CT3

ZCC
July 27, 2025

Lecture 3
- Abstract
All Covers are Projective Covers
Set-theoretic Consideration

Lecture 3

Abstract

A wide range of cotorsion pairs are constructed via covering and enveloping. We analyse the existence of projective covers at the outset, and identify where the obstacle arises. Subsequently, we introduce criteria for the existence of covers and envelopes.

All Covers are Projective Covers

We explain the general theory of (pre)covers with projective (pre)covers in a functorial approach.

The Projective Cover

For an Abelian category with enough projectives, we provide two equivalent definitions of projective cover.

(Small/superfluous). A subobject $i : K ↪ X$ is small if, for any subobject $j : L ↪ X$ with $L + K = X$, then $j$ must be an isomorphism.

A subobject is characterised by a morphism, rather than a plain object. The expression $L + K = X$ means that the following $θ$ induced by the pushout is an isomorphism

One may also write $L × _X K =: L ∩ K$ for convenience.

Show that $i : k ↪ X$ is small if there exists some pushout pullback square of the form , then $j$ must be an isomorphism.

Suppose $p : P → M$ is a projective cover if the following equivalent conditions hold (projective cover = $𝐏𝐫𝐨𝐣$-cover!):

(Projective cover). $p$ is an epimorphism, and $\ker p$ (as a morphism) is small;
($𝐏𝐫𝐨𝐣$-cover). Any morphism $Q → M$ ($Q$ projective) factors through $p$, and any $p ∘ α = p$ implies $α$ is an automorphism.

(1 → 2). Clearly, $Q → M$ factors through $p$ by the lifting property of projectives. We show that $p ∘ α = p$ implies $α$ is an automorphism. We construct the pullback square of monomorphisms:

By $p ∘ α = p$, the morphism $p|_{\operatorname{im}α }$ is epic, thus the first row is a short exact sequence. By exercise, the subobject $\operatorname{im}(α ) ↪ P$ is an isomorphism. This shows $α$ is epic. We show $α$ is also monic. Let $j$ be the right inverse of $α$ (by the lifting property of projectives). Then $p ∘ j = p ∘ α ∘ j = p$. Hence $j$ is epic. Since $j$ is also split monic, $j$ is an isomorphism. Therefore $α$ is an automorphism.
(2 → 1). $p$ is epic, since any epimorphism $Q ↠ M$ factors through $p$. We show $\ker p$ is small. If not, there exists a proper subobject $N ↪ P$ such that $N + \ker p = P$. Therefore we have the pullback pushout square of short exact sequences. We introduce $l$ by the lifting property of projectives, then $α : λ ∘ l$ is a non-automorphism such that $p ∘ α = p$, a contradiction.

Here we do not require that the objects are well-powered, which means that all subobjects of a certain object form a set.

Show another equivalent definition of projective cover: say $p : P → A$ is a projective cover if and only if it is an essential epimorphism, i.e., $p ∘ g$ is epic if and only if $g$ is epic.

Functorial Approach

The Yoneda embedding $\mathcal{A} ↪ 𝐏𝐒𝐡 (\mathcal{A})$ via $X ↦ (-, X)_\mathcal{A}$ is fully faithful and left exact. In particular, the representable functor $(-, X)_\mathcal{A}$ are projective in $𝐏𝐒𝐡 (\mathcal{A})$.

The Yoneda embedding yields a bijection $(X,Y) ↔ [(-, X), (-, Y)]_{𝐏𝐒𝐡 (\mathcal{A})}$, which identifies $f$ with a collection of natural transformations $\{(A, f)\}_{A ∈ \mathcal{A}}$.

The assignment $? ↦ (-, ?)$ preserves all limits, and is thus left exact;
For any surjection $θ : F → G$ of presheaves, we observe $[(-, ?), θ_-] ≃ θ_?$ is surjective. Hence, representable functors are projective in $𝐏𝐒𝐡 (\mathcal{A})$.

We now provide a functorial characterisation of (pre)cover. Throughout, $\mathcal{F}$ denotes a replete (= closed under isomorphism) class of objects.

