to assume that ( R, M ) is a local ring.We begin to show that R ⊆ R is chained.We may assume R = R . Since R ⊂ S is distributive, so is R ⊂ R .The assumption yields that R is local, as any element of [ R, R ]. Set T := + R R = tR R because R ⊂ S is unbranched (Proposition 2.10). Then,Corollary 4.12 shows that [ R, R ] = [
R, T ] ∪ [ T, R ] with T ⊆ R chained.Now, assume that R = T .If { S i } ni =0 is the Loewy series of R ⊂ T , then S i ⊂ S i +1 is a Booleanextension for each i ∈ { , . . . , n − } according to [25, Proposition3.8]. But S i ⊂ S i +1 is also a ∆-extension, with S i a local ring for each i ∈ { , . . . , n − } . Now S i ⊂ S i +1 is necessarily minimal ramified,because R ⊂ T is subintegral (see the conditions of [24, Theorem 3.30]in the proof of Proposition 5.2). An easy induction shows that R ⊂ T is chained: since S = R ⊂ S is minimal, R ⊂ T has only oneatom (which is S ), and so has any subextension S i ⊂ T for each i ∈ { , . . . , n − } , whence R ⊂ T is chained. Now, [ R, S ] = [
R, T ] ∪ [ T, R ] ∪ [ R, S ] according to Lemma 2.4, because there does not exist -EXTENSIONS 37 some V ∈ [ R, S ] \ ([ R, R ] ∪ [ R, S ]) since V ∩ R = R . But R ⊆ S is alsochained, because R being local, [6, Theorem 6.10] implies that R ⊂ S is chained. To conclude, R ⊂ S is chained. (cid:3) We saw in Theorem 4.16 that an infra-integral extension of length 2is a ∆-extension. The next proposition characterizes ∆-extensions oflength 2.
Proposition 5.5.
A length 2 extension R ⊂ S is a ∆ -extension, exceptwhen the three following conditions are together verified: | MSupp(
S/R ) | = 1 , R ⊂ S is t-closed and | [ R, S ] | > .Proof. We use the characterization of ring extensions of length 2 de-scribed in [23, Theorem 6.1] and keep only cases involving ∆-extension:A ring extension R ⊂ S is of length 2 if and only if one of the followingconditions hold:(1) | Supp(
S/R ) | = 2 , Supp(
S/R ) ⊆ Max( R ) and | [ R, S ] | = 4. Inthis case, for each M ∈ Supp(
S/R ) , R M ⊂ S M is minimal, sothat R ⊂ S is arithmetic, and then a ∆-extension by Proposi-tion 3.11.(2) | Supp(
S/R ) | = 2 , Supp(
S/R ) Max( R ) and | [ R, S ] | = 3.(3) R ⊂ S is a non-integral M -crucial extension and | [ R, S ] | = 3.(4) R ⊂ S is an integral M -crucial extension such that tS R = R, S and | [ R, S ] | = 3.In cases (2), (3) and (4), R ⊂ S is chained and then a ∆-extension by Proposition 3.11.(5) R ⊂ S an infra-integral M -crucial extension such that + S R = R, S and either | [ R, S ] | = 3 or ( R : S ) = M with | [ R, S ] | = 4.(6) R ⊂ S is a subintegral M -crucial extension of length 2 witheither | [ R, S ] | = 3 or | [ R, S ] | = | R/M ] + 3.(7) R ⊂ S is a seminormal infra-integral M -crucial extension suchthat | [ R, S ] | = 5.In cases (5), (6) and (7), R ⊂ S is infra-integral and then a∆-extension by Theorem 4.16.(8) R ⊂ S is a t-closed integral M -crucial extension of length 2,so that M = ( R : S ), and R/M ⊂ S/M is a field extension.According to Proposition 4.7, a t-closed extension R ⊂ S is a ∆-extension if and only if R ⊂ S is arithmetic if and only if R/M ⊂ S/M is chained for M ∈ MSupp(
S/R ). But | MSupp(
S/R ) | = 1,so that R/M ⊂ S/M is chained for M ∈ MSupp(
S/R ) if andonly if R ⊂ S is chained if and only if | [ R, S ] | = 3. To sum up, a ring extension R ⊂ S of length 2 is a ∆-extension exceptwhen R ⊂ S is a t-closed integral extension such that | MSupp(
S/R ) | =1 and | [ R, S ] | > (cid:3) Corollary 5.6. A B -extension is a ∆ -extension except if it is t-closed.Proof. Let R ⊂ S be a B -extension, that is ℓ [ R, S ] = 2 and | [ R, S ] | =4. According to the characterization of length 2 extensions recalled inthe proof of Proposition 5.5, | [ R, S ] | = 4 happens in the following casesof this proof: (1) and then R ⊂ S is a ∆-extension; (5) and R ⊂ S is a ∆-extension. In the t-closed case, when R ⊂ S is a ∆-extension, R ⊂ S is chained, so that | [ R, S ] | = 3 = 4 when ℓ [ R, S ] = 2. (cid:3)
