Flat commutative ring epimorphisms of almost Krull dimension zero
aa r X i v : . [ m a t h . A C ] S e p FLAT COMMUTATIVE RING EPIMORPHISMSOF ALMOST KRULL DIMENSION ZERO
LEONID POSITSELSKI
Abstract.
We consider flat epimorphisms of commutative rings R −→ U suchthat, for every ideal I ⊂ R for which IU = U , the quotient ring R/I is semilocalof Krull dimension zero. Under these assumptions, we show that the projectivedimension of the R -module U does not exceed 1. We also describe the Geigle–Lenzing perpendicular subcategory U ⊥ , in R – Mod . Assuming additionally thatthe ring U and all the rings R/I are perfect, we show that all flat R -modules are U -strongly flat. Thus we obtain a generalization of some results of the paper [6],where the case of the localization U = S − R of the ring R at a multiplicative subset S ⊂ R was considered. Contents
Introduction 11. F -h-locality 42. I -Contramodule R -Modules 63. Projective Dimension 1 Theorem 84. Divisible Modules 105. u -Contramodule R -Modules 126. U -Strongly Flat R -Modules 14References 16 Introduction
Let R be a commutative integral domain with the field of fractions Q . It wasdiscovered by Matlis [16] that certain commutative and homological algebra con-structions behave much better when the projective dimension of the R -module Q does not exceed 1. Commutative domains R with this property are now known as Matlis domains [9]. More generally, the same observation applies to the case when R is a commutative ring and Q is its full ring of quotients, that is Q = S − R , where S ⊂ R is the set of all regular elements in R [17].Even more generally, the case of a multiplicative subset S ⊂ R consisting of (some)regular elements in R was considered by Angeleri H¨ugel, Herbera, and Trlifaj in thepaper [1], where a number of conditions equivalent to pd R S − R ≤ R -module Q = S − R does not exceed 1if and only if Q ⊕ Q/R is a 1-tilting R -module, if and only if two natural notions f S -divisibility for R -modules coincide, and if and only if Q/R is a direct sum ofcountably presented R -modules. The next step, towards the study of localizations S − R where a multiplicative subset S ⊂ R is allowed to contain some zero-divisorsin R , was made in the present author’s papers [19, Section 13] and [20, 23, 6].From the contemporary point of view, it appears that the maximal natural general-ity in this line of thought is achieved by considering ring epimorphisms u : R −→ U .In this context, the condition that the projective dimension of the (left or right) R -module U does not exceed 1 continues to play an important role. In particular, foran injective homological ring epimorphism u (of associative, not necessarily commu-tative rings) a number of homologically relevant conditions equivalent to pd R U ≤ proving this condition. How does one show that pd R S − R ≤
1, for a specificmultiplicative subset S in a commutative ring R ? Or more generally that pd R U ≤ u : R −→ U ?Notice first of all that S − R is always a flat R -module, while for a ring epimorphism R −→ U the (left or right) R -module U does not have to be flat. The followingcondition is usually imposed on a ring epimorphism, however: a ring epimorphism u : R −→ U is said to be homological if Tor Rn ( U, U ) = 0 for all n ≥
1. In the contextof commutative rings, the following recent result of Bazzoni and the present author isrelevant in this connection: if R −→ U is an epimorphism of commutative rings suchthat Tor R ( U, U ) = 0 and pd R U ≤ , then U is a flat R -module [7, Theorem 5.2]. Ina sense, this means that one can restrict oneself to flat epimorphisms when studyingthe projective dimension 1 condition for epimorphisms of commutative rings.Flat ring epimorphisms can be described in terms of Gabriel filters (otherwiseknown as
Gabriel topologies ) [26, Chapters VI and IX–XI]. To every epimorphism ofassociative rings u : R −→ U such that U is a flat left R -module, one assigns the set G of all right ideals I ⊂ R such that R/I ⊗ R U = 0, or equivalently, IU = U . Thering U then can be recovered as the ring of quotients U = R G of the ring R withrespect to the Gabriel filter G . In particular, if R is commutative and U = S − R ,then G is the set of all ideals in R intersecting S .The simplest class of ring epimorphisms R −→ U for which the projective dimen-sion of the R -module U does not exceed 1 is the following one. Let S ⊂ R be a countable multiplicative subset. Then pd R S − R ≤
1. A far-reaching generaliza-tion of this elementary observation was obtained in the paper [21]. Specifically, if u : R −→ U is a left flat ring epimorphism and the related Gabriel filter of right idealsin R has a countable base, then the projective dimension of the flat left R -module U does not exceed contramodules over a topological ring(specifically, over the completion of R with respect to G ).Another basic observation is that pd R S − R ≤ R is a Noetherian com-mutative ring of Krull dimension 1. In fact, any flat module over a Noetherian ommutative ring of Krull dimension ≤ ≤ for anycommutative ring R with a multiplicative subset S ⊂ R such that R/sR is a semilocalring of Krull dimension zero for every s ∈ S , one has pd R S − R ≤ u : R −→ U . Let G denote the Gabriel filter of ideals in R related to u . We show that pd R U ≤ whenever the quotient ring R/I is semilocalof Krull dimension zero for every ideal I ∈ G . The argument is based on the notionof I -contramodule R -modules for an ideal I in a commutative ring R . This result ofours has already found its uses in the work of Bazzoni and Le Gros on envelopes andcovers in the tilting cotorsion pairs related to 1-tilting modules over commutativerings; see [4, Remark 8.6 and Theorem 8.7] and [5, Remark 8.2 and Theorem 8.17].The proofs of the pd R U ≤ u : R −→ U , one considers the two-term complex K • u = ( R → U ). When R and U are associative rings, K • u is acomplex of R - R -bimodules; in the commutative case, it is simply a complex of R -modules. To any (left) R -module B , one assigns the sequence of (left) R -modulesExt nR ( K • u , B ) = Hom D ( R – Mod ) ( K • u , B [ n ]) of homomorphisms in the derived category D ( R – Mod ), indexed by the integers n ≥