((Pre)cover). Let $X ∈ \mathcal{A}$ and $F ∈ \mathcal{F}$. We say $f : F → X$ is an $\mathcal{F}$-precover if the following equivalent conditions hold:

Any morphism $g : G → X$ with $G ∈ \mathcal{F}$ factors through $f$;
There exists an epimorphism $f_∗ : (- , F)_\mathcal{F} ↠ (-, X)_\mathcal{A}$ from the projective object $(-, F)_\mathcal{F} ∈ 𝐏𝐒𝐡 (\mathcal{F})$.

A precover $f : F → X$ is termed a cover if the following equivalent conditions hold:

Any endomorphism $α : F → X$ such that $f ∘ α = f$ implies $α$ is an automorphism;
The epimorphism $f_∗ : (-, F)_\mathcal{F} ↠ (-, X)_\mathcal{A}$ is a projective cover.

Note that for $2$, $\mathrm{End}_\mathcal{F}(F) ≃ \mathrm{End}_{𝐏𝐒𝐡 (\mathcal{F})}((-, F)_\mathcal{F})$ via $α ↦ α _∗$.

From a categorical perspective, to study covers, it suffices to study projective covers.

The functorial construction yields remarkable results for AR theory, e.g., this note, or a general theory of Auslander algebra.

A Tautological Comparison

From the above, we deduce that

both projective precovers and covers are epic;
if $M$ admits a projective cover, then it is unique up to isomorphism, and is always a summand of $Q ↠ M$;
$p : P ↠ M$ is a (pre)cover whenever $p_∗ : (-, P)_{𝐏𝐫𝐨𝐣} ↠ (-|_{𝐏𝐫𝐨𝐣}, M)_{\mathcal{A}}$ is.

We are pleased to compare the projective covers with those arising in the functorial context.

(Idempotent obstacle). There exist cases where an idempotent morphism $e : X → X$ does not split. In this case, $\mathrm{im}(-, e)$ is a non-representable finitely generated projective object.

Assuming $\mathcal{A}$ is idempotent complete, we begin with a well-known lemma.

(Special Yoneda lemma). This is a standard result in representation theory. Let $T'$ be a summand of finite biproducts of $T$. There is an isomorphism

$$\begin{equation} (T', M)_{\mathcal{A}} → ((T,T')_\mathcal{A}, (T, M)_\mathcal{A})_{\mathrm{End}(T)}, \quad f ↦ f_∗ . \end{equation}$$

The right module structure of $(T,M)$ is given by $f ⋅ α = f ∘ α ^∗$ for $α ∈ \mathrm{End}(T)$.

(Projectivisation). Let $𝐚𝐝𝐝 (T)$ denote the full subcategory containing $T$ and closed under isomorphisms, finite biproducts and summands. Then there is an equivalence:

$$\begin{equation} (T,-) : 𝐚𝐝𝐝 (T) → 𝐩𝐫𝐨𝐣 (𝐌𝐨𝐝 _{\mathrm{End}(T)}),\quad T_0 ↦ (T, T_0). \end{equation}$$

By the special Yoneda lemma.

The tautological comparison is as follows.

We assume idempotent completeness. Let $p : P → M$ be a $\mathcal{P}$-precover, then the following are equivalent.

$p : T → M$ admits a $\mathcal{P}$-cover;
$p_∗ : (-, P)_{\mathcal{P}} ↠ (-|_{\mathcal{P}}, M)$ admits a projective cover;
$p_∗ : (-, P)_{𝐚𝐝𝐝 (P)} ↠ (-|_{𝐚𝐝𝐝 (P)}, M)$ admits a projective cover;
$p_∗ : (P, P)_{\mathrm{End}(P)} ↠ (P, M)_{\mathrm{End}(P)}$ admits a projective cover.

Note that the last three are projective precovers as they are epimorphisms.

(1 ↔ 2) follows from the definition. In the case of $1$, a $\mathcal{P}$-cover must be a summand of $p$. Since there is no differences between $\mathcal{P}$-cover and $𝐚𝐝𝐝 (P)$-cover for $M$, we show (1 ↔ 3). (3 → 4) is given by the special Yoneda lemma. It remains to show (4 → 1).
When $p_∗ : (P, P)_{\mathrm{End}(P)} ↠ (P, M)_{\mathrm{End}(P)}$ admits a projective cover, it takes the form

$$\begin{equation} (P, P)_{\mathrm{End}(P)} ↠ Q ↠ (P, M)_{\mathrm{End}(P)}. \end{equation}$$

