Covering classes and 1-tilting cotorsion pairs over commutative rings

1 Introduction

The classification problem for the category of modules over arbitrary rings is in general hopeless. Approximation theory was developed as a tool to approximate arbitrary modules by modules in certain classes where the classification is more manageable. Left and right approximations were studied in the case of modules over finite-dimensional algebras by Auslander, Reiten and Smalø and independently by Enochs and Xu for modules over arbitrary rings using the terminology of preenvelopes and precovers. An important problem in approximation theory is when minimal approximations, that is covers or envelopes, over certain classes exist. In other words, for a class 𝒞 of modules, the aim is to characterise the rings over which every module has a minimal approximation in 𝒞 and furthermore to characterise the class 𝒞 itself.

In this paper, we study when, for a 1-tilting cotorsion pair (𝒜,𝒯) over a commutative ring R, the class 𝒜 provides for covers.

One of the first examples of the power of studying the module category using the existence of minimal approximations was done for the class of projective modules by Bass [4]. Bass introduced and characterised the class of perfect rings which are exactly the rings over which every module has a projective cover. Moreover, he showed that this is equivalent to the class of projective modules being closed under direct limits. It is interesting to study this closure property for the general case of covering classes.

In fact, a famous theorem of Enochs says that if a class 𝒞 in Mod⁢-⁢R is closed under direct limits, then any module that has a 𝒞-precover has a 𝒞-cover [13]. The converse problem, that is if a covering class 𝒞 is necessarily closed under direct limits, is still an open problem which is known as Enochs Conjecture.

Some significant advancements have been made towards Enochs Conjecture in recent years. In 2017, Angeleri Hügel–Šaroch–Trlifaj in [3] proved that Enochs Conjecture holds for a large class of cotorsion pairs. Explicitly, they proved that, for a cotorsion pair (𝒜,ℬ) such that ℬ is closed under direct limits, 𝒜 is covering if and only if it is closed under direct limits. In particular, this holds for all tilting cotorsion pairs. The result of Angeleri Hügel–Šaroch–Trlifaj is based on methods developed in Šaroch’s paper [25], which uses sophisticated set-theoretical methods in homological algebra.

Moreover, in a very recent paper by Bazzoni–Positselski–Šťovíček in [10], it is proved using an algebraic approach that, for a cotorsion pair (𝒜,ℬ) such that ℬ is closed under direct limits and a module M∈𝒜∩ℬ, if Add⁢(M) is covering, then Add⁢(M) is closed under direct limits. In particular, it follows that Enochs Conjecture holds for tilting cotorsion pairs.

We are interested in giving ring-theoretic characterisations of the commutative rings for which the class 𝒜 of a 1-tilting cotorsion pair provides for covers, by purely algebraic methods. As mentioned in the previous paragraphs, Enochs Conjecture is known to hold for these cotorsion pairs, although via our characterisation of such rings, we find yet another proof of Enochs Conjecture.

In our investigation, we use extensively the bijective correspondence between 1-tilting cotorsion pairs over commutative rings and faithful finitely generated Gabriel topologies, as demonstrated by Hrbek in [16]. More precisely, Hrbek associates to a 1-tilting class 𝒯 the collection of ideals J≤R which “divide” 𝒯, that is {J∣J⁢T=T⁢for all⁢T∈𝒯}, and in the converse direction, he associates to a faithful finitely generated Gabriel topology 𝒢 the 1-tilting class, denoted 𝒟𝒢, of the 𝒢-divisible modules, that is the modules M such that J⁢M=M for every J∈𝒢.

We are interested in a particular type of Gabriel topology. The ring of quotients of R with respect to 𝒢, denoted R𝒢, is 𝒢-divisible if and only if 𝒢 arises from a perfect localisation, that is ψR:R→R𝒢 is a flat ring epimorphism and 𝒢={J≤R∣ψR⁢(J)⁢R𝒢=R𝒢}. Following Stenström’s terminology [24], these Gabriel topologies are called perfect Gabriel topologies.

