Enveloping classes over commutative rings

The classification problem for classes of modules over arbitrary rings is in general very difficult, or even hopeless. Nonetheless, approximation theory was developed as a tool to approximate arbitrary modules by modules in classes where the classification is more manageable. Left and right approximations were first studied for finite dimensional modules by Auslander, Reiten, and Smalø and 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 certain class $\mathcal{C}$, the aim is to characterise the rings over which every module has a minimal approximation in $\mathcal{C}$ and furthermore to characterise the class $\mathcal{C}$ itself. The most famous positive result of when minimal approximations exist is the construction of an injective envelope for every module. Instead, Bass proved in [Bas60] that projective covers rarely exist. In his paper, Bass introduced and characterised the class of perfect rings which are exactly the rings over which every module admits a projective cover. Among the many characterisations of perfect rings, the most important from the homological point of view is the closure under direct limits of the class of projective modules.

A class $\mathcal{C}$ of modules is called covering, respectively enveloping, if every module admits a $\mathcal{C}$-cover, respectively a $\mathcal{C}$-envelope.

A cotorsion pair $(\mathcal{A},\mathcal{B})$ admits (special) $\mathcal{A}$-precovers if and only if it admits (special) $\mathcal{B}$-preenvelopes. This observation lead to the notion of complete cotorsion pairs, that is cotorsion pairs admitting approximations.

Results by Enochs and Xu ([Xu96, Theorem 2.2.6 and 2.2.8]) show that a complete cotorsion pair $(\mathcal{A},\mathcal{B})$ such that $\mathcal{A}$ is closed under direct limits admits both $\mathcal{A}$-covers and $\mathcal{B}$-envelopes. Note that in the case of the cotorsion pair $(\mathcal{P}_{0},\mathrm{Mod}\textrm{-}{R})$, where $\mathcal{P}_{0}$ is the class of projective modules, Bass’s results state that $\mathcal{P}_{0}$ is a covering class if and only if $\mathcal{P}_{0}$ is closed under direct limits.

In this paper we are interested in the conditions under which a class $\mathcal{C}$ is enveloping. We will deal with classes of modules over commutative rings and in particular with $1$-tilting classes.

An important starting point is the bijective correspondence between faithful finitely generated Gabriel topologies $\mathcal{G}$ and $1$-tilting classes over commutative rings established by Hrbek in [Hrb16]. The tilting class can then be characterised as the class $\mathcal{D}_{\mathcal{G}}$ of $\mathcal{G}$-divisible modules, that is, the modules $M$ such that $JM=M$ for every $J\in\mathcal{G}$.

We prove in Section 5 that if a $1$-tilting class is enveloping, then $R_{\mathcal{G}}$, the ring of quotients with respect to the Gabriel topology $\mathcal{G}$, is $\mathcal{G}$-divisible, so that $R\to R_{\mathcal{G}}$ is a flat injective ring epimorphism.

It is well known that every flat ring epimorphism gives rise to a finitely generated Gabriel topology. We will consider the case of a flat injective ring epimorphism $u\colon R\to U$ between commutative rings and show that if the module $R$ has a $\mathcal{D}_{\mathcal{G}}$-envelope, then $U$ has projective dimension at most one.

From results by Angeleri Hügel and Sánchez [AHS11] and also [Hrb16, Proposition 5.4], we infer that the module $U\oplus K$, where $K$ is the cokernel of $u$, is a $1$-tilting module with $\mathcal{D}_{\mathcal{G}}$ the associated tilting class. In other words, $\mathcal{D}_{\mathcal{G}}$ arises from the perfect localisation $u$, so it coincides with the class of modules generated by $U$, that is epimorphic images of direct sums of copies of $U$ or also with $K^{\perp}$, the right Ext-orthogonal of $K$. Assuming furthermore that the class $\mathcal{D}_{\mathcal{G}}$ is enveloping, we prove that all the quotient rings $R/J$, for $J\in\mathcal{G}$ are perfect rings and so are all the discrete quotient rings of the topological ring $\mathfrak{R}=\operatorname{End}(K)$ (Theorems  7.13 and  7.14). In the terminology of Positselski and Bazzoni-Positselski (for example [BP18]) this means that $\mathfrak{R}$ is a pro-perfect topological ring. Moreover, the converse holds, that is if $\mathfrak{R}=\operatorname{End}(K)$ is a pro-perfect topological ring then the class of $\mathcal{G}$-divisible modules is enveloping (Theorem 8.5).