We now consider the ∆-properties for pointwise minimal extensions.A ring extension R ⊂ S is pointwise minimal if R ⊂ R [ t ] is minimalfor each t ∈ S \ R . We characterized these extensions in a joint workwith Cahen in [3]. The properties of pointwise minimal extensions R ⊂ S allow to assume that ( R, M ) is a local ring. In [3, Theorems 3.2and 5.4 and Proposition 3.5], we gave the different conditions for anextension R ⊂ S to be pointwise minimal. Firstly, a pointwise minimalextension is either integral or integrally closed, and, in this last case,is minimal. As minimal extensions are ∆-extensions, we need only toconsider pointwise minimal integral extensions which are not minimal,that we describe as follows. Let R ⊂ S be an integral extension suchthat ( R, M ) is a local ring and which is not minimal. Then R ⊂ S is apointwise minimal extension if and only if M = ( R : S ) and one of thefollowing condition is satisfied:( α ) R/M = Z / Z and S/M ∼ = ( R/M ) n for some integer n ≥
3. Inthis case, R ⊂ S is seminormal infra-integral.( β ) M ∈ Max( S ) , c( R/M ) = p , a prime integer, and R/M ⊂ S/M is a purely inseparable extension where x p ∈ R for each x ∈ S . In thiscase, R ⊂ S is t-closed.( γ ) R ⊂ S is subintegral with Max( S ) = { N } such that x ∈ M foreach x ∈ N .( δ ) Max( S ) = { N } is such that x ∈ M for each x ∈ N, R ⊂ R + N ⊂ S, c( R/M ) = p , a prime integer and x p ∈ R for each x ∈ S .In this case, the spectral map of R ⊂ S is bijective. Proposition 5.7.
A pointwise minimal FCP extension R ⊂ S over thelocal ring ( R, M ) is a ∆ -extension if and only if one of the followingconditions holds: (1) R ⊂ S is minimal. (2) R ⊂ S is seminormal infra-integral with | Max( S ) | = 3 . (3) R ⊂ S is subintegral with N ⊆ M , where Max( S ) = { N } . -EXTENSIONS 39 Proof.
Since we have recalled just before the characterization of point-wise minimal extensions R ⊂ S , we examine, for each case, a necessaryand sufficient condition in order that R ⊂ S be a ∆-extension. So, inthe following, we assume that R ⊂ S is a pointwise minimal extension.If R ⊂ S is integrally closed, R ⊂ S is minimal, and then a ∆-extension and we recover case (1).If R ⊂ S is not integrally closed, then R ⊂ S is a integral extension.If R ⊂ S is minimal, then R ⊂ S is a ∆-extension and we recover case(1).If R ⊂ S is a integral extension which is not minimal, then R ⊂ S satisfies one of conditions ( α )–( δ ) and moreover, M = ( R : S ).In case ( α ), we have R/M = Z / Z and S/M ∼ = ( R/M ) n for someinteger n ≥
3. In this case, R ⊂ S is seminormal infra-integral, and byCorollary 4.20, R ⊂ S is a ∆-extension if and only if | V S ( M S ) | ≤ M ∈ MSupp(
S/R ), which is equivalent, since (
R, M ) is a local ring,to | Max( S ) | ≤
3. But | Max( S ) | = 2 implies that R ⊂ S is minimal, acontradiction. To sum up, if R ⊂ S is seminormal infra-integral nonminimal, then R ⊂ S is a ∆-extension if and only if | Max( S ) | = 3which gives case (2).The condition ( β ) is not involved for a t-closed non-minimal ∆-extension R ⊂ S where ( R, M ) is a local ring since it is chainedby Proposition 4.7, and so is
R/M ⊂ S/M . In fact, in this case,
R/M ⊂ S/M is chained if and only if
R/M ⊂ S/M is minimal: assumethat the contrary holds. Then
R/M ⊂ S/M is a non-minimal chainsatisfying condition ( β ). Let S ′ ∈ [ R/M, S/M ] be such that
R/M ⊂ S ′ is minimal, which implies that S ′ = S/M . Let x ∈ ( S/M ) \ S ′ , so that x p ∈ R/M , giving
R/M ⊂ ( R/M )[ x ] is minimal, with S ′ = ( R/M )[ x ],a contradiction. It follows that R ⊂ S needs to be a minimal extensionand we recover case (1).If case ( γ ) holds, R ⊂ S is subintegral with Max( S ) = { N } suchthat x ∈ M for each x ∈ N . In particular, R [ x ] = R + Rx foreach x ∈ N . We claim that R ⊂ S is a ∆-extension if and only if N ⊆ M . But N ⊆ M if and only if xy ∈ M for each x, y ∈ N .According to Proposition 3.1, R ⊂ S is a ∆-extension if and only if R [ s, t ] = R [ s ] + R [ t ] for all s, t ∈ S . Since R ⊂ S is subintegral, theisomorphism R/M ∼ = S/N shows that S = R + N .Assume that R ⊂ S is a ∆-extension and let x, y ∈ N \ R = N \ M .Then R ⊂ R [ x ] is minimal ramified by Theorem 2.3 because x ∈ M and xM ⊆ M . For the same reason, R ⊂ R [ y ] is minimal ramified. Let P be the maximal