In Section 4, we first prove that if the class 𝒜 in a 1-tilting cotorsion pair (𝒜,𝒟𝒢) over a commutative ring is covering, then 𝒢 arises from a perfect localisation and that the projective dimension of R𝒢 is at most one as an R-module (see Lemma 4.3 and Proposition 4.4). Then, by work of Angeleri Hügel–Sánchez in [2], R𝒢⊕R𝒢/R is a 1-tilting module with associated cotorsion pair (𝒜,𝒟𝒢).

In this situation, we have a much larger range of theories to use, in particular the works of Bazzoni–Positselski and Positselski.

Indeed, in [7], using the theory of contramodules and the tilting-cotilting correspondence from [22], Bazzoni–Positselski give classification results for some 1-tilting cotorsion pairs satisfying Enochs Conjecture – those which arise from injective homological ring epimorphisms u:R→U in the sense of [2].

In [6], we prove that if a 1-tilting class over a commutative ring R is enveloping, then the tilting module arises from an injective flat ring epimorphism and give a ring-theoretic characterisation of the ring in terms of perfectness of the quotient rings R/J for every ideal J in the associated Gabriel topology.

This paper concerns the covering side. We briefly summarise the results of this paper as well as some consequences. Consider a 1-tilting cotorsion pair (𝒜,𝒯) over a commutative ring and the associated Gabriel topology 𝒢. After proving that if 𝒜 is covering, then 𝒢 is a perfect Gabriel topology and the projective dimension of R𝒢 is at most one, we show the following characterisation (Theorem 8.19):

It is interesting to note that if R𝒢 is a perfect ring, then it follows that 𝒢 is a perfect Gabriel topology (see Lemma 8.1).

The above characterisation uses [9, Theorem 1.2], where Bazzoni and Positselski state that, for a (not necessarily injective) ring epimorphism u:R→U such that Tor1R⁡(U,U)=0 and K:=U/u(R), there is a Matlis equivalence between the full subcategory of u-divisible u-comodules and the full subcategory of u-torsion-free u-contramodules via the adjunction pair ((-⊗RK),HomR(K,-)). When u:R→U is a flat injective ring epimorphism of commutative rings as in our case, this becomes an equivalence between the 𝒢-divisible 𝒢-torsion modules and the 𝒢-torsion-free u-contramodules.

This paper is structured as follows. We begin with some general preliminaries where we introduce minimal approximations and 1-tilting classes in Section 2. Section 3 makes up the background to our main results. We introduce Gabriel topologies before relating them to some more recent advancements with tilting classes, as well as providing some of our own results. Next, in Section 4, we have an initial main result for a 1-tilting cotorsion pair (𝒜,𝒟𝒢) over a commutative ring R such that 𝒜 is covering. We show that 𝒢 is a perfect localisation and that R𝒢⊕R𝒢/R is a 1-tilting module for the 1-tilting cotorsion pair (𝒜,𝒟𝒢). Next we introduce topological rings and u-contramodules for a ring epimorphism u in Section 5, which will be used for our main results. These results are collected from various papers of Positselski and Bazzoni–Positselski. In Section 6, we again consider a 1-tilting cotorsion pair (𝒜,𝒟𝒢) over a commutative ring R such that 𝒜 is covering and show that R is 𝒢-almost perfect. In Section 7, we introduce ℋ-h-local rings with respect to a linear topology ℋ over a commutative ring, as a generalisation of results in [8]. In Section 8, we show the converse of the combination of Sections 4 and 6. That is, if (𝒜,𝒟𝒢) is a 1-tilting cotorsion pair over a commutative ring R and R is 𝒢-almost perfect, we show that 𝒜 is covering.

2 Preliminaries

In this section, we recall some definitions and some notation.

All rings will be associative with a unit, Mod⁢-⁢R (R⁢-⁢Mod) the category of right (left) R-modules over the ring R, and mod⁢-⁢R the full subcategory of Mod⁢-⁢R which is composed of all the modules which have a projective resolution consisting of finitely generated projective modules.