In conclusion, we obtain that a $1$-tilting class over a commutative ring is enveloping if and only if it arises from a flat injective ring epimorphism with associated Gabriel topology $\mathcal{G}$ such that the factor rings $R/J$ are perfect rings for every $J\in\mathcal{G}$ (Theorem 8.7). This provides a partial answer to Problem 1 of [GT12, Section 13.5] and generalises the result proved in [Baz10] for the case of commutative domains and divisible modules.

Applying results from [BP18, Section 19] or [BP19a, Section 13], we obtain that $\mathrm{Add}(K)$, the class of direct summands of direct sums of copies of $K$, is closed under direct limits. Since $\mathcal{D}_{\mathcal{G}}$ coincides with the right Ext-orthogonal of $\mathrm{Add}(K)$, we have an instance of the necessity of the closure under direct limits of a class whose right Ext-orthogonal admits envelopes.

Therefore in our situation we prove a converse of the result by Enochs and Xu ([Xu96, Theorem 2.2.6]) which states that if a class $\mathcal{A}$ of modules is closed under direct limits and extensions and whose right Ext-orthogonal $\mathcal{A}^{\perp}$ admits special preenvelopes with cokernel in $\mathcal{A}$, then $\mathcal{A}^{\perp}$ is enveloping.

The case of a non-injective flat ring epimorphism $u\colon R\to U$ is easily reduced to the injective case, since the class of $\mathcal{G}$-divisible modules is annihilated by the kernel $I$ of $u$, so all the results proved for $R$ apply to the ring $R/I$ and to the cokernel $K$ of $u$.

The paper is organised as follows. After the necessary preliminaries, in Section 3 we state some general results concerning properties of envelopes with respect to classes of modules.

In Section 4 we recall the notion of a Gabriel topology and outline the properties of the related ring of quotients. In Subsection 4.1 we provide some of our own results for general Gabriel topologies which we will use in the later sections. Next in Subsection 4.2 we review the relationship between Gabriel topologies and $1$-tilting classes as well as silting classes as done by Hrbek. Finally in Subsection 4.3 we recall the classical notion of a Gabriel topology arising from a perfect localisation, as well as a lemma.

In Section 5, we consider a $1$-tilting class over a commutative ring and its associated Gabriel topology via Hrbek’s results  [Hrb16]. We prove that if the $1$-tilting class is enveloping, then the ring of quotients with respect to the Gabriel topology $\mathcal{G}$, $R_{\mathcal{G}}$, is $\mathcal{G}$-divisible, hence $\mathcal{G}$ arises from a flat injective ring epimorphism $\psi_{R}\colon R\to R_{\mathcal{G}}$.

In Section 6 we introduce the completion of a ring with respect to a Gabriel topology and the endomorphism ring of a module as a topological ring. Considering the particular case of a perfect localisation corresponding to a flat injective ring epimorphism $u\colon R\to U$ between commutative rings, we show the isomorphism between the completion of $R$ with respect to the associated Gabriel topology and the topological ring $\mathfrak{R}=\operatorname{End}(K)$.

In the main Sections 7 and  8, we prove a ring theoretic and topological characterisation of commutative rings for which the class of $\mathcal{G}$-divisible modules is enveloping where $\mathcal{G}$ is the Gabriel topology associated to a flat injective ring epimorphism. Namely, the characterisation in terms of perfectness of the factor rings $R/J$, for every $J\in\mathcal{G}$ and the pro-perfectness of the topological ring $\mathfrak{R}=\operatorname{End}(K)$.

In Section 9 we extend the results proved in Sections 7 and  8 to the case of a non-injective flat ring epimorphism

The ring $R$ will always be associative with a unit and $\mathrm{Mod}\textrm{-}{R}$ the category of right $R$-modules.

Let $\mathcal{C}$ be a class of right $R$-modules. The right $\operatorname{Ext}^{1}_{R}$-orthogonal and right $\operatorname{Ext}^{\infty}_{R}$-orthogonal classes of $\mathcal{C}$ are defined as follows.