ideal of R [ x ]. If R [ x ] = R [ y ], it follows that x, y ∈ N ∩ R [ x ] = P , so that xy ∈ P ⊆ M . Assume R [ x ] = R [ y ]. Then, xy ∈ R [ x, y ] = R [ x ] + R [ y ] = R + Rx + Ry , so that xy = a + bx + cy ( ∗ ), with a, b, c ∈ R . Then, a = xy − bx − cy ∈ R ∩ N = M . By ( ∗ ), wehave a + bx = y ( x − c ) ( ∗∗ ). If c M , then c N , which leads to c − x is a unit in R [ x ] and in S . It follows that y = ( a + bx )( x − c ) − in S .Since ( a + bx )( x − c ) − ∈ R [ x ], we get y ∈ R [ x ], a contradiction. Then, c ∈ M and cy ∈ M because M = ( R : S ). A similar proof shows that bx ∈ M , giving xy ∈ M , so that N ⊆ M .Conversely, assume that N ⊆ M . We claim that R [ s, t ] = R [ s ]+ R [ t ]for all s, t ∈ S . It is enough to prove that st ∈ R [ s ]+ R [ t ] for all s, t ∈ S .Since S = R + N , we can write s = λ + x and t = µ + y , with λ, µ ∈ R and x, y ∈ N . In particular, xy ∈ N ⊆ M . But, st = λµ + λy + µx + xy = λ ( µ + y ) + µ ( λ + x ) + xy − λµ = λt + µs + xy − λµ ∈ R [ s ] + R [ t ]. Toconclude, R ⊂ S is a ∆-extension.We do not consider the case ( δ ) for the following reason. An exten-sion verifying ( δ ) cannot be a ∆-extension, because it is not catenarian:there exist two minimal extensions R ⊂ R [ x ] and R ⊂ R [ y ], with, forinstance, R ⊂ R [ x ] ramified, R ⊂ R [ y ] inert and M = ( R : S ) (seeProposition 2.15). In fact, y ∈ S \ ( R + N ). (cid:3) We saw in Proposition 5.7 that in the seminormal infra-integral case,we deal with extensions of the form
R/M ⊂ ( R/M ) . We are going tolook at infra-integral extensions of the type R ⊂ R n . Lemma 5.8.
Let R be an Artinian ring and n an integer with n > .If R ⊂ R n is a ∆ -extension, then n ≤ .Proof. Since ( R n ) M = ( R M ) n for any maximal ideal M of R , we mayassume that R is a local ring by Proposition 3.6. Set S := R n and T := + S R . Then, R ⊂ S is FCP infra-integral by [21, Proposition 1.4],with | Max( S ) | = n and ℓ [ T, S ] = n − n ≤ (cid:3) Proposition 5.9.
Let R be an absolutely flat ring. Then R ⊂ R n is a ∆ -extension if and only if n ≤ .Proof. By Proposition 3.6, we may assume that R is local, so that R is a field. Then [21, Proposition 1.4] shows that R ⊂ R n is an infra-integral seminormal FCP extension. By Proposition 4.20, R ⊂ R n is a∆-extension if and only if ℓ [ R, R n ] ≤ n ≤ (cid:3) According to [21, Proposition 1.4], when R is not reduced and n = 3,there is a subintegral part R ⊂ + R R of R ⊂ R , so that we cannot useCorollary 4.20. We next give an example of a ∆-extension R ⊂ R ,where R is an Artinian local and non-reduced ring (which is not afield). Moreover, R ⊂ R has FCP and is infra-integral by the above -EXTENSIONS 41 reference. We leave the easy but tedious calculations of this exampleto the reader. Example 5.10.
Set R := ( Z / Z )[ T ] / ( T ) = ( Z / Z )[ t ], where t isthe class of T in R . Then R is an Artinian local ring which is notreduced and whose maximal ideal M := Rt = 0 is such that M = 0.Moreover, | R/M | = | Z / Z | = 2. Let B := { e , e , e } be the canonicalbasis of R . According to [21, Proposition 2.8], S := + R R = R + N = R ( e + e + e ) + Rte + Rte (for instance), where N := M × M × M = Rte + Rte + Rte is the maximal ideal of the local ring S . Since | Max( R ) | = 3, we get ℓ [ S, R ] = 2 according to [6, Lemma 5.4]; so that, S ⊂ R is a ∆-extension by Corollary 4.20, and ℓ [ R, S ] = L R ( N/M )by [6, Lemma 5.4]. But
M N = 0 yields L R ( N/M ) = L
R/M ( N/M ) =dim
R/M [( M × M × M ) /M ] = 2 thanks to [17, Corollary 2 of Proposition24, page 66]. It follows that ℓ [ R, S ] = 2 and ℓ [ R, R ] = 4, always by[6, Lemma 5.4]. In particular, Theorem 4.16 shows that R ⊂ S is a ∆-extension. Hence, from Proposition 4.25, we deduce that R ⊂ R is a∆-extension if and only if ℓ [ T, U V ] = 2, for each
T, U, V ∈ [ R, R ] suchthat T ⊂ U is minimal ramified and T ⊂ V is minimal decomposed. Tofind such extensions, we are going to determine all elements of [ R, R ].For each i = 1 , ,
3, set x i := te i , so that R [ x i ] = R + Rte i isa local ring with maximal ideal N i := Rte i + Rt ( e j + e l ), for j, l = i and j = l . Then, { R, R [ x ] , R [ x ] , R [ x ] , S } ⊆ [ R, S ], with the R [ x i ]’s all distinct, so that | [ R, S ] | >