0. In particular, for n ≥ R -module isomorphism Ext nR ( K • u , B ) ≃ Ext nR ( U, B ) [7, Lemma 2.1(b)].In order to prove that Ext nR ( U, B ) = 0 for n ≥
2, one shows that the two classes of R -modules of the form Ext nR ( K • u , B ) and Ext nR ( U, B ) (where n ≥ B ∈ R – Mod )only intersect at zero. The two kinds of Ext modules just have incompatible proper-ties. In fact, Ext nR ( U, B ) is the underlying R -module of a U -module. The R -modulesExt nR ( K • u , B ) are described in terms of some kind of contramodules (it depends onthe setting which specific contramodule category is more convenient to work with).One wants to show that no nonzero R -module of the form C = Ext nR ( K • u , B ) admitsan extension of its R -module structure to a U -module structure. In fact, one usuallyproves that Hom R ( U, C ) = 0; this is certainly enough.The
Geigle–Lenzing perpendicular subcategory U ⊥ , to the R -module U in thecategory of R -modules R – Mod consists all R -modules C such that Hom R ( U, C ) =0 = Ext R ( U, C ) [11]. We use the name u -contramodules for R -modules C ∈ U ⊥ , ,and the notation R – Mod u - ctra = U ⊥ , ⊂ R – Mod for the full subcategory formed bysuch modules. Another result of this paper is a description of the abelian category R – Mod u - ctra in terms of the abelian categories of m -contramodule R -modules, where m ranges over the maximal ideals of R belonging to G . The same assumption that R/I is semilocal of Krull dimension zero for all I ∈ G is made here.Finally, let us recall that a module F over a commutative domain R is said to be strongly flat if it is a direct summand of an R -module G appearing in a short exactsequence of R -modules 0 −→ V −→ G −→ W −→ V is a free R -module nd W is a Q -vector space [27, 8]. A series of generalizations of this concept wasdeveloped in the papers [10, 6, 23, 21]. We refer to the introduction to [23] for adetailed discussion with further references.In particular, let R be a commutative ring and S ⊂ R be a multiplicative subset.Then an R -module F is called S -strongly flat [6] if it is a direct summand of an R -module G appearing in a short exact sequence of R -modules 0 −→ V −→ G −→ W −→ V is a free R -module and W is a free S − R -module. In theterminology of [6], a ring R is said to be S -almost perfect if S − R is a perfect ringand R/sR is a perfect ring for every s ∈ S . According to [6, Theorem 7.9], R is S -almost perfect if and only if all flat R -modules are S -strongly flat.More generally, given a left flat epimorphism of associative rings u : R −→ U , aleft R -module F is called U -strongly flat [21, Section 9] if it is a direct summand of aleft R -module G appearing in a short exact sequnce of left R -modules 0 −→ V −→ G −→ W −→ V is a free left R -module and W is a free left U -module.Let u : R −→ U be a flat epimorphism of commutative rings and G be the relatedGabriel filter of ideals in R . In the terminology of [5, Sections 6 and 8], a ring R issaid to be G -almost perfect if U = R G is a perfect ring and R/I is a perfect ring for all I ∈ G . We show that all flat R -modules are U -strongly flat whenever R is G -almostperfect (the converse assertion also holds). More generally, in the assumption that R/I is semilocal of Krull dimension zero for all I ∈ G , we characterize U -stronglyflat R -modules F as flat R -modules for which the U -module U ⊗ R F is projectiveand the R/I -modules
F/IF are projective.
Acknowledgement.
I am grateful to Silvana Bazzoni and Michal Hrbek for numer-ous very helpful discussions and communications. The author is supported by theGA ˇCR project 20-13778S and research plan RVO: 67985840.1. F -h-locality Recall that a commutative ring T is said to be semilocal if the set of all maximalideals in T is finite. The ring T is said to have Krull dimension zero if all the primeideals in T are maximal. The following description is standard and easy. Lemma 1. (a)
A commutative ring T is isomorphic to a finite product of local ringsif and only if T is semilocal with the additional property that every prime ideal of T is contained in a unique maximal ideal. (b) A commutative ring T is semilocal of Krull dimension zero if and only if itis isomorphic to a finite product of local rings T × · · · × T m such that, for every ≤ j ≤ m , the maximal ideal in T j consists of nilpotent elements. (cid:3) A filter of ideals (or a “linear topology”) F in a commutative ring R is a set ofideals in R such that R ∈ F , and K ⊃ I ∩ J , I , J ∈ F implies K ∈ F .Let R be a commutative ring with a filter of ideals F . An R -module N is said tobe F -torsion if, for every element x ∈ N , the annihilator of x in R belongs to F . e will say that the ring R is F -h-local [9, Section IV.3], [5, Section 7] if everyideal I ∈ F is contained only in finitely many maximal ideals of R and every primeideal of R belonging to F is contained in a unique maximal ideal. In other words, R is F -h-local if and only if for every I ∈ F the quotient ring R/I is semilocal and everyprime ideal of
R/I is contained in a unique maximal ideal. Equivalently, this meansthat the ring
R/I is a finite product of local rings (by Lemma 1(a)).As usually, we denote by Max R the set of all maximal ideals of the ring R . Givena maximal ideal m ⊂ R and an R -module N , one can consider the local ring R m and the R m -module N m , that is, the localizations of R and N at m . Conversely,any R m -module can be considered as an R -module. Notice that if N is F -torsionand m / ∈ F , then N m = 0 (indeed, for any x ∈ N , the annihilator of x in R is notcontained in m ). Lemma 2.