Note that $Q ≃ (P, P_0)_{\mathrm{End}(P)}$ for $P_0 ∈ 𝐒𝐦𝐝 (P)$ by projectivisation. With the special Yoneda lemma, we may write the mapping sequence as

$$\begin{equation} (P, P)_{\mathrm{End}(P)} \overset {a_∗ } ↠ (P, P_0)_{\mathrm{End}(P)} \overset {b_∗ } ↠ (P, M)_{\mathrm{End}(P)}. \end{equation}$$

We show that $b : P_0 → M$ is a $\mathcal{P}$-cover. By special Yoneda lemma, $(ba)_∗ = p _∗$ yields $ba = p$. As $a$ is split epic, $b$ and $p$ factors through each other, which shows that $b$ is a $\mathcal{P}$-precover. If there exists $b ∘ γ = b$, we have $b_∗ ∘ γ _∗ = b_∗$. Now $γ _∗$ is an isomorphism as $b_∗$ is a projective cover. By projectivisation, $γ$ is an automorphism. Hence $b$ is a $\mathcal{P}$-cover.

It is a local property to exhibit a projective cover. For idempotent complete cases, a $\mathcal{P}$-precover $P → M$ admits a $\mathcal{P}$-cover when $\mathrm{End}(P)$ is right semi-perfect (where f.g. modules admits a projective cover).

Example: $𝐏𝐫𝐨𝐣$-cover of Flat Objects

Not every object has a projective cover. In particular, a flat module with projective cover is projective.

The author is pleased to introduce the equivalent definitions of a perfect ring; for the sake of coherence in this lecture, we shall only focus on flat modules and projective covers.

A flat module with projective cover is projective.

We provide a functorial proof which shows that it is not a non-set-theoretic property. We write a flat object $F$ as the filtered colimit of finitely generated projective objects, i.e., $F = \varinjlim ^{fil} P_∙$ along with structure morphisms $f_i : P_i → F$. Let $π : P → F$ denote the projective cover.
Since $f_i$ factors through $π$, we obtain a collection of split pullback squares $P_i ⊕ Q_i := P × _F P_i$. This defines a new system for $\{(P ⊕ Q)_∙ , g_∙\}$:

In particular, $ψ _{j,i} = φ _{j,i} ⊕ μ _{j,i}$ is diagonal. The universal property of pullback shows $\{(P⊕ Q)_∙ , g_∙ \}$ is again a filtered system. The universal property of colimit yields the retraction $$\begin{equation} \mathrm{id} = \left[\varinjlim {}^{fil} P_∙ → \varinjlim {}^{fil} P_∙⊕ Q_∙ → \varinjlim {}^{fil} P_∙\right]. \end{equation}$$

It remains to show $q : \varinjlim {}^{fil} P_∙⊕ Q_∙ ≃ P$. We firstly prove the injection $\varinjlim {}^{fil} P_∙⊕ Q_∙ ↪ P$. We take any $α : K → \varinjlim {}^{fil} P_∙⊕ Q_∙$ such that $K$ is compact, and $K \xrightarrow {α } \varinjlim {}^{fil} P_∙⊕ Q_∙ → P$ composes to $0$; then show that $α = 0$. Notice that $α$ admits a factorisation

Since $α_i$ and $0$ are both solutions to the pullback problem, we see $α _i = 0$. Hence $[q ∘ α = 0] → [α = 0]$ for any $α$ coming from a compact object. Note that every object is a filtered colimit of compact objects, so we conclude that $q$ is injective.
This gives morphisms between short exact sequences

Since $\ker π$ is small, $\varinjlim^{fil}(P_\bullet \oplus Q_\bullet) ↪ P$ must be an isomorphism. It shows that $F$ is a retract of $P$, and thus is projective.

The proof is valid for finitely presented categories.

Example: vNR Rings

A ring $R$ with such that $𝐏𝐫𝐨𝐣 = 𝐌𝐨𝐝_R$ is known as a semisimple ring. In comparison with this, the ring $R$ with $𝐅𝐥𝐚𝐭 = 𝐌𝐨𝐝 _R$ is called vNR (von Neumann regular).

The property of being either semisimple or vNR is a symmetric property, i.e., there is no need to distinguish left and right modules.
This is not the original definition in sense of von Neumann; but it is a beautiful coincidence that these two definitions coincide.