For an element x of a right R-module M, let Ann⁢(x) denote the annihilator ideal of x, that is

For a right R-module M and a right ideal I of R, we let M⁢[I] denote the submodule of M of elements which are annihilated by the ideal I. That is, M[I]:={x∈M∣xI=0}.

Let 𝒞 be a class of right R-modules. The right ExtR1-orthogonal and right ExtR∞-orthogonal classes of 𝒞 are defined as follows:

The left Ext-orthogonal classes 𝒞⊥1 and 𝒞⊥ are defined symmetrically.

If the class 𝒞 has only one element, say 𝒞={X}, we write X⊥1 instead of {X}⊥1, and similarly for the other Ext-orthogonal classes.

We denote by 𝒫n⁢(R) (ℱn⁢(R), ℐn⁢(R)) the class of right R-modules of projective (flat, injective) dimension at most n, or simply 𝒫n (ℱn, ℐn) when the ring is clear from the context. We let 𝒫n⁢(mod⁢-⁢R) denote the intersection of mod⁢-⁢R and 𝒫n⁢(R). The projective dimension (weak or flat dimension, injective dimension) of a right R-module M is denoted p⁢.⁢dim⁡MR (w⁢.⁢dim⁡MR, inj⁢.⁢dim⁡MR).

Given a ring R, the right big finitistic dimension, F⁢.⁢dim⁡R, is the supremum of the projective dimension of right R-modules with finite projective dimension. The right little finitistic dimension, f.dim R, is the supremum of the projective dimension of right R-modules in mod⁢-⁢R with finite projective dimension.

For an R-module C, we let Add⁢(C) denote the class of R-modules which are direct summands of direct sums of copies of C, and Gen⁡(C) the class of R-modules which are homomorphic images of direct sums of copies of C.

Recall that A is a pure submodule of a right R-module B, or A⊆∗B, if for each finitely presented right module F, the functor HomR⁡(F,-) is exact when applied to the short exact sequence

or equivalently, when for every left R-module M, the functor (-⊗RM) is exact when applied to sequence (2.1). The embedding A↪B is called a pure embedding, the epimorphism B↠B/A a pure epimorphism and the short exact sequence (2.1) a pure exact sequence.

Short exact sequences arising from the canonical presentation of a direct limit form an important class of examples of pure exact sequences.

A module X is called Σ-pure-split if every pure embedding A⊆∗B with B∈Add⁢(X) splits.

2.2 Covers, precovers and cotorsion pairs

For this section, 𝒞 will be a class of right R-modules closed under isomorphisms and direct summands. We recall the definitions of precovers and covers as well as some properties of covers and covering classes.

Many of the results in this section are taken from Xu’s book [26], which generalises work based on Enochs’ paper [13], where he works mainly in the setting where 𝒞 is the class of injective modules or flat modules. For this reason, many results are attributed to Enochs–Xu rather than just Enochs.

Definition 2.2.

A 𝒞-precover of M is a homomorphism ϕ:C→M, where C∈𝒞 with the property that, for every homomorphism f:C′→M, where C′∈𝒞, there exists f′:C′→C such that ϕ⁢f′=f:

A 𝒞-cover of M is a 𝒞-precover with the additional property that, for every homomorphism f:C→C such that ϕ⁢f=ϕ, f is an isomorphism:

A 𝒞-precover ϕ:C→M of M is called a special 𝒞-precover if ϕ is an epimorphism and Ker⁡ϕ∈𝒞⊥.

If every R-module has a 𝒞-cover (𝒞-precover, special 𝒞-precover), the class 𝒞 is called covering (respectively, precovering, special precovering). If a cover does exist for a module M, we can describe the relationship between a 𝒞-cover and a 𝒞-precover of M.

Theorem 2.3 ([26, Theorem 1.2.7]).

Suppose C is a class of modules and M admits a C-cover and ϕ:C→M is a C-precover. Then C=C′⊕K for submodules C′,K of C such that the restriction ϕ↾C′ gives rise to a C-cover of M and K⊆Ker⁡ϕ.