The left Ext-orthogonal classes ${}^{\perp_{1}}\mathcal{C}$ and ${}^{\perp}\mathcal{C}$ are defined symmetrically.

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

We will now recall the notions of $\mathcal{C}$-preenvelope, special $\mathcal{C}$-preenvelope and $\mathcal{C}$-envelope for a class $\mathcal{C}$ of $R$-modules.

The notions of $\mathcal{C}$-precover (right approximation), special $\mathcal{C}$-precover and of $\mathcal{C}$-cover (minimal right approximation) (see [Xu96]) are defined dually.

A class $\mathcal{C}$ of $R$-modules is called enveloping (covering) if every module admits a $\mathcal{C}$-envelope ($\mathcal{C}$-cover).

A pair of classes of modules $(\mathcal{A},\mathcal{B})$ is a cotorsion pair provided that $\mbox{$\mathcal{A}$}={}^{\perp_{1}}\mbox{$\mathcal{B}$}$ and $\mbox{$\mathcal{B}$}=\mbox{$\mathcal{A}$}^{\perp_{1}}$.

We consider preenvelopes and envelopes for particular classes of modules, that is classes which form the right-hand class of a cotorsion pair.

A cotorsion pair $(\mathcal{A},\mathcal{B})$ is complete provided that every $R$-module $M$ admits a special $\mathcal{B}$-preenvelope or equivalently, every $R$-module $M$ admits a special $\mathcal{A}$-precover.

Results by Enochs and Xu ([Xu96, Theorem 2.2.6 and 2.2.8]) show that a complete cotorsion pair $(\mathcal{A},\mathcal{B})$ such that $\mathcal{A}$ is closed under direct limits admits both $\mathcal{B}$-envelopes and $\mathcal{A}$-covers.

A cotorsion pair $(\mathcal{A},\mathcal{B})$ is hereditary if for every $A\in\mathcal{A}$ and $B\in\mathcal{B}$, $\operatorname{Ext}^{i}_{R}(A,B)=0$ for all $i\geq 1$.
Given a class $\mathcal{C}$ of modules, the pair $(^{\perp}(\mathcal{C}^{\perp}),\mathcal{C}^{\perp})$ is a (hereditary) cotorsion pair called the cotorsion pair generated by $\mathcal{C}$, while $(^{\perp}\mathcal{C},(^{\perp}\mathcal{C})^{\perp})$ is a (hereditary) cotorsion pair called the cotorsion pair cogenerated by $\mathcal{C}$.

Examples of complete cotorsion pairs are abundant. In fact, by [ET01, Theorem 10] or [GT12, Theorem 6.11] a cotorsion pair generated by a set of modules is complete.

For an $R$-module $C$, we let $\mathrm{Add}(C)$ denote the class of $R$-modules which are direct summands of direct sums of copies of $C$, and $\operatorname{Gen}(C)$ denote the class of $R$-modules which are homomorphic images of direct sums of copies of $C$.

We now define $1$-tilting and silting modules.
A right $R$-module $T$ is $1$-tilting ([CT95]) if the following conditions hold.

1. (T1)

   $\operatorname{p.dim}T\leq 1$.

2. (T2)

   $\operatorname{Ext}_{R}^{i}(T,T^{(\kappa)})=0$ for every cardinal $\kappa$ and every $i>0$.

3. (T3)

   There exists an exact sequence of the following form where each $T_{i}$ is in $\mathrm{Add}(T)$.

   $$0\to R\to T_{0}\to T_{1}\to 0$$

Equivalently, $T$ is $1$-tilting if and only if $T^{\perp_{1}}=\operatorname{Gen}(T)$ ([CT95, Proposition 1.3]). The cotorsion pair $({}^{\perp}(T^{\perp}),T^{\perp})$ is called a $1$-tilting cotorsion pair and the torsion class $T^{\perp}$ is called a $1$-tilting class. Two $1$-tilting modules $T$ and $T^{\prime}$ are equivalent if they define the same $1$-tilting class $T^{\perp}=T^{\prime\perp}$ (equivalently, if $\mathrm{Add}(T)=\mathrm{Add}(T^{\prime})$).

A $1$-tilting class can be generalised in the following way. For a homomorphism $\sigma:P_{-1}\to P_{0}$ between projective modules in $\mathrm{Mod}\textrm{-}R$, consider the following class of modules.