The followings are equivalent definition for $R$ to be a vNR ring. The first definition shows being vNR is a symmetric property. We proof for right modules only.

(von Neumann’s original definition). For any $x ∈ R$, there exists some $y ∈ R$ s.t. $xyx=x$.
All modules are flat.
All f.g. ideals are projective.

(2 → 3). Clear.
(3 → 2). All modules are filtered colimit of f.p. modules. Note that $𝐦𝐨𝐝 ⊆ 𝐏𝐫𝐨𝐣$ in this case.
(3 → 1). Any $x ∈ R$ induces a retract of cyclic modules $Rx \overset {⋅ y}↪ R \overset {⋅ x} ↠ Rx$.
(1 → 3). Clearly, principal ideals are projective. By induction, it suffices to show all $2$-generated ideals are principal and projective. Let $(x,y)$ be such ideal. Then $(x)$ and $(y)$ are projective. Since $(x,y) = (x,(1-x)y)$, we assume $xy = 0$. We take $\overline y$ s.t. $y \overline y y = y$. We take idempotent element

$$\begin{equation} e := y \overline y (1-x) ,\quad e^2 = e. \end{equation}$$

We show $(e + x) = (x,y)$. $e,x ∈ (x,y)$ is clear. Conversely, $y = (e+x) y$, and $x = (e+x)x$. Since $(e+x)$ is principal,it is projective.

We show a class of examples of vNR rings.

Let $V$ be a vector space over a field $𝔽$, then $\mathrm{End}_𝔽(V)$ is vNR. Since $φ$ has the matrix form $\binom {0 \ 0}{0 \ 1}$ over $V ≃ (\ker φ ⊕ \operatorname{im}(φ ))$.
In particular, once $\dim_𝔽 (\operatorname{im}(φ )) = κ$ is an infinite cardinal, then the left (right) ideal generated by $φ$ contains precisely the endomorphisms with $\dim_𝔽 (\operatorname{im}(φ )) ≤ κ$. Therefore,

$\mathrm{End}_𝔽 (V)$ is simple iff $\dim _𝔽 V < ∞$;
$\mathrm{End}_𝔽 (V)$ has a unique non-trivial ideal iff $\dim _𝔽 V = ω$.

This example shows the non-projective f.g. flat modules.

We show some vNR-invariant operations.

Show that for idempotent $e ∈ R$. $eRe$ is vNR if $R$ is vNR.

(Kaplansky). When $R$ is a vNR ring, then so is the matirx ring $\mathrm{M}_n (R)$.

Hence, being vNR is Morita invariant (i.e., it passes up from $R ↦ 𝐌𝐨𝐝 _R$).

Let $P_k$ denote the proposition that $\mathrm M_n (R)$ is vNR. The previous problem shows that $P_{n+1} → P_n$. Once we show $P_{n} → P_{2n}$, the lemma is proved by induction.
It suffices to show $P_2$. We write $[p; q] := p - pqp$ for simplicity. Note that $R$ is vNR iff $\ker [x; (-)] ≠ ∅$ for each $x$. Note that

$$\begin{equation} [[a;b];c] = [a; b+c-cab-bac+bacab]. \end{equation}$$

Hence, if $[a ; b]$ is a vNR element, then so is $a$. We shall show that

$$\begin{equation} \binom {∗ \ ∗ }{∗ \ ∗}\ \text{are vNR} \overset {α}← \binom {∗ \ ∗ }{0 \ ∗}\ \text{are vNR} \overset {β}← \binom {0 \ ∗ }{0 \ 0}\ \text{are vNR} \overset {γ} ← \ \text{True}. \end{equation}$$

(α). If we take arbitrary $[c; \overline c] = 0$, then

$$\begin{equation} \left[\binom {a \ b}{c \ d} ; \binom {0 \ \overline c}{0 \ 0} \right] ∈ \binom {∗ \ ∗ }{0 \ ∗ }. \end{equation}$$

(β). If we take arbitrary $[a; \overline a] = [d; \overline d] = 0$, then

$$\begin{equation} \left[\binom {a \ b}{0 \ d} ; \binom {\overline a \ 0}{0 \ \overline d} \right] ∈ \binom {0 \ ∗ }{0 \ 0 }. \end{equation}$$

(γ). By the same procedure as α.