Let R be a commutative ring with a filter of ideals F such that the ring R is F -h-local. Given an F -torsion R -module N , consider all the maximal ideals m of thering R such that m ∈ F . Then the natural map N −→ Q m N m whose components arethe localization maps N −→ N m factorizes through the submodule L m N m ⊂ Q m N m and induces an isomorphism N ≃ L m N m .Conversely, if F is a filter of ideals in R such that for every F -torsion R -module N there is an isomorphism of R -modules N ≃ L m N m , then the ring R is F -h-local.Proof. This is [5, Proposition 7.4]. (cid:3)
It follows from Lemma 2 that, for any F -h-local ring R , the category of F -torsion R -modules is equivalent to the Cartesian product of the categories of F -torsion R m -modules, taken over the maximal ideals m of the ring R belonging to the fil-ter F . The direct sum functor( N ( m )) m ∈ F ∩ Max R M m ∈ F ∩ Max R N ( m )establishes the equivalence, with an inverse equivalence provided by the localizationfunctor N ( N m ) m ∈ F ∩ Max R . In particular, the decomposition of an R -module N into a direct sum of F -torsion R m -modules N ( m ) is unique and functorial if it exists (and it exists if and only if N is F -torsion). All these assertions can be found in the discussion in [5, Section 7].Alternatively, they can be deduced directly from Lemma 2 using the dual versionof [7, Lemma 8.6].Let R be a commutative ring with a filter of ideals F . We will say that R is F -h-nil [6, Section 6], [5, Section 7] if every ideal I ∈ F is contained only in finitely manymaximal ideals of R and all the prime ideals of R belonging to F are maximal. In otherwords, R is F -h-nil if and only if for every I ∈ F the quotient ring R/I is semilocalof Krull dimension zero. Equivalently, this means that
R/I is a finite product oflocal rings whose maximal ideals consist of nilpotent elements (by Lemma 1(b)).Obviously, any F -h-nil ring is F -h-local. e refer to the book [26, Chapter VI] for the definition of a Gabriel filter of ideals (also known as a Gabriel topology) in a ring R . A filter of ideals F in a ring R is saidto have a base of finitely generated ideals if for any ideal J ∈ F there exists a finitelygenerated ideal I ⊂ R such that I ∈ F and I ⊂ J .In the rest of this section (and partly in the next one) we will be working in thefollowing setup. Setup 3.
Let R be a commutative ring and G be a Gabriel filter of ideals in R witha base of finitely generated ideals. We assume that the ring R is G -h-nil.Given a commutative ring R and an ideal I ⊂ R , we say that an R -module N is I -torsion if for every s ∈ I and x ∈ N there exists an integer n ≥ s n x = 0in N . Lemma 4.
Assume Setup 3, and let m be a maximal ideal of R belonging to G .Then an R -module N is a G -torsion R m -module if and only if N is an m -torsion R -module.Proof. “If”: clearly, any m -torsion R -module N is an R m -module. To show that N is G -torsion, choose a finitely generated ideal I such that I ⊂ m and I ∈ G . Thenfor every x ∈ N there exists n ≥ I n x = 0 in N . Since I n ∈ G [26,Lemma VI.5.3], it follows that the annihilator of x in R belongs to G .“Only if”: let N be a G -torsion R m -module and x ∈ N be an element. Choose anideal I ∈ G annihilating x . Then the cyclic submodule Rx ⊂ N is a module over thering R m /R m I = ( R/I ) m . The ring ( R/I ) m is local and, by Setup 3 and Lemma 1(b),its maximal ideal R m m /R m I consists of nilpotent elements. Hence every module over( R/I ) m is m -torsion. (cid:3) Corollary 5.
Assuming Setup 3, an R -module N is G -torsion if and only if it isisomorphic to a direct sum of some m -torsion R -modules N ( m ) over the maximalideals m ∈ G ∩ Max R , N ≃ M m ∈ G ∩ Max R N ( m ) . Such a direct sum decomposition is unique and functorial when it exists, and the R -modules N ( m ) can be recovered as the localizations N ( m ) = N m .Proof. The ring R in Setup 3 is G -h-local (as G -h-nil implies G -h-local). Hence theassertion follows from Lemmas 2 and 4. (cid:3) I -Contramodule R -Modules Let R be a commutative ring. Given an element t ∈ R , an R -module C is saidto be a t -contramodule if Hom R ( R [ t − ] , C ) = 0 = Ext R ( R [ t − ] , C ). Given an ideal I ⊂ R , an R -module C is said to be an I -contramodule if C is a t -contramodule forevery t ∈ I . The class of all I -contramodule R -modules is closed under the kernels,cokernels, extensions, and infinite products in R – Mod [6, Lemma 5.1(1)]. The fullsubcategory of I -contramodule R -modules is denoted by R – Mod I - ctra ⊂ R – Mod . emma 6. Let R be a commutative ring and m ⊂ R be a maximal ideal. Then any m -contramodule R -module is an R m -module.Proof. This is [6, Lemma 5.1(2)]. (cid:3)
Lemma 7.
Assuming Setup 3, for any G -torsion R -module N , any R -module B ,and any integer n ≥ , the R -module P = Ext nR ( N, B ) can be presented as a productof some m -contramodule R -modules P ( m ) over the maximal ideals m ∈ G ∩ Max R , Ext nR ( N, B ) ≃ Y m ∈ G ∩ Max R P ( m ) . Proof.