An $R$-module $T$ is said to be silting if it admits a projective presentation

such that $\operatorname{Gen}(T)=D_{\sigma}$. In the case that $\sigma$ is a monomorphism, $\operatorname{Gen}(T)$ is a $1$-tilting class.

A ring $R$ is left perfect if every module in $R\textrm{-}\mathrm{Mod}$ has a projective cover. By [Bas60], $R$ is left perfect if and only if all flat modules in $R\textrm{-}\mathrm{Mod}$ are projective.

An ideal $I$ of $R$ is said to be left T-nilpotent if for every sequence of elements $a_{1},a_{2},...,a_{i},...$ in $I$, there exists an $n>0$ such that $a_{1}a_{2}\cdots a_{n}=0$. The following proposition for the case of commutative perfect rings is well known.

The following fact will be useful. Let ${}_{R}F$ be a left $R$-module ${}_{S}G_{R}$ be an $S$-$R$-bimodule such that $\operatorname{Tor}_{1}^{R}(G,F)=0$. Then, for every left $S$-module $M$ there is an injective map of abelian groups

In this section we discuss some useful results in relation to envelopes.

The following result is crucial in connection with the existence of envelopes.

We will consider $\mathcal{C}$-envelopes where $\mathcal{C}$ is a class closed under direct sums and therefore we will make use of the following result which is strongly connected with the notion of T-nilpotency of a ring.

For a complete cotorsion pair $(\mathcal{A},\mathcal{B})$, we investigate the properties of $\mathcal{B}$-envelopes of arbitrary $R$-modules. First of all we state two straightforward lemmas.

In this section we briefly introduce Gabriel topologies and discuss some advancements that relate Gabriel topologies to $1$-tilting classes and silting classes over commutative rings as done in [Hrb16] and [AHH17]. Furthermore, we include some of our own results Subsection 4.1. For a more detailed discussion on torsion pairs and Gabriel topologies, refer to [Ste75, Chapters VI and IX].

We will start by giving definitions in the case of a general ring with unit (not necessarily commutative).

Recall that a torsion pair $(\mathcal{E},\mathcal{F})$ in $\mathrm{Mod}\textrm{-}R$ is a pair of classes of modules in $\mathrm{Mod}\textrm{-}R$ which are mutually orthogonal with respect to the $\operatorname{Hom}$-functor and maximal with respect to this property. The class $\mathcal{E}$ is called a torsion class and $\mathcal{F}$ a torsion-free class.

A class $\mathcal{C}$ of modules is a torsion class if and only if it is closed under extensions, direct sums, and epimorphic images. A torsion pair $(\mathcal{E},\mathcal{F})$ is called hereditary if $\mathcal{E}$ is also closed under submodules.

A torsion pair $(\mathcal{E},\mathcal{F})$ is generated by a class $\mathcal{C}$ if $\mathcal{F}$ consists of all the modules $F$ such that $\operatorname{Hom}_{R}(C,F)=0$ for every $C\in\mathcal{C}$.

A (right) Gabriel topology on $R$ is a filter of right ideals of $R$, denoted $\mathcal{G}$, such that the following conditions hold. Recall that for a right ideal $I$ in $R$ and an element $t\in R$, $(I:t):=\{r\in R:tr\in I\}$.

• (i)

  If $I\in\mathcal{G}$ and $r\in R$ then $(I:r)\in\mathcal{G}$.

• (ii)

  If $J$ is a right ideal of $R$ and there exists a $I\in\mathcal{G}$ such that $(J:t)\in\mathcal{G}$ for every $t\in I$, then $J\in\mathcal{G}$.

Right Gabriel topologies on $R$ are in bijective correspondence with hereditary torsion pairs in $\mathrm{Mod}\textrm{-}R$. Indeed, to each right Gabriel topology $\mathcal{G}$, one can associate the following hereditary torsion class.

Then, the corresponding torsion pair $(\mathcal{E}_{\mathcal{G}},\mathcal{F}_{\mathcal{G}})$ is generated by the cyclic modules $R/J$ where $J\in\mathcal{G}$. The classes $\mathcal{E}_{\mathcal{G}}$ and $\mathcal{F}_{\mathcal{G}}$ are referred to as the $\mathcal{G}$-torsion and $\mathcal{G}$-torsion-free classes, respectively.