Corollary 2.4 ([26, Corollary 1.2.8]).

Suppose M admits a C-cover. Then a C-precover ϕ:C→M is a C-cover if and only if there is no non-zero direct summand K of C contained in Ker⁡ϕ.

The following two theorems will be useful when working with covers.

Theorem 2.5 ([26, Theorem 1.4.7, Theorem 1.4.1]).

Suppose for each integer n≥1, ϕn:Cn→Mn is a C-cover.

1. If ⊕nϕn:⊕nCn→⊕nMn is a 𝒞-precover, then it is also a 𝒞-cover.

2. The direct sum ⊕μn:⊕nCn→⊕nMn is a 𝒞-cover of ⊕nMn if and only if, for fn:Cn→Cn+1 a family of homomorphisms such that Im⁡fn⊆Ker⁡ϕn+1, for each x∈C1, there is an integer m such that

   fm⁢fm-1⁢…⁢f1⁢(x)=0.

A pair of classes of modules (𝒜,ℬ) is a cotorsion pair provided that 𝒜=ℬ⊥1 and ℬ=𝒜⊥1. A cotorsion pair is called complete if ℬ is special preenveloping or equivalently 𝒜 is special precovering. A famous result due to Eklof–Trlifaj states that if 𝒮 is a set, the cotorsion pair ((𝒮⊥1)⊥1,𝒮⊥1) generated by 𝒮 is complete (see [15, Theorem 6.11]).

A cotorsion pair (𝒜,ℬ) is called hereditary if ExtRi⁡(A,B)=0 for every A∈𝒜, B∈ℬ and i>0 (see [15, Lemma 5.24]). Thus if a cotorsion pair (𝒜,ℬ) is hereditary, then 𝒜=ℬ⊥ and ℬ=𝒜⊥; thus there is no need to differentiate between ⊥1 and ⊥.

A cotorsion pair (𝒜,ℬ) is of finite type if there is a set 𝒮 of modules in mod⁢-⁢R such that 𝒮⊥=ℬ (recall mod⁢-⁢R denotes the class of modules admitting a projective resolution consisting of finitely generated projective modules). In other words, (𝒜,ℬ) is of finite type if and only if ℬ=(𝒜∩mod⁢-⁢R)⊥.

3 Gabriel topologies

In this section, we recall the notions of torsion pairs and Gabriel topologies as well as proving some results that will be useful to us later on. We will conclude by discussing some advancements that relate Gabriel topologies to 1-tilting classes over commutative rings. The reference for this section, in particular for torsion pairs and Gabriel topologies, is Stenström’s book [24, Chapters VI, IX and XI].

We will start by giving definitions in the case of a general ring with unit (not necessarily commutative). Everything will be done with reference to right R-modules (and right Gabriel topologies), but everything can be done analogously for left R-modules.

A torsion pair(ℰ,ℱ) in Mod⁢-⁢R is a pair of classes of modules in Mod⁢-⁢R which are mutually orthogonal with respect to the Hom-functor and maximal with respect to this property. That is,

The class ℰ is called a torsion class and ℱ a torsion-free class.

Torsion and torsion-free classes are characterised by closure properties: A class 𝒞 of modules is a torsion class if and only if it is closed under extensions, direct sums and epimorphic images, and 𝒞 is a torsion-free class if and only if it is closed under extensions, direct products and submodules [24, Propositions VI.2.1 and VI.2.2]. A torsion pair (ℰ,ℱ) is called hereditary if the torsion class ℰ is also closed under submodules, which is equivalent to ℱ being closed under injective envelopes.

A ring R is a topological ring if it has a topology such that the ring operations are continuous. A topological ring R is right linearly topological if it has a topology with a basis of neighbourhoods of zero consisting of right ideals of R. A ring R with a right Gabriel topology is an example of a right linearly topological ring (see [24, Section VI.4]).