Show that over vNR,

f.g modules $=$ f.g. $𝐏𝐫𝐨𝐣$;
there are no nilpotent ideals;
prime ideals $=$ maximal ideals. Hint: $R / I$.

Example: Torsion Theory

The covering need not be epimorphisms! A trivial example is that every additive category admits a $\{0\}$-covering. A general kind of examples is known as the torsion theory.

Show that every f.g. Abelian group is a unique direct sum of finite Abelian groups and a f.g. free group.

Let $t : 𝐚𝐛 → 𝐚𝐛^{\text{finite}}$ be the functor of taking maximal torsion subgroup. Then $t(M) ↪ M$ is an $𝐚𝐛^{\text{finite}}$-cover.

The f.g. torsion groups ($𝐚𝐛^{\text{finite}}$) and f.g. free Abelian groups ($𝐚𝐛^{\text{f.g. free}}$) are orthogonal with respect to the $\mathrm{Hom}$-functor. We write $X ⟂_o Y$ for $\mathrm{Hom}(X,Y) = 0$, and denote $(-)^o := (-)^{⟂_o}$ and $^o(-) := {}^{⟂_o}(-)$ for simplicity.

Unwinding a brief introduction, we enumerate some closure properties of a torsion class as follows.

(Torsion theory). Let $\mathcal{A}$ be an Abelian category which is well-powered and subobjects are closed under transfinite filtrations. Then the following are equivalent for defining a torsion theory $(\mathcal{T}, \mathcal{F})$:

There is a subfunctor $t ⊆ 1_\mathcal{A}$ such that $t ∘ t = t$ and $t(\frac{M}{tM}) = 0$. Here $t ∘ t = t$ means that $t(tM) = tM$ when identifying subobjects of $tM$ as that of $M$. Set $\mathcal{T} = \operatorname{im} t$ and $\mathcal{F} = \ker t$.
$\mathcal{T}⟂ _o \mathcal{F}$, and every object $M$ fits into a short exact sequence $0 → tM → M → M/tM → 0$. In this case, such a sequence is functorial in $M$.
$\mathcal{T}^o = \mathcal{F}$ and $\mathcal{T} = {}^o \mathcal{F}$, or other equivalent form given by Galois connection.
$\mathcal{F} = \mathcal{T}^o$, and $\mathcal{T}$ is closed under taking transfinite filtrations (including extensions) and quotient objects.
$\mathcal{F} = \mathcal{T}^o$, and there is a class of objects $\mathcal{X}$ such that $\mathcal{T}$ consists of objects which are transfinite filtrations of quotient objects in $\mathcal{X}$.

We explain $1 \leftrightarrow 2 \rightarrow 3 \rightarrow 4 \leftrightarrow 5$ in a few words, and show $5 \rightarrow 1$ (which requires supplementary conditions).
($1 \leftrightarrow 2$). A canonical injection refers to a canonical short exact sequence $t$. ($2 \rightarrow 3$). For instance, we show $\mathcal{F} = \mathcal{T} ^o$. We claim $\mathcal{T} ⟂ _o \mathcal{F}$; if there is a non-zero $f : T → F$, then $\operatorname{im}(f) ∈ \mathcal{T}$ is a subobject of $F$, yielding $tF ≠ 0$. Conversely, we show $\mathcal{T} ^o ⊆ \mathcal{F}$; notice that every $X ∈ \mathcal{T} ^o$ has no subobjects in $\mathcal{T} ∖ \{0\}$. ($3 \rightarrow 4$). Note that any $^o\mathcal{X}$ is closed under extensions and quotients. ($4 \leftrightarrow 5$). Clearly, $→$ is by taking $\mathcal{X} = \mathcal{T}$. Conversely, any particular class $𝐅𝐢𝐥 (\text{quotients of } \ \mathcal{X})$ is closed under transfinite filtrations and quotient objects.
($4 \rightarrow 2$). We show $X ∈ \mathcal{A}$ has a maximal subobject in $\mathcal{T}$, which is also the greatest. Let $\mathcal{S}$ denote the set of subobjects of $X$ lying in $\mathcal{T}$. $0 ∈ \mathcal{S}$ is non-empty. Notice that,

Every chain in $\mathcal{S}$ has an upper bound, as $\mathcal{T}$ is closed under transfinite filtrations.
Any two maximal objects $T_1, T_2 ∈ \mathcal{S}$ coincide. If not, then $(T_1 + T_2) ∈ \mathcal{T}$, a contradiction.