3. Then, [23, Theorem 6.1]says that | [ R, S ] | = 5, because R ⊂ S is subintegral. In particu-lar, [ R, S ] = { R, R [ x ] , R [ x ] , R [ x ] , S } . Moreover, | [ S, R ] | = 5 be-cause S ⊂ R is seminormal infra-integral. Indeed, R ⊂ R is infra-integral. A short calculation shows that there does not exist any y ∈ R such that R ⊂ R [ y ] is minimal decomposed. By defini-tion of S = + R R , for each i = 1 , ,
3, the only minimal ramified ex-tension starting from R [ x i ] is R [ x i ] ⊂ S . For each i = 1 , ,
3, set R i := R [ e i ] = R + Re i = R [ x i ] + e i R [ x i ]. We get that R [ x i ] ⊂ R i is minimal decomposed and another calculation shows that this is theonly minimal decomposed extension starting from R [ x i ]. It followsthat the maximal ideals of R i are M i := M + Re i = N i + e i R [ x i ] and M ′ i := M + R (1 − e i ). By definition of S , there does not exist any y ∈ R such that S ⊂ S [ y ] is minimal ramified. Moreover, for each i = 1 , , S ⊂ SR i ⊂ R , with SR i = R + Re i + Rte j + Rte l , for j, l = i and j = l . Since S ⊂ R is seminormal infra-integral, then SR i = + R R i because R i ⊂ SR i is minimal ramified (use SR i = R i [ te j ] with i = j ,for instance) and R i ⊂ SR i is the only minimal ramified extension starting from R i . In particular, [ S, R ] = { S, SR , SR , SR , R } be-cause | [ S, R ] | = 5 as we have seen just above. We claim that there doesnot exist any y ∈ R such that R i ⊂ R i [ y ] is minimal decomposed (ifthe contrary holds, then R i [ y ] ⊂ R is minimal ramified, contradicting[8, Lemma 17], because M R = M × M × M is a radical ideal of R containing 0 = ( R : R )). To sum up, the only minimal extensions of R are the R [ x i ], for a given i , the only minimal extensions of R [ x i ] are R i and S , the only minimal extension of R i is SR i , and the only minimalextensions of S are the SR i . At last, SR i ⊂ R is minimal for each i . Then, we get the following commutative diagram corresponding toonly one i ∈ { , , } , where r , (resp. d ) indicates a minimal ramified(resp. decomposed) extension: R i d ր r ց R r → R [ x i ] SR i d → R r ց d ր S To conclude, the only extensions such that T ⊂ U is minimal ramifiedand T ⊂ V is minimal decomposed are gotten for T = R [ x i ] , U = S and V = R i for each i = 1 , ,
3. Since
U V = SR i and ℓ [ T, U V ] = ℓ [ R [ x i ] , SR i ] = 2, we get that R ⊂ R is a ∆-extension by Proposi-tion 4.25 as we claimed before.In the context of extensions of the form R ⊂ R n , for a positive integer n , we now consider extensions of the form R ⊂ R [ t ], where t is eitheridempotent or nilpotent of index 2. We recall that a ring R is called a SPIR if R is a principal ideal ring with a nonzero prime ideal M suchthat M is nilpotent of index p > Proposition 5.11.
An extension R ⊂ R [ t ] , where t is either idem-potent or nilpotent of index 2, and such that R is a SPIR, is an FIP ∆ -extension.Proof. Set S := R [ t ]. If t is either idempotent or nilpotent of index2, then S is a finitely generated R -module. Moreover, R/ ( R : S ) isan Artinian ring because R is a SPIR. Then, R ⊂ S has FCP by [6,Theorem 4.2]. In both cases, we can write t − rt = 0 ( ∗ ), with either r = 0 or r = 1. This implies t − rt = 0, then R ⊂ R [ t ] is an t -elementary extension [18, Definition 1.1] and [18, Proposition 2.18]shows that R/ ( R : S ) ⊂ S/ ( R : S ) can be identified with R/ ( R : S ) ⊂ ( R/ ( R : S ))[ X ] / ( X − ¯ rX ), where ¯ r is the class of r in R/ ( R : S ) .But R ⊂ S is a ∆-extension if and only if R/ ( R : S ) ⊂ S/ ( R : S ) is -EXTENSIONS 43 a ∆-extension by Proposition 3.6. Then, there is no harm to assumethat ( R : S ) = 0, so that S = R [ t ] = R + Rt ∼ = R [ X ] / ( X − rX ), with r ∈ { , } .Since R is a SPIR, ( R, M ) is an Artinian local ring and M + Rt isa maximal ideal of S lying above M . If M = 0, then R is a field andeither S ∼ = R or S ∼ = R [ X ] / ( X ). In both cases, R ⊂ S is a minimalextension, decomposed when r = 1 and