Follows from Corollary 5 and [6, Lemma 6.5(1)]. (cid:3)
Lemma 8.
Let R be a commutative ring, t ∈ R be an element, D be an R [ t − ] -module,and Q be a t -contramodule R -module. Then Ext iR ( D, Q ) = 0 for all integers i ≥ .Proof. The element t acts invertibly in the R -module E = Ext iR ( D, Q ), since it actsinvertibly in D . On the other hand, the R -module E is a t -contramodule, since the R -module Q is [6, Lemma 6.5(2)]. Thus E ≃ Hom R ( R [ t − ] , E ) = 0. (cid:3) Given a filter of ideals F in a commutative ring R , we will say that the ring R is F -h-semilocal if the ring R/I is semilocal for all I ∈ F . This captures “a half of” theconditions from the definition of an F -h-local ring in Section 1. Proposition 9.
Let F be a filter of ideals in a commutative ring R such that F hasa base of finitely generated ideals and the ring R is F -h-semilocal. Let P ( m ) and Q ( m ) be two families of m -contramodule R -modules, indexed over the maximal ideals m ∈ F ∩ Max R . Then, for every i ≥ , the natural map Y m ∈ F ∩ Max R Ext iR (cid:0) P ( m ) , Q ( m ) (cid:1) −−→ Ext iR Y m ∈ F ∩ Max R P ( m ) , Y m ∈ F ∩ Max R Q ( m ) ! is an isomorphism.Proof. Fix n ∈ F ∩ Max R , and denote by P the product of P ( m ) over all m ∈ F ∩ Max R , m = n . We have to show that Ext iR ( P, Q ( n )) = 0.Let I be a finitely generated ideal in R such that I ⊂ n and I ∈ F . Since R is F -h-semilocal, the set of all maximal ideals m of the ring R containing the ideal I isfinite.Let s , . . . , s m be a finite set of generators of the ideal I . For any ideal m ∈ F ∩ Max R such that m does not contain I , there exists an integer 1 ≤ j ≤ m suchthat m does not contain s j . Denote by P j the product of the R -modules P ( m ) overall the maximal ideals m ∈ F ∩ Max R such that s j / ∈ m .By Lemma 6, P ( m ) is an R m -module. Hence s j acts invertibly in P j . On the otherhand, Q ( n ) is an s j -contramodule. By Lemma 8, it follows that Ext iR ( P j , Q ( n )) = 0.It remains to show that Ext iR ( P ( m ) , Q ( n )) = 0 for each one among the finite setof maximal ideals m ∈ F ∩ Max R such that I ⊂ m , m = n . Let s ∈ R be anelement such that s ∈ n and s / ∈ m . Then s acts invertibly in P ( m ), while Q ( n ) is an s -contramodule. Once again, it follows that Ext iR ( P ( m ) , Q ( n )) = 0. (cid:3) orollary 10. For any filter of ideals F in a commutative ring R such that F has abase of finitely generated ideals and the ring R is F -h-semilocal, the full subcategoryof all R -modules of the form Q m ∈ F ∩ Max R C ( m ) , where C ( m ) are m -contramodule R -modules, is closed under the kernels, cokernels, extensions, and infinite productsin R – Mod .Proof.
Follows from Proposition 9 (for i = 0 and 1). (cid:3) Given a complex of R -modules M • , an R -module B , and an integer n , we willdenote by Ext nR ( M • , B ) the derived category Hom module Hom D ( R – Mod ) ( M • , B [ n ]).Here D ( R – Mod ) is the derived category of R -modules.The following proposition is the key technical assertion of this paper, on which theproofs of the main results are based. Proposition 11.
Assume Setup 3. Let M • = ( M − → M ) be a two-term complexof R -modules whose cohomology modules H − ( M • ) and H ( M • ) are G -torsion. Thenfor any R -module B and any integer n ≥ , the R -module C = Ext nR ( M • , B ) can bepresented as the product of some m -contramodule R -modules C ( m ) over the maximalideals m ∈ G ∩ Max R , Ext nR ( M • , B ) ≃ Y m ∈ G ∩ Max R C ( m ) . Proof.
By Lemma 7, the R -modules Ext nR ( H − ( M • ) , B ) and Ext nR ( H ( M • ) , B ) canbe decomposed as direct products of m -contramodule R -modules for all n ≥
0. Theexistence of a similar direct product decompositions of the R -modules Ext nR ( M • , B )follows from the natural long exact sequence of R -modules · · · −−→ Ext n − ( H − ( M • ) , B ) −−→ Ext nR ( H ( M • ) , B ) −−→ Ext nR ( M • , B ) −−→ Ext n − R ( H − ( M • ) , B ) −−→ Ext n +1 ( H ( M • ) , B ) −−→ · · · in view of Corollary 10 (cf. [6, proof of Lemma 6.11] or [21, proof of Lemma 8.2]). (cid:3) An obvious generalization of Proposition 11 holds for any bounded above complexof R -modules M • with G -torsion cohomology modules and any bounded below com-plex of R -modules B • , as one can see from the related spectral sequence. We do notinclude the details, as we do not need them.3. Projective Dimension Theorem
Let u : R −→ U be a flat epimorphism of commutative rings. This means that u is a homomorphism of commutative rings such that U is a flat R -module and theinduced ring homomorphisms U ⇒ U ⊗ R U −→ U are isomorphisms.Denote by G the set of all ideals I ⊂ R such that R/I ⊗ R U = 0. Then G is aGabriel filter of ideals in R , and the ring U can be recovered as the ring of quotients U = R G of the ring R with respect to the Gabriel filter G . Gabriel filter is said to be perfect it corresponds to a flat epimorphism of ringsin this way [26, Chapter XI]. Notice that any perfect Gabriel filter G has a base offinitely generated ideals [26, Section XI.3].The main results of this paper presume the following setup. Setup 12.