Let $tX$ denote such a maximal subobject in $\mathcal{T}$; we complete the proof.

We call $\mathcal{T}$ a torsion class, and $\mathcal{F}$ a torsion-free class.

The above are equivalent for either $𝐌𝐨𝐝 _R$ or $C(R)$. Note that these categories are well-powered and cocomplete.

Show that subobjects in either $𝐌𝐨𝐝 _R$ or $C(R)$ are closed under transfinite filtrations. Hint:

the forgetful functor $U : \mathcal{A} → 𝐒𝐞𝐭𝐬$ creates all filtered colimits, and
$i_∙ : M_∙ ↪ X$ induces a monomorphism $\varinjlim ^{fil} M_∙ ↪ X$.

Another almost-example for torsion theory derives from complete hereditary cotorsion pair $(\mathcal{C}, \mathcal{F})$. Recall that $\mathcal{C}^o = \mathcal{F}$ and ${}^o\mathcal{C} = \mathcal{F}$ holds in $\frac{\mathcal{A}}{\mathcal{C} ∩ \mathcal{F}}$. When $\frac{\mathcal{A}}{\mathcal{C} ∩ \mathcal{F}}$ has exact structure, we almost done.

Envelopes and Examples

This is the dual statement of covers. The general theory of envelopes also translates to that of projective covers.

Let $\mathcal{E}$ be a replete class of objects. For any $E ∈ \mathcal{E}$, say $j : X ↪ E$ is

a preenvelope, provided there is an epimorphism from projective objects $j ^∗ : (E, -)_\mathcal{E} ↠ (X, -|_\mathcal{E})_{\mathcal{A}}$;
an envelope, provided the projective cover $j ^∗ : (E, -)_\mathcal{E} ↠ (X, -|_\mathcal{E})_{\mathcal{A}}$.

There might (the writer has never encountered it before) be a dual concept called co-small quotient module or co-superfluous quotient module. Say $p: X ↠ C$ is co-small, provided for any quotient object $q: X ↠ C'$ such that $X \xrightarrow ∼ C' × _{C ⊔ _X C'} C$, $q$ is an isomorphism.

For set-theoretic consideration, we substitute the expression co-small with essential extension.

$p : X ↠ C$ is co-small, if and only if $\ker p ↪ X$ is an essential extension which is defined as follows:

any non-zero submodule $L ↪ X$ has a non-zero intersection with $\ker p$.

($\rightarrow$). Suppose $p : X ↠ C$ is co-small, while there exists a non-zero submodule $L ↪ X$ such that $L ∩ \ker p = 0$. Then we shall see contradictions. By definition, one has the following pushout-pullback square

By definition of co-small, $X = \frac{X}{L \cap \ker p} ↠ \frac{X}{L}$ is an isomorphism, yielding $L = 0$, a contradiction.
($\leftarrow$). Conversely, suppose that $\ker p ↪ X$ is essential yet $p : X ↠ \frac{X}{\ker p}$ is not co-small. Then it is possible to find a non-zero submodule $L$ such that the above diagram is a pushout-pullback square. Now $L ∩ \ker p = 0$, a contradiction.

Show $i : X → E$ is an injective envelope, provided the following equivalent conditions:

$i$ is injective with a co-small cokernel;
$i ^∗ : (E, -)_\mathcal{E} ↠ (X, -|_\mathcal{E})_{\mathcal{A}}$ is a projective cover.

Unlike projective covers, the injective envelopes always exist.

Every module $X ∈ 𝐌𝐨𝐝 _R$ has an injective envelope.

Set $\mathcal{S} : \{i : X ↪ Y ∣ i \ \text{is essential}\}$. We claim every ascending chain in $\mathcal{S}$ has an upper bound. Note that

$$\begin{equation} ⋃ _{i ∈ I} (L ∩ Y_i) = L ∩ (⋃_{i ∈ I} Y_i)\qquad (⇔ \text{AB5 condition}). \end{equation}$$

We take any maximal object $Q ∈ \mathcal{S}$. Clearly $Q$ has only trivial essential extensions. We claim that $Q$ is injective by showing any short exact sequence $0 → Q → E → E/Q → 0$ splits. Again by AB5, there is a maximal subobject $T ↪ E$ such that $T ∩ Q = 0$. By definition, $Q ↪ \frac{E}{T}$ is essential, thus $Q ↪ E ↠ \frac{E}{T}$ is an isomorphism. Now $Q+T = E$ and $Q ∩ T = 0$, hence $E = Q ⊕ T$.
Finally, $X ↪ Q$ is an injective envelope as it is monic with co-small cokernel (= essential extension).