ramified when r = 0, and thena ∆-extension.If M = 0, then M is nilpotent and principal. Let n be its indexof nilpotency, so that M n = 0, with M n − = 0. Set M := Rx . Webegin by showing a result that holds for both values of r . For k ∈ N n ,set R k := R + M k S = R + Rx k t , which is obviously a local ring withmaximal ideal M k := M + M k S = Rx + Rx k t . In particular, for k < n , we have M k +1 = M R k and R k = R k +1 [ x k t ]. Set y k := x k t and C := { R k } k ∈ N n ; so that, R k = R k +1 [ y k ].Next we show that C is a finite maximal chain. Observe that y k = x k t = x k rt ∈ R k +1 , M k +1 y k = ( Rx + Rx k +1 t ) x k t = Rx k +1 t + Rx k +1 rt ⊆ M k +1 and L R k +1 ( M k /M k +1 ) = L R k +1 /M k +1 ( M k /M k +1 ) =dim R/M ( M k /M k +1 ) = 1, since M k = M k +1 + Rx k t . It follows that R k +1 ⊂ R k is minimal ramified by [21, Theorem 1.1]. Therefore, C is afinite maximal chain and R ⊂ R is subintegral.Now, we intend to prove that [ R, R ] = C . Assume that the contraryholds; so that there exists some U ∈ [ R, R ] \ C . Since { R k } nk =1 is adecreasing chain going from R to R , there exists some k ∈ N n − suchthat U ⊂ R k , with U R k +1 . Moreover, R ⊂ S is an FCP extension.Since U = R k , there exists V ∈ [ U, R k ] so that V ⊂ R k is minimal.In particular, V is a local ring. Let N be its maximal ideal. Then, N := ( V : R k ) ∈ Max( V ), which is also an ideal of R k . Since R ⊂ V issubintegral, we get that V = R + N . Then, M ⊆ N implies M k +1 = M R k ⊆ N R k = N , so that R k +1 = R + M k +1 ⊆ R + N = V ⊂ R k ,which leads to V = R k +1 . Then, U ⊆ R k +1 , a contradiction. Therefore,[ R, R ] = C .To show that R ⊂ S is a ∆-extension, we split the proof in two cases,according to the value of r .Assume that r = 1. It follows that we are reduced to show that R ⊂ R =: S is a ∆-extension.According to [21, Proposition 1.4], R ⊂ R is an infra-integral FCPextension, and, setting T := + S R , [21, Proposition 2.8] says that T = R + ( M × M ) = R + M S = R . Since | Max( S ) | = 2, it followsthat R ⊂ S is minimal decomposed by [6, Lemma 5.4]. Proposition4.25 asserts that R ⊂ S is a ∆-extension if and only if the following statements hold: R ⊂ R is a ∆-extension (1), R ⊂ S is a ∆-extension(2), and for each W, U, V ∈ [ R, S ] such that W ⊂ U is minimal ramifiedand W ⊂ V is minimal decomposed, ℓ [ W, U V ] = 2 (3). Since R ⊂ S is minimal, (2) is satisfied. We have proved above that [ R, R ] = C is a finite maximal chain, and then is a ∆-extension. To show (3),we prove that there do not exist W, U, V ∈ [ R, S ] such that W ⊂ U isminimal ramified and W ⊂ V is minimal decomposed. Assume that thecontrary holds, then in particular, 2 ≤ | Max( W ) | + 1 = | Max( V ) | ≤| Max( S ) | = 2 implies that | Max( V ) | = 2. Assume first that W [ R, R ] ∪ [ R , S ] and set W ′ := W ∩ R ∈ [ R, R [ with W ′ = W , sothat W ′ ⊂ W is seminormal infra-integral with | Max( W ) | ≥
2. But2 = | Max( V ) | = | Max( W ) | + 1 ≥
3, a contradiction. It follows that W ∈ [ R, R ] ∪ [ R , S ]. If W ∈ [ R , S ], then W ⊂ U ⊆ S together with W ⊂ U minimal ramified leads to a contradiction. Then, W ∈ [ R, R ],which implies U ∈ [ R, R ]. But | Max( V ) | = 2 = | Max( S ) | implies thateither V = S , which is impossible since V ⊂ U V ⊆ S , or V ⊂ S issubintegral. In this case, there exists V ′ ∈ [ R, S ] such that V ′ ⊂ S isminimal ramified. Now, [8, Lemma 17] gives that M S is not a radicalideal of S , a contradiction with M S = M × M . To conclude, (3) is aninvalid statement and R ⊂ S is a ∆-extension when r = 1. Moreover, R ⊂ S has FIP by [21, Corollary 2.5] because R has finitely manyideals.Assume now that r = 0. Then S = R [ t ] = R [ t ], where t = 0and M t = Rxt + Rxt = Rxt ⊆ M show that R ⊂ S is minimalramified. In fact, [ R, S ] = [
R, R ] ∪ { S } . It is enough to use the proofshowing that [ R, R ] is a maximal chain with k = 0, setting R := S and M := M + Rt . Since R ⊂ S is chained, it is a ∆-extension andhas FIP, because it has FCP. (cid:3) In the same spirit as the previous Proposition, we get the followingresult:
Proposition 5.12.