A right Gabriel topology 𝒢 on a ring R is a filter of open right ideals in a right linear topology on R satisfying an extra condition. This condition is such to guarantee that there is a bijective correspondence between right Gabriel topologies 𝒢 on R and hereditary torsion classes in Mod⁢-⁢R (see [24, Theorem VI.5.1]).

The bijection is given by the following assignments:

The torsion pair corresponding to a Gabriel topology 𝒢 will be denoted by (ℰ𝒢,ℱ𝒢). It is generated by the cyclic modules R/J, where J∈𝒢, so ℱ𝒢 consists of the modules N such that HomR⁡(R/J,N)=0 for every J∈𝒢. The classes ℰ𝒢 and ℱ𝒢 are referred to as the 𝒢-torsion and 𝒢-torsion-free classes, respectively.

For a right R-module M, let t𝒢 denote the associated (left exact) radical; thus t𝒢⁢(M) is the maximal 𝒢-torsion submodule of M, or sometimes t⁢(M) when the Gabriel topology is clear from context.

3.1 Modules of quotients

A right Gabriel topology allows us to generalise localisations of commutative rings with respect to a multiplicative subset to the non-commutative setting.

The module of quotients of the Gabriel topology 𝒢 of a right R-module M is the module M𝒢 defined as follows, where the morphisms in the direct system {HomR⁡(J,M/t𝒢⁢(M))}J∈𝒢 are the restriction morphisms:

M𝒢:=lim→J∈𝒢HomR(J,M/t𝒢(M)).

Furthermore, there is the canonical homomorphism

ψM:M≅HomR⁡(R,M)→M𝒢.

For each R-module M, the homomorphism ψM is part of the following exact sequence, where both the kernel and cokernel of the map ψM are 𝒢-torsion R-modules:

0→t𝒢⁢(M)→M→ψMM𝒢→M𝒢/ψM⁢(M)→0.

By substituting M=R, the assignment gives a ring homomorphism ψR:R→R𝒢, and furthermore, for each R-module M, the module M𝒢 is both an R-module and an R𝒢-module. Also, for any R-homomorphism f:M→N, ψ induces an R𝒢-homomorphism f𝒢:M𝒢→N𝒢.

Let q:Mod⁢-⁢R→Mod⁢-⁢R𝒢 denote the functor that maps each M to its module of quotients M𝒢. Then the ψ can be considered a natural transformation of endofunctors of Mod⁢-⁢R, that is the following diagram commutes:

A right R-module M is 𝒢-closed if the natural homomorphism

M≅HomR⁡(R,M)→HomR⁡(J,M)

is an isomorphism for each J∈𝒢. This amounts to saying that HomR⁡(R/J,M)=0 for every J∈𝒢 (i.e. M is 𝒢-torsion-free) and ExtR1⁡(R/J,M)=0 for every J∈𝒢 (i.e. M is 𝒢-injective). Moreover, if M is 𝒢-closed, then M is isomorphic to its module of quotients M𝒢 via ψM. Conversely, every R-module of the form M𝒢 is 𝒢-closed. The 𝒢-closed modules form a full subcategory of both Mod⁢-⁢R and Mod⁢-⁢R𝒢. Additionally, every R-linear morphism of 𝒢-closed modules is also R𝒢-linear.

A left R-module N is called 𝒢-divisible if J⁢N=N for every J∈𝒢. Equivalently, N is 𝒢-divisible if and only if R/J⊗RN=0 for each J∈𝒢. We denote the class of 𝒢-divisible modules by 𝒟𝒢. It is straightforward to see that 𝒟𝒢 is a torsion class in R⁢-⁢Mod.

A right Gabriel topology is faithful if HomR⁡(R/J,R)=0 for every J∈𝒢, or equivalently if R is 𝒢-torsion-free, that is the natural map ψR:R→R𝒢 is injective.