We provide a non-trivial example that projective cover does not exist, yet the injective envelope is easy to construct. We begin with a well-known theorem.

(Specker’s theorem). $\mathrm{Hom}(∏_ω ℤ , ℤ ) ≃ ∐_ω ℤ$.

We show that any $f : ∏_ω ℤ → ℤ$ vanishes in the ultraproduct part, that is, $f = 0$ if $f|_{∐_ω ℤ } = 0$.

Suppose $f|_{∐_ω ℤ } = 0$, but there is some infinite sequence $(x_n)$ such that $A := f((x_n)) ≠ 0$. There is a way to write $x_n = 2^n ⋅ b_n+ 3^n ⋅ c_n$. Set $B := f((2^n ⋅ b_n))$ and $C := f((3^n ⋅ c_n))$. Now $A = B + C$. By cancelling the first finite terms, we see $2^n ∣ B$ for any $n$. Hence $B = 0$. Similarly $C = 0$. Now $A = 0$.

It suffices to show $f(e_k)$ vanishes for some large $k$. If not, then we assume (without loss of generality) every $f(e_k) ≠ 0$. We construct a sharply increasing sequence $s := (2^{n_0}, 2^{n_1}, \ldots )$, so that $|f(s_{≤ k})| ≪ n_{k+1}$ for $k ≥ 1$. Now we have the triangle inequality

$$\begin{equation} |f(s)| + |f(s_{≤ k})| ≥ |f((s_{≥ k+1}))| = 2^{n_{k+1}} ⋅ (\text{something in } \ ℕ_+) \quad (∀ k). \end{equation}$$

Since $f(s)$ is constant, and $\lim_{k → ∞ }\frac{|f(s_{≤ k})|}{|f((s_{≥ k+1}))|} = 0$, a contradiction.

We show a non-trivial example of a module with injective envelope but no projective cover. Take $R = ℤ$ which is a PID (hence coherent), set $X = ℤ^ω$.

$X$ is flat, since flatness is preserved by products over coherent rings.
$X$ is not projective (= free over PID). If there is $ℤ ^{(λ)} ≃ ℤ^ω $, then $ℤ ^λ ≃ \mathrm{Hom}(ℤ^{(λ)}, ℤ ) ≃ \mathrm{Hom}(ℤ^ω, ℤ ) ≃ ℤ^{(ω)}$. Since $λ$ is uncountable, we see the contradiction.
$X$ has an injective envelope $ℤ ^ω ↪ ℚ ^ω$. $ℚ ^ω$ is clearly injective. Since $ℤ ↪ ℚ$ is essential, $ℤ ^ω ↪ ℚ ^ω$ is also essential.

For Noetherian rings, the assignment $M ↦ E(M)$ preserves both products and coproducts. Matlis’s theory shows the $1:1$ correspondence between indecomposable injective modules and prime ideals, namely $𝔭 ↦ E(R / 𝔭)$.

(Injective modules and Noetherian rings). For arbitrary ring $R$, the followings are equivalent.

$R$ is Noetherian.
$𝐈𝐧𝐣$ is closed under coproducts.
$𝐈𝐧𝐣$ is closed under filtered colimits.

(1 → 2 and 3). Note that any inclusion of ideal ($i$) is a monomorphism between f.p. modules. Let $I_∙$ denote a filtered system in $𝐈𝐧𝐣$, we see

$$\begin{equation} \varinjlim{}^{fil}(ι , I_∙) ≃ (ι , \varinjlim{}^{fil} I_∙) ,\quad ∐ (ι , I_∙ ) ≃ (ι , ∐ I_∙) \end{equation}$$

are surjections.
(2 → 1). When there exists some non-f.g. ideals, one can find a countably ascending chain of ideals

$$\begin{equation} 0 = I_0 ⊊ I_1 ⊊ I_2 ⊊ \cdots,\quad I_ω = \bigcup_{n < ω} I_n. \end{equation}$$

Note that $(π _n)_{n ∈ ℕ} : I_ω → ∐_{n ∈ ω} \frac{I}{I_n}$ is well-defined. Since $∐_{n ∈ ω} \frac{I}{I_n} ↪ ∐_{n ∈ ω} E(\frac{I}{I_n} )$ is an essential extension towards injective, we obtain $φ$:

Hence, $φ ∘ i$ factors through some $∐ _{n < n_0} E(\frac{I}{I_0})$ while $(∐ ι _n) ∘ (π)_∙$ does not. A contradiction.