Let R ⊂ S be an FCP subintegral extension, where ( R, M ) is a local Artinian ring, so that ( S, N ) is also a local ring. Let n > be the index of nilpotency of M . Set R k := R + M k S , for any k ∈ { , . . . , n } , with R = S and R n = R . Assume that N ⊆ M S andthat [ R, S ] is pinched at { R k } n − k =1 . Then, R ⊂ S is a ∆ -extension.Proof. For each k ∈ { , . . . , n } , set M k := M + M k S and M := N .Mimicking the proof of Proposition 5.11, we get that ( R k , M k ) is alocal ring such that M k +1 = M R k = ( R k +1 : R k ) ( ∗ ) for each k ∈{ , . . . , n − } . Since R ⊂ S is subintegral, so are each R k +1 ⊂ R k .Then R k = R + M k = R + M k S . We begin to show that each R k +1 ⊂ R k -EXTENSIONS 45 is pointwise minimal. For k ∈ { , . . . , n − } , let x = a + y ∈ R k \ R k +1 , a ∈ R, y ∈ M k S , so that R k +1 [ x ] = R k +1 [ y ]. For k = 0, wechoose y ∈ N \ M S . Then, M k +1 y ⊆ M k +1 by ( ∗ ), which also holds for k = 0. Moreover, y ∈ M k = M + M k +1 S + M k S ⊆ M k +1 shows that R k +1 ⊂ R k +1 [ y ] is minimal, so that R k +1 ⊂ R k is pointwise minimal.This also holds for k = 0 since N ⊆ M S . In particular, this showsthat R ⊂ S is a ∆-extension by Proposition 5.7. The same Propositionshows that R k +1 ⊂ R k is a ∆-extension because M k ⊆ M k +1 as we havejust seen. Then, R ⊂ S is a ∆-extension because [ R, S ] is pinched at { R k } n − k =1 according to Corollary 3.5. (cid:3) Proposition 5.13.
An FIP subintegral extension k ⊂ S over the field k is a ∆ -extension if either (1) | k | = ∞ or (2) k ⊂ S is chained.Proof. We use [1, Theorem 3.8] together with the fact that k ⊂ S isFIP subintegral. This last condition implies that either (a) : | k | < ∞ with S a finite dimensional vector-space such that k ⊂ S is subintegral,or (b) | k | = ∞ with S = k [ α ] for some α ∈ S which satisfies α = 0.If (2) holds, then k ⊂ S is a ∆-extension by Proposition 3.11.Assume now (1), that is | k | = ∞ . Since k ⊂ S is subintegral, S isa local ring with maximal ideal M := kα + kα . If α = 0, it followsthat k ⊂ S is minimal ramified, and then a ∆-extension. If α = 0, [6,Lemma 5.4] shows that ℓ [ k, S ] = L k ( M ) = dim k ( M ) = 2. Hence k ⊂ S is a ∆-extension according to Theorem 4.16. (cid:3) Remark 5.14.
Even if | k | < ∞ , then, | k ( X ) | = ∞ and we may use(1) of Proposition 5.13 for the extension k ( X ) ⊂ S ( X ). Because ofProposition 3.10, then k ⊂ S is a ∆-extension if so is k ( X ) ⊂ S ( X ).But, in order to use (1), we need that k ( X ) ⊂ S ( X ) has FIP, thislast property being equivalent to k ⊂ S is an FIP chained extensionaccording to [20, Theorem 4.2]. Corollary 5.15.
Let R := Q ni =1 k i be a product of infinite fields and R ⊂ S be an FIP subintegral extension. Then, R ⊂ S is a ∆ -extension.Proof. Proposition 3.15 says that for each i ∈ N n , there exists ringextensions k i ⊆ S i such that S ∼ = Q ni =1 S i . Moreover R ⊆ S is asubintegral ∆-extension if and only if so is k i ⊆ S i for each i ∈ N n (seethe proof of Proposition 3.15). Conclude with Proposition 5.13. (cid:3) Here is an example of an infra-integral ∆-extension of number fieldorders whose length is >
2. Its ∆-property is proved by checking thehypotheses of Proposition 4.25 and in particular the condition (2) ofProposition 4.25.
Example 5.16.