A right Gabriel topology is finitely generated if it has a basis consisting of finitely generated right ideals. Equivalently, 𝒢 is finitely generated if the 𝒢-torsion radical preserves direct limits (that is there is a natural isomorphism t𝒢⁢(lim→i⁡Mi)≅lim→i⁡(t𝒢⁢(Mi))) if and only if the 𝒢-torsion-free modules are closed under direct limits (that is, the associated torsion pair is of finite type). The first of these two equivalences was shown in [24, Proposition XIII.1.2], while the second was noted by Hrbek in the discussion before [16, Lemma 2.4].

3.4 More properties of Gabriel topologies

We refer to [6] for more properties of right Gabriel topologies. Many of them hold in the non-commutative case. In particular, we will use the following results.

Lemma 3.3 ([6, Lemma 4.1, Lemma 4.2]).

Suppose R is a commutative ring and G is a Gabriel topology. Then the following statements hold.

1. If an R-module D is both 𝒢-divisible and 𝒢-torsion-free, then D is an R𝒢-module and D≅D⊗RR𝒢 via the natural map idD⊗RψR:D⊗RR→D⊗RR𝒢.

2. If M is an R-module with p⁢.⁢dimR⁡M≤1, then Tor1R⁡(M,R𝒢)=0 and Tor1R⁡(R𝒢,M)=0.

We will often refer to the following exact sequence, where ψR is the ring of quotients homomorphism discussed in Subsection 3.1:

0→t𝒢⁢(R)→R→ψRR𝒢→R𝒢/ψR⁢(R)→0.

We will denote t𝒢⁢(M) simply by t⁢(M) and when clear from the context, ψ instead of ψR. We add the following result.

Lemma 3.4.

Consider a right Gabriel topology G. Let M be a G-torsion module and N a G-closed module in Mod⁢-⁢R. Then ExtR1⁡(M,N)=0.

Proof.

Let 𝒢 be a Gabriel topology of right ideals and ℰ its associated hereditary torsion class in Mod⁢-⁢R which is generated by the cyclic modules R/J, where J∈𝒢. Therefore, for M a 𝒢-torsion module, there exists a presentation of M as follows:

(3.1)0→H→⊕Jα∈𝒢R/Jα→M→0.

The module H is 𝒢-torsion since ℰ is a hereditary torsion class. Take a 𝒢-closed module N, and apply the functor HomR⁡(-,N) to (3.1):

(3.2)0=HomR⁡(H,N)→ExtR1⁡(M,N)→ExtR1⁡(⊕R/Jα,N)=0.

The first abelian group of sequence (3.2) vanishes since H is 𝒢-torsion, and the last abelian group vanishes since ExtR1⁡(R/Jα,N)=0 for every Jα∈𝒢. Therefore, ExtR1⁡(M,N)=0 as desired. ∎

The following lemma is taken from [24, Exercise IX.1.4], although we state the result in a slightly more convenient way for us and include (iii) and (iv). We let E⁢(M) denote the injective envelope of M.

Lemma 3.5 ([24, Exercise IX.1.4]).

Let G be a right Gabriel topology on R. Then the following are equivalent.

1. The functor q:Mod⁢-⁢R→Mod⁢-⁢R𝒢 which maps each module M to the 𝒢-closed module M𝒢 is exact.

2. The module E⁢(M)/M is 𝒢-closed for every 𝒢-closed module M.

3. For every 𝒢-closed module M and each J∈𝒢, ExtR2⁡(R/J,M)=0.

4. For every 𝒢-closed module M and basis element J∈𝒢, ExtR2⁡(R/J,M)=0.

Sketch of proof.

We will prove (i) ⇒ (ii) ⇔ (iii) ⇒ (i) and (iii) ⇔ (iv). The equivalence of (ii) and (iii) follows from the isomorphism ExtR1⁡(R/J,E⁢(M)/M)≅ExtR2⁡(R/J,M).

For the implication (i) ⇒ (ii), we begin by assuming that q is exact. Fix a J∈𝒢 and take a 𝒢-closed R-module M. Then, since M is essential in its injective envelope E⁢(M), E⁢(M) must be 𝒢-torsion-free and so is 𝒢-closed. Thus we have the following commuting diagram, where the exactness of the bottom row follows by our assumption that q is exact:

It follows by the snake lemma that E⁢(M)/M is isomorphic to its module of quotients, so is 𝒢-closed.