Let u : R −→ U be a flat epimorphism of commutative rings, and let G be the related perfect Gabriel filter of ideals in R . We assume that, for every I ∈ G ,the quotient ring R/I is semilocal of Krull dimension zero (in other words, the ring R is G -h-nil in our terminology).Following [19, Section 13], [20, Section 1], [23, Section 3], [6, Section 4], [21, Sec-tion 8], [7, Section 2], etc., we will denote by K • u the two-term complex of R -modules R −→ U , with the term R placed in the cohomological degree − U placed in the cohomological degree 0. The next corollary is the particular case ofProposition 11 which we will use. Corollary 13.
Assuming Setup 12, for any R -module B and any integer n ≥ , the R -module C = Ext nR ( K • u , B ) can be presented as the product of some m -contramodule R -modules C ( m ) over the maximal ideals m ∈ G ∩ Max R , Ext nR ( K • u , B ) ≃ Y m ∈ G ∩ Max R C ( m ) . Proof.
The complex of U -modules U ⊗ R K • u is contractible, hence the cohomologymodules H − ( K • u ) and H ( K • u ) of the complex K • u are G -torsion R -modules. There-fore, Proposition 11 is applicable. (cid:3) The aim of this section is to prove the following theorem, which is our most im-portant result.
Theorem 14.
Assuming Setup 12, the projective dimension of the R -module U can-not exceed . The argument is based on Corollary 13. We need several more lemmas beforeproceeding to prove the theorem.
Lemma 15.
Let R be a commutative ring, t ∈ R be an element, and C be a t -contramodule R -module. Then C/tC = 0 whenever C = 0 .Proof. Assume that C = tC for an R -module C . Then one easily observes that thenatural R -module map Hom R ( R [ t − ] , C ) −→ C is surjective. If C is a t -contramodule,then Hom R ( R [ t − ] , C ) = 0, and it follows that C = 0. (cid:3) Lemma 16.
Let R be a commutative ring and I = ( s , . . . , s m ) ⊂ R be a finitelygenerated ideal. Let C be an I -contramodule R -module. Then C/IC = 0 whenever C = 0 .Proof. Assume that C = 0. Then, by Lemma 15, we have C/s C = 0. Furthermore, C/s C is still an I -contramodule, since it is the cokernel of the R -module morphism s : C −→ C between two I -contramodules. Applying Lemma 15 again, we see that C/ ( s C + s C ) = 0, etc. (cid:3) emma 17. Let u : R −→ U be a flat epimorphism of commutative rings, and let G be the related perfect Gabriel filter of ideals in R . Suppose that D is a U -moduleand C is an I -contramodule R -module, where I ∈ G . Suppose further that there is asurjective R -module morphism D −→ C . Then C = 0 .Proof. Without loss of generality we can assume the ideal I ∈ G to be finitely gener-ated (because the filter G has a base of finitely generated ideals). We have D/ID = 0,since
R/I ⊗ R U = 0. As the morphism D/ID −→ C/IC is surjective, it follows that
C/IC = 0. By Lemma 16, we can conclude that C = 0. (cid:3) Proof of Theorem 14.
Let B be an R -module. Then, for every integer n ≥ R -modules Ext nR ( K • u , B ) ≃ Ext nR ( U, B ) (see [7,Lemma 2.1(b)]; cf. [6, Lemma 4.8(3)]). By Corollary 13, we have Ext nR ( K • u , B ) ≃ Q m ∈ G ∩ Max R C ( m ), where C ( m ) are some m -contramodule R -modules. On the otherhand, D = Ext nR ( U, B ) is a U -module. For every m , the R -module C ( m ) is a quotient(in fact, a direct summand) of the R -module D . By Lemma 17, it follows that C ( m ) = 0, hence Ext nR ( U, B ) ≃ Ext nR ( K • u , B ) = 0. (cid:3) Divisible Modules
The following definitions, generalizing the classical notions of S -divisible and S -h-divisible modules for a multiplicative subset S ⊂ R and the related ring epimorphism R −→ S − R [20, Section 1], are quite natural. Definition 18.
Given a filter of ideals F in a commutative ring R and an R -module D , we say that D is F -divisible [26, Section VI.9] if R/I ⊗ R D = 0 for all I ∈ F . Definition 19.
Let u : R −→ U be an epimorphism of commutative rings. We saythat an R -module D is u -divisible (or u -h-divisible ) [7, Remark 1.2(1)] if D is aquotient R -module of a U -module.For a flat epimorphism u and the related Gabriel filter G , one can immediatelysee that any u -divisible R -module is G -divisible. The converse is not true in general,even when U = S − R [1, Proposition 6.4], [20, Lemma 1.8(b)]. Theorem 20.
Let u : R −→ U be a flat epimorphism of commutative rings, andlet G be the related perfect Gabriel filter of ideals in R . Assume that the projectivedimension of the R -module U does not exceed . Then the classes of u -divisible and G -divisible R -modules coincide.Proof. The following argument based on the results of the paper [2] was communi-cated to the author by S. Bazzoni. Denote the R -module coker( u ) = U/u ( R ) simplyby U/R . Then, by [15, Example 6.5], the direct sum U ⊕ U/R is a silting R -module.The related silting class is the class of all quotient modules of direct sums of copiesof U ⊕ U/R , which means the class of all u -divisible R -modules. By [2, Theorem 4.7],for any silting class over a commutative ring R there exists a Gabriel filter G (witha base of finitely generated ideals) in R such that the silting class consists of all the -divisible R -modules. So the class of all u -divisible R -modules coincides with theclass of all G -divisible R -modules for some Gabriel filter G . It follows immediatelythat G is the perfect Gabriel filter related to u , as desired. (cid:3) Remark 21.