In [10, Example 3.7 (5)], El Fadil, Chillali and Akhar-raz consider the quartic number field K defined by the irreducible poly-nomial X + 22 X + 66. Let S be the ring of integers of K . It is shownin this example that 3 S = P P , where P and P are the maximalideals of S lying above 3 Z . Set R := Z + 3 S . We are going to provethat R ⊂ S is a ∆-extension.Since [ K : Q ] = 4, the fundamental formula [27, Theorem 1, page193] gives 4 = P gi =1 e i f i , where g is the decomposition number of 3 Z in the extension Q ⊂ K, e i is the ramification index of P i and f i is the inertial degree of P i . It follows that g = 2 , e = 1 , e = 3and f i = 1 for each i . Observe that S/P i ∼ = k for each i , where k := Z / Z . In particular, R ⊂ S is an infra-integral extension because3 S = ( R : S ) is a maximal ideal of R and R/ S ∼ = Z / Z = k ∼ = S/P i foreach i , where P and P are the only maximal ideals of S containing( R : S ) = 3 S . Set W := + S R . Then W = S because V S (3 S ) = { P , P } and 3 S ∈ Max( R ), so that R ⊂ S is not subintegral. Since | V S (3 S ) | = 2, it follows that W ⊂ S is minimal decomposed accordingto Theorem 2.3 and Proposition 2.10 and then a ∆-extension. Hence,the second part of condition (1) of Proposition 4.25 is satisfied. Now,3 S = P P ⊆ ( W : S ) ⊆ P P implies ( W : S ) = P P by the samereference. Since ( W : S ) ∈ Max( W ) with Z / Z ∼ = ( Z + P P ) /P P ⊆ W/P P ∼ = S/P i ∼ = Z / Z for each i = 1 ,
2, this shows that W = Z + P P . Moreover, 3 S = ( R : W ). Let N be the maximal ideal ofthe local ring W/ S .Because ( R : S ) = 3 S , [6, Lemma 5.4] gives ℓ [ R, W ] = ℓ [ R/ S, W/ S ]= L k ( N ) = dim k ( N ) < dim k ( W/ S ) < dim k ( S/ S ) = 4, so that ℓ [ R, W ] ≤
2. It follows that R ⊂ W is a ∆-extension by Theorem4.16 and condition (1) of Proposition 4.25 is satisfied. In particular, ℓ [ R, S ] ≤ R ⊂ S is infra-integral. We show that condition (2)of Proposition 4.25 is also satisfied.Set U := R + P P , V := R + P , V := R + P and U := R + P P .We have the following commutative diagram with W = U so that ℓ [ R, W ] = 2 and ℓ [ R, S ] = 3: U → U ր ց R ց S ց ր V → V Since ( R : S ) = P P , for any T ∈ [ R, S ], we have P P ⊆ ( T : S ) ( ∗ ),so that P and P are the only maximal ideals of S that may contain ( T : S ). Moreover, ( T : S ) = P α P β , for some ( α, β ) ∈ { , } × { , , , } , -EXTENSIONS 47 because S is a Dedekind domain. In particular, if T ⊂ S is minimal,it is either ramified, and in this case, M ⊆ ( T : S ) ⊂ M for somemaximal ideal M of S . This leads to ( T : S ) = P and T = V . If T ⊂ S is decomposed, the only possible case is ( T : S ) = P P and T = W = U .Let T, U, V ∈ [ R, S ] be such that T ⊂ U is minimal ramified and T ⊂ V is minimal decomposed. Since ℓ [ R, S ] = 3, we may have ℓ [ T, U V ] > T = R and U V = S . We are going to show that V = V . Ofcourse, the diagram shows that R ⊂ V is minimal and P = ( V : S )is a maximal ideal of V . Since P is the only maximal ideal of S lyingabove ( V : S ), it follows that V ⊂ S is subintegral. Now R ⊂ V isminimal decomposed, because if we suppose that the contrary holds,then R ⊂ S is subintegral, a contradiction. Assume that there isanother V ′ ∈ [ R, S ] such that R ⊂ V ′ is minimal decomposed. Therewould be in V V ′ , 3 maximal ideals lying above 3 S because ℓ [ R, V V ′ ] ≥ R ⊂ V V ′ seminormal infra-integral, a contradictionsince only 2 maximal ideals of S lie above 3 S . Then, V is the only V ∈ [ R, S ] such that R ⊂ V is minimal decomposed. Now, let U ∈ [ R, S ] be such that R ⊂ U is minimal ramified, so that there is onlyone maximal ideal N of U lying above 3 S = ( R : S ). Since P and P lie above 3 S in R , they both lie above N in U , so that N ⊆ P P .It follows that N P ⊆ P P ⊂ P P , which shows that ℓ [ R, U V ] = 2according to Proposition 2.15. Since condition (2) of Proposition 4.25is also satisfied, R ⊂ S is a ∆-extension.We now introduce a property linked to ∆-extensions and to [13] (seeProposition 3.1). Let R ⊂ S be a ring extension. We say that R ⊂ S is a δ -extension if R [ x ] + R [ y ] = R [ x + y ] for any x, y ∈ S such that R [ x ] = R [ y ]. Proposition 5.17.