Now we show that (iii) ⇒ (i). Assume that, for every 𝒢-closed module M and every J∈𝒢, ExtR2⁡(R/J,M)=0; therefore, it follows that the 𝒢-closed modules are closed under cokernels of monomorphisms. Now consider q applied to the exact sequence 0→L→𝑓M→𝑔N→0. Recall that q is left exact, so it remains only to show that the induced map g𝒢 is a surjection. We have the following commuting diagram, where the bottom row is also in Mod⁢-⁢R𝒢:

By assumption, Coker⁡f𝒢 is 𝒢-closed. By the property of cokernels, ψN factors through N→Coker⁡f𝒢→N𝒢. By [24, Proposition IX.1.11], for any R-module N and any 𝒢-closed module X, there is an isomorphism ψN∗:HomR⁡(N𝒢,X)→≅HomR⁡(N,X), so N→Coker⁡f𝒢 extends to N𝒢→Coker⁡f𝒢. Since the homomorphism N𝒢→Coker⁡f𝒢→N𝒢 preserves ψN⁢(N), it is the identity on N𝒢; thus Coker⁡f𝒢→N𝒢 is surjective, so also g𝒢 is.

That (iii) ⇒ (iv) is trivial. For the converse, for every ideal J∈𝒢, there exists a basis element J0∈𝒢 such that J0⊆J. Thus, for M𝒢-closed, one applies HomR⁡(-,M) to 0→J/J0→R/J0→R/J→0. As J/J0 is 𝒢-torsion, the conclusion follows by applying Lemma 3.4. ∎

The following lemma will be useful when working with a faithful Gabriel topology over a commutative ring that arises from a perfect localisation.

Lemma 3.6 ([6, Lemma 4.5]).

Let R be a commutative ring, u:R→U a flat injective ring epimorphism, and G the associated Gabriel topology. Then the annihilators of the elements of U/R form a sub-basis for the Gabriel topology G. That is, for every J∈G, there exist z1,z2,…,zn∈U such that

⋂0≤i≤nAnnR⁢(zi+R)⊆J.

4 When 𝒜 is covering 𝒢 arises from a perfect localisation and R𝒢⊕R𝒢/R is 1-tilting

In this section, we consider the following setting.

First we show that the Gabriel topology 𝒢 arises from a perfect localisation and that p⁢.⁢dim⁡R𝒢≤1 so that 𝒟𝒢=Gen⁡(R𝒢); in other words, we show that the equivalent conditions of Proposition 3.2 hold.

We begin by describing 𝒜-covers of modules annihilated by some J∈𝒢.

Next we show that 𝒢 must arise from a perfect localisation using Lemma 3.5.

The above lemma allows us to use the equivalent conditions of [24, Proposition XI.3.4]. In particular, we have that ψR:R→R𝒢 is a flat injective ring epimorphism and that R𝒢 is 𝒢-divisible, so R𝒢∈𝒟𝒢. It remains to see that if 𝒜 is covering in (𝒜,𝒟𝒢), then R𝒢⊕R𝒢/R is the associated 1-tilting module, that is the equivalent conditions of Proposition 3.2. This amounts to showing that p⁢.⁢dim⁡R𝒢≤1.

Proof.

We know that 𝒢 is perfect, so that R𝒢 is 𝒢-divisible by Lemma 4.3. We prove that p⁢.⁢dim⁡R𝒢≤1 by showing that R𝒢∈𝒜. Let the following be an 𝒜-cover of R𝒢:

(4.1)0→D→A→ϕR𝒢→0.