For injective flat epimorphisms of commutative rings u : R −→ U (i. e.,when the map u is injective), the following converse assertion to Theorem 20 holds.If all G -divisible R -modules are u -divisible, then the projective dimension of the R -module U does not exceed 1. This is a part of [13, Theorem 5.4].However, there do exist noninjective flat commutative ring epimorphisms u ofprojective dimension more than 1 for which the class of u -divisible modules coincideswith that of G -divisible ones. In fact, such examples exist already among the maps oflocalization by multiplicative subsets u : R −→ S − R = U . Consequently, neither [1,Proposition 6.4] nor [20, Lemma 1.8(b)] hold true for multiplicative subsets S ⊂ R containing zero-divisors. The following transparent construction of counterexampleswas communicated to the author by M. Hrbek.Let R be a von Neumann regular commutative ring and I ⊂ R be an ideal. Then I is generated by some set of idempotent elements e j ∈ R . Let S ⊂ R be themultiplicative subset generated by the complementary idempotents f j = 1 − e j . Then S − R ≃ R/I . Furthermore, an R -module M is annihilated by e j for a given index j if and only if f j acts invertibly in M , and if and only if f j acts by a surjectiveendomorphism of M . Hence an R -module M is annihilated by the whole ideal I ifand only if all the elements of S act invertibly in M , and if and only if all the elementsof S act in M by surjective maps.Consequently, all S -divisible R -modules are S -h-divisible (moreover, all of themare S − R -modules). In other words, if u : R −→ S − R = R/I = U is the naturalsurjective flat epimorphism of rings and G is the related perfect Gabriel filter in R (i. e., the filter of all ideals intersecting S ), then the classes of u -divisible and G -divisible R -modules coincide. (Cf. the discussion of silting and cosilting classesover von Neumann regular commutative rings in [14, Section 1.5.3].)On the other hand, it is well-known that there exist von Neumann regular commu-tative rings of arbitrary homological dimension (see, e. g., [18]). So one can choose R and I so as to make the projective dimension of the R -module R/I to be equal toany chosen nonnegative integer or infinity.
Corollary 22.
Assuming Setup 12, an R -module is u -divisible if and only if it is G -divisible.First proof. Follows immediately from Theorems 14 and 20. (cid:3)
In order to illustrate the workings of contramodule techniques, in the rest of thissection we present an alternative proof of Corollary 22. For this purpose, we needthe following strengthening of the lemmas from Section 3.
Proposition 23.
Let R be a commutative ring and I = ( s , . . . , s m ) ⊂ R be afinitely generated ideal. Let B be an R -module such that B/IB = 0 and C be an I -contramodule R -module. Then Hom R ( B, C ) = 0 . roof. In other words, the proposition says that an I -contramodule R -module C hasno R -submodules D for which ID = D . This is provable using infinite summationoperations, as hinted in [19, Lemma 4.2].Alternatively, one can use the reflector ∆ I : R – Mod −→ R – Mod I - ctra onto the fullsubcategory of I -contramodule R -modules R – Mod I - ctra ⊂ R – Mod (that is, the leftadjoint functor to the inclusion R – Mod I - ctra −→ R – Mod ). The functor ∆ I wasconstructed in [19, Theorem 7.2]. Any R -module morphism B −→ C into an I -contramodule R -module C factorizes uniquely as B −→ ∆ I ( B ) −→ C , where B −→ ∆ I ( B ) is the adjunction morphism.The functor ∆ I is R -linear and right exact, so applying it to a surjective morphism( s , . . . , s m ) : B m −→ B produces a surjective morphism ( s , . . . , s m ) : ∆ I ( B ) m −→ ∆ I ( B ). Hence B/IB = 0 implies ∆ I ( B ) /I ∆ I ( B ) = 0. But ∆ I ( B ) is an I -contra-module, so by Lemma 16 it follows that that ∆ I ( B ) = 0. Thus any R -modulemorphism B −→ C vanishes. (cid:3) Second proof of Corollary 22.
For any R -module B , there is a natural 5-term exactsequence of R -modules( ∗ ) 0 −−→ Ext R ( K • u , B ) −−→ Hom R ( U, B ) −−→ B −−→ Ext R ( K • u , B ) −−→ Ext R ( U, B ) −−→ D ( R – Mod ) ( − , B ) to the distinguished triangle R −−→ U −−→ K • u −−→ R [1]in D ( R – Mod ) (cf. [6, formula ( ∗∗ ) in Section 4], [21, formula (8.2)], or [7, for-mula (9)]). By Corollary 13, we have an isomorphism of R -modules Ext R ( K • u , B ) ≃ Q m ∈ G ∩ Max R C ( m ), where C ( m ) are some m -contramodule R -modules.Now assume that the R -module B is G -divisible. Given a maximal ideal m ∈ G ∩ Max R , choose a finitely generated ideal I ∈ G such that I ⊂ m . Then C ( m )is an I -contramodule R -module and B/IB = 0. By Proposition 23, it follows thatHom R ( B, C ( m )) = 0. As this holds for all m ∈ G ∩ Max R , we can conclude that themap B −→ Ext R ( K • u , B ) vanishes. Hence the map Hom R ( U, B ) −→ B is surjectiveand the R -module B is u -divisible. (cid:3) u -Contramodule R -Modules In this section we obtain a description of the Geigle–Lenzing perpendicular sub-category R – Mod u - ctra = U ⊥ , in the category of R -modules R – Mod .An R -module C is said to be a u -contramodule [7, Section 1] if C ∈ U ⊥ , , that isHom R ( U, C ) = 0 = Ext R ( U, C ). Assuming Setup 12, the R -module U has projectivedimension at most 1 by Theorem 14. By [11, Proposition 1.1] or [19, Theorem 1.2(a)],it follows that the full subcategory of u -contramodule R -modules R – Mod u - ctra is closedunder the kernels, cokernels, extensions, and infinite products in R – Mod .The functor ∆ u = Ext R ( K • u , − ) plays an important role as the reflector onto thefull subategory R – Mod u - ctra ⊂ R – Mod [7, Proposition 3.2(b)]. emma 24. Let u : R −→ U be a flat epimorphism of commutative rings and I ⊂ R be an ideal such that IU = U . Then any R/I -module is a u -contramodule.Proof. Since U is a flat R -module, by [24, Lemma 4.1(a)] we have Ext iR ( U, D ) ≃ Ext iR/I ( U/IU, D ) = 0 for any
R/I -module D and all i ≥ (cid:3) Proposition 25.