Let R ⊂ S be a ring extension. (1) R ⊂ S is a δ -extension if and only if P ni =1 R [ x i ] = R [ P ni =1 x i ] for any x , . . . , x n ∈ S and any integer n such that R [ x i ] = P j ∈ I R [ x j ] , for any I ⊆ N n \ { i } and for any i ∈ N n . (2) If R ⊂ S is a δ -extension, then R ⊂ S is a ∆ -extension. (3) If R ⊂ S is an FCP δ -extension, then R ⊂ S is simple.Proof. (1) One implication is obvious. Conversely, assume that R ⊂ S is a δ -extension, that is R [ x ] + R [ y ] = R [ x + y ] for any x, y ∈ S such that R [ x ] = R [ y ], and let x , . . . , x n ∈ S be such that R [ x i ] P j ∈ I R [ x j ], for any I ⊆ N n \ { i } and for any i ∈ N n . We show that P ni =1 R [ x i ] = R [ P ni =1 x i ] by induction on n . The induction hypothesisis obviously satisfied for n = 2. Let n > y := P n − i =1 x i , so that P ni =1 x i = y + x n . Assume that the induction hypothesis holdsfor n −
1. It follows that R [ y ] = P n − i =1 R [ x i ], with R [ x n ] R [ y ]. Then, R [ P ni =1 x i ] = R [ y + x n ] = R [ y ] + R [ x n ] by the hypothesis, which leadsto R [ P ni =1 x i ] = P n − i =1 R [ x i ] + R [ x n ] = P ni =1 R [ x i ]. Then, it holds forany n .(2) If R ⊂ S is a δ -extension, then R [ x ] + R [ y ] = R [ x + y ] for any x, y ∈ S such that R [ x ] = R [ y ], so that R [ x ] + R [ y ] ∈ [ R, S ] for any x, y ∈ S , which shows that R ⊂ S is a ∆-extension by Proposition 3.1since x, y ∈ R [ x ] + R [ y ] implies R [ x, y ] ⊆ R [ x ] + R [ y ] ⊆ R [ x, y ].(3) Recall that an extension R ⊂ S is called strongly affine if eachelement of [ R, S ] is a finite-type R -algebra. Assume that R ⊂ S isan FCP δ -extension. Then, R ⊂ S is strongly affine by [6, Proposition3.12], so that S = R [ x , . . . , x n ] for some x , . . . , x n ∈ S . We can choosethe x i ’s as a minimal generating set, so that R [ x i ] P j ∈ I R [ x j ], forany I ⊆ N n \ { i } and for any i ∈ N n . Moreover, R ⊂ S is a ∆-extension by (2). Then, we get S = P ni =1 R [ x i ] by Proposition 3.1, sothat S = R [ P ni =1 x i ] and R ⊂ S is simple. (cid:3) Proposition 5.18.
A chained extension is a δ -extension.Proof. Let R ⊂ S be a chained extension, and let x, y ∈ S be such that R [ x ] = R [ y ]. Since R [ x ] and R [ y ] are comparable, assume R [ x ] ⊂ R [ y ],so that x ∈ R [ y ] which implies R [ x + y ] ⊆ R [ y ]. Moreover, R [ x + y ]and R [ x ] are comparable. If R [ x + y ] ⊆ R [ x ], then x + y ∈ R [ x ] whichgives y ∈ R [ x ], a contradiction. It follows that R [ x ] ⊂ R [ x + y ], andthen x ∈ R [ x + y ], which gives y ∈ R [ x + y ]. Then, R [ x + y ] = R [ y ] = R [ y ] + R [ x ] and R ⊂ S is a δ -extension. (cid:3) Corollary 5.19.
A Pr¨ufer extension is a δ -extension.Proof. Let R ⊂ S be a Pr¨ufer extension, and so is R M ⊂ S M forany M ∈ MSupp(
S/R ). Let M ∈ MSupp(
S/R ). According to [22,Proposition 1.2], R M ⊂ S M is chained. Indeed, R M ⊂ S M is Pr¨ufer foreach M ∈ Supp(
S/R ) by [22, Proposition 1.1], and since R M is local,there exists P ∈ Spec( R M ) such that S M = ( R M ) P , P = P S M , with R M /P a valuation domain with quotient field S M /P . Then, R M /P ⊂ S M /P is chained, and so is R M ⊂ S M . It follows that R M ⊂ S M is a δ -extension by Proposition 5.18. Let x, y ∈ S . Then, R M [ x ] + R M [ y ] = R M [ x + y ]. Since this holds for any M ∈ MSupp(
S/R ), we get that R [ x ] + R [ y ] = R [ x + y ], so that R ⊂ S is a δ -extension. (cid:3) Example 5.20. (1) Let R ⊂ T and R ⊂ U be two minimal extensionssuch that S := T U exists. If R ⊂ S satisfies one of the two followingconditions, then R ⊂ S is a δ -extension: -EXTENSIONS 49 (a) C ( R, T ) = C ( R, U ).(b) R ⊂ T and R ⊂ U are two minimal infra-integral extensions ofdifferent types such that ℓ [ R, S ] = 2.In both cases, we can set T = R [ x ] and U = R [ y ].If case (a) holds, let M := C ( R, T ) and N := C ( R, U ). Then, U M = R M and T N = R N imply ( T + U ) M = T M = S M and ( T + U ) N = U N = S N , so that T + U = S . Since x + y T, U , we have S = R [ x, y ] = R [ x + y ] because [ R, S ] = { R, T, U, S } by Proposition 2.15.Then, R [ x ] + R [ y ] = R [ x + y ]. Finally, any z ∈ S is such that R [ z ] ∈{ R, T, U, S } , so that R ⊂ S is a δ -extension.If case (b) holds, we can assume M := C ( R, T ) = C ( R, U ) (if not,then (a) holds). Now, ℓ [ R, S ] = 2 implies [
R, S ] = { R, T, U, S } by [23,Theorem 6.1 (5)] because | [ R, S ] | ≥
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