Conversely, if $(\mathcal{E},\mathcal{F})$ is a hereditary torsion pair in $\mathrm{Mod}\textrm{-}R$, the set

is a right Gabriel topology.

For a right $R$-module $M$ let $t_{\mathcal{G}}(M)$ denote the $\mathcal{G}$-torsion submodule of $M$, or sometimes $t(M)$ when the Gabriel topology is clear from context.

The module of quotients of the Gabriel topology $\mathcal{G}$ of a right $R$-module $M$ is the module

Furthermore, there is a canonical homomorphism

By substituting $M=R$, the assignment gives a ring homomorphism $\psi_{R}:R\to R_{\mathcal{G}}$ and furthermore, for each $R$-module $M$ the module $M_{\mathcal{G}}$ is both an $R$-module and an $R_{\mathcal{G}}$-module. Both the kernel and cokernel of the map $\psi_{M}$ are $\mathcal{G}$-torsion $R$-modules, and in fact $\operatorname{Ker}(\psi_{M})=t_{\mathcal{G}}(M)$.

Let $q:\mathrm{Mod}\textrm{-}R\to\mathrm{Mod}\textrm{-}R_{\mathcal{G}}$ denote the functor that maps each $M$ to its module of quotients. Let $\psi^{\ast}$ denote the right exact functor $\mathrm{Mod}\textrm{-}R\to\mathrm{Mod}\textrm{-}R_{\mathcal{G}}$ where $\psi^{\ast}(M):=M\otimes R_{\mathcal{G}}$. In general, there is a natural transformation $\Theta:\psi^{\ast}\to q$ with $\Theta_{M}:M\otimes R_{\mathcal{G}}\to M_{\mathcal{G}}$ which is defined as $m\otimes\eta\mapsto\psi_{M}(m)\cdot\eta$. That is, for every $M$ the following triangle commutes.

A right $R$-module is $\mathcal{G}$-closed if the following natural homomorphisms are all isomorphisms for every $J\in\mathcal{G}$.

This amounts to saying that $\operatorname{Hom}_{R}(R/J,M)=0$ for every $J\in\mathcal{G}$ (i.e. $M$ is $\mathcal{G}$-torsion-free) and $\operatorname{Ext}^{1}_{R}(R/J,M)=0$ for every $J\in\mathcal{G}$ (i.e. $M$ is $\mathcal{G}$-injective). Thus if $M$ is $\mathcal{G}$-closed then $M$ is isomorphic to its module of quotients $M_{\mathcal{G}}$. Conversely, every $R$-module of the form $M_{\mathcal{G}}$ is $\mathcal{G}$-closed. The $\mathcal{G}$-closed modules form a full subcategory of both $\mathrm{Mod}\textrm{-}R$ and $\mathrm{Mod}\textrm{-}R_{\mathcal{G}}$.

A left $R$-module $N$ is called $\mathcal{G}$-divisible if $JN=N$ for every $J\in\mathcal{G}$. Equivalently, $N$ is $\mathcal{G}$-divisible if and only if $R/J\otimes_{R}N=0$ for each $J\in\mathcal{G}$. We denote the class of $\mathcal{G}$-divisible modules by $\mathcal{D}_{\mathcal{G}}$. It is straightforward to see that $\mathcal{D}_{\mathcal{G}}$ is a torsion class in $R\textrm{-}\mathrm{Mod}$.

A right Gabriel topology is faithful if $\operatorname{Hom}_{R}(R/J,R)=0$ for every $J\in\mathcal{G}$, or equivalently if $R$ is $\mathcal{G}$-torsion-free, that is the natural map $\psi_{R}\colon R\to R_{\mathcal{G}}$ is injective.

A right Gabriel topology is finitely generated if it has a basis consisting of finitely generated right ideals. Equivalently, $\mathcal{G}$ is finitely generated if the $\mathcal{G}$-torsion radical preserves direct limits (that is there is a natural isomorphism $t_{\mathcal{G}}(\varinjlim_{i}M_{i})\cong\varinjlim_{i}(t_{\mathcal{G}}(M_{i}))$) if and only if the $\mathcal{G}$-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 [Ste75, Proposition XIII.1.2], while the second was noted by Hrbek in the discussion before [Hrb16, Lemma 2.4].