Let R −→ U be a flat epimorphism of commutative rings, andlet G be the related perfect Gabriel filter of ideals in R . Assume that the projectivedimension of the R -module U does not exceed . Then, for any ideal I ∈ G , any I -contramodule R -module is a u -contramodule.Proof. Without loss of generality we can assume the ideal I to be finitely generated.Let C be an I -contramodule R -module. Then Hom R ( U, C ) = 0 by Proposition 23.There are several ways to prove that Ext R ( U, C ) = 0. One can observe thatany I -contramodule R -module is obtainable as an extension of two quotseparated I -contramodule R -modules [24, Section 5.5], [22, Section 1]; any quotseparated I -contramodule R -module is the cokernel of an injective morphism between two I -adically separated and complete modules; and finally any I -adically separatedand complete module is the kernel of a morphism of R -modules of the formHom Z ( N, Q / Z ), where N ranges over the G -torsion R -modules [21, Proposition 5.6].Since R -modules of the latter form belong to R – Mod u - ctra , it follows that all the I -contramodule R -modules do.Alternatively, one can say that all I -contramodule R -modules C are simply rightobtainable from R/I -modules (i. e., obtainable using the passages to extensions, cok-ernels of monomorphisms, infinite products, and infinitely iterated extensions in thesense of the projective limit [24, Section 3], [23, Section 2]). This is the assertionof [24, Lemma 8.2] (see also [19, proof of Theorem 9.5]). Since Ext iR ( U, D ) = 0 forany
R/I -module D and all i > iR ( U, C ) = 0 for i > (cid:3)
Theorem 26.
Assuming Setup 12, an R -module C is a u -contramodule if and onlyif it is isomorphic to a product of some m -contramodule R -modules C ( m ) over themaximal ideals m ∈ G ∩ Max R , C ≃ Y m ∈ G ∩ Max R C ( m ) . Such an infinite product decomposition is unique and functorial when it exists, andthe R -modules C ( m ) can be recovered as the colocalizations C ( m ) = Hom R ( R m , C ) .Proof. The assertion “if” is provided by Proposition 25. To prove the “only if”,notice that the natural morphism C −→ Ext R ( K • u , C ) is an isomorphism for any u -contramodule R -module C (in view of the exact sequence ( ∗ ) from Section 4).Now Corollary 13 provides the desired direct product decomposition.Let us compute the colocalizations. Any m -contramodule R -module C ( m ) is an R m -module by Lemma 6, hence Hom R ( R m , C ( m )) = C ( m ). On the other hand, forany maximal ideal n = m of the ring R , we have Hom R ( R n , C ( m )) = 0 by Lemma 8 choose any element t ∈ m \ n ). Finally, our direct product decomposition is uniqueand functorial by Proposition 9 (for i = 0). (cid:3) Thus, assuming Setup 12, the category of u -contramodule R -modules is equivalentto the Cartesian product of the categories of m -contramodule R -modules, taken overthe maximal ideals m of the ring R belonging to the filter G . The product functor( C ( m )) m ∈ G ∩ Max R Y m ∈ G ∩ Max R C ( m )establishes the equivalence, with an inverse equivalence provided by the colocalizationfunctor C ( C m = Hom R ( R m , C )) m ∈ G ∩ Max R . U -Strongly Flat R -Modules We refer to [21, beginning of Section 9] for a general discussion of U -weakly co-torsion and U -strongly flat left R -modules for a left flat epimorphism of associativerings u : R −→ U . In this section, as in the rest of this paper, we restrict ourselvesto the commutative case.Let u : R −→ U be a flat epimorphism of commutative rings. A left R -module C is said to be U -weakly cotorsion if Ext R ( U, C ) = 0. A left R -module F is said to be U -strongly flat if Ext R ( F, C ) = 0 for all U -weakly cotorsion R -modules C .By [11, Theorem 4.4], we have Ext iR ( U, U ( X ) ) ≃ Ext iU ( U, U ( X ) ) = 0 for all i > X ; in particular, Ext R ( U, U ( X ) ) = 0 (where U ( X ) denotes the direct sumof X copies of U ). Hence [12, Corollary 6.13] provides the following description of U -strongly flat R -modules. An R -module F is U -strongly flat if and only if it is adirect summand of an R -module G appearing in a short exact sequence of R -modules0 −−→ V −−→ G −−→ W −−→ , where V is a free R -module and W is a free U -module.The notion of a simply right obtainable module (from a given class of modules) wasalready mentioned in the proof of Proposition 25. We refer to [24, Section 3] or [23,Section 2] for a detailed discussion. Proposition 27.