Note that A is 𝒢-divisible since both R𝒢 and D are 𝒢-divisible. We will first show that A must be 𝒢-torsion-free, and therefore an R𝒢-module. Fix a finitely generated J∈𝒢 with generators x1,…,xn. We will show that A⁢[J]=0, that is the only element of A annihilated by J is 0. Since R𝒢 is 𝒢-divisible, one can write 1R=1R𝒢=∑xi⁢ηi for some fixed ηi∈R𝒢, 1≤i≤n. Let 𝐱, 𝐬 and 𝐬A be the following homomorphisms:

By the definition of 𝐬 and 𝐬A, the lower square of (4.2) commutes. Clearly ϕn is a precover of R𝒢n as Dn∈𝒟𝒢 and An∈𝒜 (it is in fact a cover). Therefore, there exists a map f such that the upper square of (4.2) commutes:

(4.2)

The map 𝐬𝐱 is the identity on R𝒢, so we have that ϕ⁢𝐬A⁢f=ϕ, and by the 𝒜-cover property of ϕ, 𝐬A⁢f is an automorphism of A. Consider an element a∈A⁢[J], and let f⁢(a)=(f1⁢(a),…,fn⁢(a))∈An. Then

𝐬A⁢(f⁢(a))=∑xi⁢fi⁢(a)=∑fi⁢(xi⁢a)=0 as⁢xi∈J,

and by the injectivity of 𝐬A⁢f, a=0.

We have shown that A,D are both 𝒢-torsion-free and 𝒢-divisible, so by Lemma 3.3 (i), they are R𝒢-modules. Then sequence (4.1) is a sequence in Mod⁢-⁢R𝒢 as R→R𝒢 is a ring epimorphism and Mod⁢-⁢R𝒢→Mod⁢-⁢R is fully faithful. Thus (4.1) splits, so R𝒢∈𝒜 and p⁢.⁢dim⁡R𝒢≤1, as required.

The last statement then follows by Proposition 3.2. ∎

5 Topological rings and u-contramodules

The material covered in this section is a combination of ideas from [18, 9, 7] and covers mostly methods using contramodules and topological rings.

An abelian group is a topological group if it has a topology such that the group operations are continuous. A topological abelian group is said to be linearly topological if there is a basis of neighbourhoods of zero consisting of subgroups.

For a linearly topological abelian group A with basis 𝔅 of subgroups of A, there is the following canonical homomorphism of abelian groups:

When λA is a monomorphism, or equivalently when ⋂V∈𝔅V=0, A is said to be separated. When λA is an epimorphism, A is said to be complete.

For a ring R and M,N∈Mod⁢-⁢R, the abelian group HomR⁡(M,N) can be considered a linearly topological abelian group as follows. Take a finitely generated submodule F of M, and consider the subgroup formed by the elements of HomR⁡(M,N) which annihilate F. Such subgroups form a base of neighbourhoods of zero in HomR⁡(M,N). Note that this is the same as considering HomR⁡(M,N) with the subspace topology of the product topology on NM, where the topology on N is the discrete topology. We will consider HomR⁡(M,N) endowed with this topology which we will call the finite topology. The topological abelian group HomR⁡(M,N) is separated and complete with respect to this topology.

Recall from Section 3 that a topological ring R is right linearly topological if it has a topology with a basis of neighbourhoods of zero consisting of right ideals of R, and that a ring R with a right Gabriel topology is an example of a right linearly topological ring.

Let ℋ be a topology of a linearly topological commutative ring R with basis 𝔅. The ℋ-topology on an R-module M is the topology where the base of neighbourhoods of 0 are the submodules MJ for J∈𝔅. For every R-module M, {M/M⁢J∣J∈𝔅} is an inverse system. The completion of M with respect to the ℋ-topology is the module

There is a canonical map λM:M→Λ𝔅⁢(M) which sends the element x∈M to (x+M⁢J)J∈𝔅. Each element in Λ𝔅⁢(M) is of the form (xJ+M⁢J)J∈𝔅 with the relation that, for J⊆J′, xJ-xJ′∈M⁢J′. The module M is called ℋ-separated if the homomorphism λM is injective, which is equivalent to ⋂J∈𝔅M⁢J=0. The module M is called ℋ-complete if the map λM is surjective.