Assuming Setup 12, the class of all u -contramodule R -modules R – Mod u - ctra ⊂ R – Mod coincides with the class of all R -modules simply right obtain-able from R/I -modules, where I ranges over the Gabriel filter G .Proof. By Theorem 26, any u -contramodule R -module is obtainable as a product of m -contramodules over the maximal ideals m ∈ G ∩ Max R . For any m , there existsa finitely generated ideal I ⊂ m such that I ∈ G . Then any m -contramodule is an I -contramodule. Finally, all I -contramodule R -modules are simply right obtainablefrom R/I -modules by [23, Lemma 8.2] or [19, proof of Theorem 9.5].Conversely, the class of all u -contramodule R -modules is closed under the kernels,cokernels, extensions, and infinite products in R – Mod by [11, Proposition 1.1] or [19,Theorem 1.2(a)]. Since all the
R/I -modules with I ∈ G belong to R – Mod u - ctra y Lemma 24, it follows that all the R -modules simply right obtainable from R/I -modules, I ∈ G , belong to R – Mod u - ctra . (cid:3) Proposition 28.
Assuming Setup 12, the class of all U -weakly cotorsion R -modulescoincides with the class of all R -modules simply right obtainable from U -modules and R/I -modules with I ∈ G .Proof. For any R -module B , it is clear from Corollary 13 and Proposition 25 thatboth the R -modules Ext R ( K • u , B ) and Ext R ( K • u , B ) are u -contramodules (cf. [7,Lemma 2.6(a) and (c)]). Now it follows from the exact sequence ( ∗ ) from Sec-tion 4 that any U -weakly cotorsion R -module C is obtainable as an extension ofa u -contramodule Ext R ( K • u , C ) and the cokernel of an injective morphism from a u -contramodule to a U -module Ext R ( K • u , C ) −→ Hom R ( U, C ). It remains to useProposition 27 for obtainability of u -contramodules.Conversely, all R/I -modules are U -weakly cotorsion R -modules by Lemma 24,and all U -modules D are U -weakly cotorsion R -modules, since Ext iR ( U, D ) ≃ Ext iU ( U, D ) = 0 for i > R -modules simply right obtainable from U -modules and R/I -modules are U -weakly cotorsion. (cid:3) The following theorem can be viewed as confirming a version of [23, OptimisticConjecture 1.1] for flat epimorphisms of commutative rings (cf. [21, Corollary 9.7]).It is also a generalization of [6, Proposition 7.13].
Theorem 29.
Assume Setup 12, and let F be a flat R -module. Then F is U -stronglyflat if and only if it satisfies the following two conditions: (i) the U -module U ⊗ R F is projective; (ii) for every ideal I ∈ G , the R/I -module
F/IF is projective.Proof.
The “only if” assertion holds for any flat epimorphism of commutative rings u : R −→ U and the related perfect Gabriel filter G . It is clear from the description of U -strongly flat R -modules F as the direct summands of the R -modules G appearingin short exact sequences of R -modules 0 −→ V −→ G −→ W −→ R -module V and a free U -module W that conditions (i–ii) are satisfied for F .Our proof of the “if” depends on the assumption of Setup 12. Let F be a flat R -module satisfying (i–ii), and let C be a U -weakly cotorsion R -module. We needto show that Ext R ( F, C ) = 0, but it will be more convenient for us to prove thatExt iR ( F, C ) = 0 for all i >
0. By Proposition 28, C is simply right obtainable from U -modules and R/I -modules with I ∈ G . According to [24, Lemma 3.4], it sufficesto consider the cases when C is either a U -module or an R/I -module.For any U -module D , we have Ext iR ( F, D ) ≃ Ext iU ( U ⊗ R F, D ) by [24,Lemma 4.1(a)]. In view of condition (i), Ext iU ( U ⊗ R F, D ) = 0. Similarly, forany
R/I -module D , we have Ext iR ( F, D ) ≃ Ext iR/I ( F/IF, D ) by [24, Lemma 4.1(a)].In view of condition (ii), Ext iR/I ( F/IF, D ) = 0. (cid:3)
Let u : R −→ U be a flat epimorphism of commutative rings and G be the relatedperfect Gabriel filter of ideals in R . Following [5, Sections 6 and 8], we will say hat the ring R is G -almost perfect if the ring U = R G is perfect and the rings R/I are perfect for all I ∈ G . Clearly, the ring epimorphism u : R −→ U satisfiesSetup 12 whenever R is G -almost perfect for the related Gabriel filter G (as all perfectcommutative rings are semilocal of Krull dimension zero). Corollary 30.
Let G be the perfect Gabriel filter of ideals related to a flat epimor-phism of commutative rings u : R −→ U . Then the ring R is G -almost perfect if andonly if all the flat R -modules are U -strongly flat.Proof. The “if” assertion is provable similarly to [6, Lemma 7.8]. One shows thatall flat U -modules are projective whenever all flat R -modules are U -strongly flat,and all Bass flat R/I -modules are projective whenever all Bass flat R -modules are U -strongly flat. We skip the (straightforward) details.The implication “only if” is a simple corollary of Theorem 29. Assume that thering R is G -almost perfect; then Setup 12 is satisfied. Let F be a flat R -module.Then the U -module U ⊗ R F is flat, and the R/J -module
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Institute of Mathematics of the Czech Academy of Sciences, ˇZitn´a 25, 115 67Praha 1 (Czech Republic); andLaboratory of Algebra and Number Theory, Institute for Information Transmis-sion Problems, Moscow 127051 (Russia)
E-mail address : [email protected]@math.cas.cz