(3 → 1). Consider $∐ _{k ∈ ω } E_k = \varinjlim^{fil}_{n ∈ ω }(∐ _{k ≤ n}E_k)$. Then use (2 → 1).

Set-theoretic Consideration

We show few results of the existence of (special) cover/envelope, provided the existence of (special) precover/preenvelope. The analysis holds for general well-powered Abelian category closed under some (co)limits. To conclude, we have the following theorem.

Suppose $(\mathcal{C},\mathcal{F})$ is a complete cotorsion theory such that either

$\mathcal{C}$ is closed under filtered colimits, or
$\mathcal{F}$ is closed under cofiltered limits.

Then $\mathcal{C}$ is a covering class and $\mathcal{F}$ is an enveloping class.

Flat Cotorsion Pair

The example comes from flat cotorsion pair.

Flat modules are closed under transfinite filtrations.

We prove with transfinite induction. Suppose $\{M_γ\}_{γ < α}$ is a transfinite filtration of flat modules.

$M_0 = 0$ is flat.
If $M_γ$ and $M_{γ + 1} / M_γ$ are flat, then the extension $M_{γ + 1}$ is also flat (by taking $\mathrm{Tor}_1$).
If $\{M_{β}\}_{γ < β}$ are flat for some limit ordinal $β$, then for any ses $θ$, the filtered colimit of ses $\varinjlim^{fil}_{γ < β} (M_γ ⊗ θ)$ is also a ses.

$𝐅𝐥𝐚𝐭$ is closed under filtered colimits, as $(\varinjlim^{fil}F_∙ ⊗ -) ≃ \varinjlim^{fil}(F_∙ ⊗ -)$ is exact.

It is straightforward to show that $(𝐅𝐥𝐚𝐭 , 𝐅𝐥𝐚𝐭 ^⟂ )$ is a cotorsion theory. Since

$𝐏𝐫𝐨𝐣 ⊆ 𝐅𝐥𝐚𝐭$;
$𝐅𝐥𝐚𝐭$ is closed under taking summands;
$𝐅𝐥𝐚𝐭$ is closed under transfinite filtrations.

Hence, $𝐅𝐥𝐚𝐭 = 𝐒𝐦𝐝 (𝐏𝐫𝐨𝐣 ∗ 𝐅𝐢𝐥 (𝐅𝐥𝐚𝐭)) = {}^⟂ (𝐅𝐥𝐚𝐭^⟂ )$.

We call $(𝐅𝐥𝐚𝐭 , 𝐅𝐥𝐚𝐭 ^⟂)$ a flat cotorsion theory.

We call $𝐅𝐥𝐚𝐭 ^⟂$ Enochs cotorsion modules. There are analogous definitions, e.g., Warfield cotorsion modules stands for the right orthogonal class of torsion-free modules.

It has long been a conjecture that whether the flat cotorsion pair is cogenerated by a set. The existence of flat covers (Enochs’s flat covering theorem) solves the problem.

Precover ⇒ Cover

Let $\mathcal{P}$ is a replete class of modules or chain complexes. We assume $\mathcal{P}$ is closed under filtered colimits and summands, e.g. $\mathcal{P} = 𝐅𝐥𝐚𝐭$ or the entire category.

In this case, it is feasible to weaken the definition of $\mathcal{P}$-cover.

Let $p : P → M$ be a $\mathcal{P}$-precover. It is a $\mathcal{P}$-cover, iff the following equivalent conditions holds:

for any other morphism $q$ from $\mathcal{P}$ to $M$, $q ∘ α = p$ implies $α$ is monic,
for any other $\mathcal{P}$ precover $p$, $p' ∘ α = p$ implies $α$ is monic,
$p ∘ α = p$ implies $α$ is monic,
$p ∘ α = p$ implies $α$ is an isomorphism.

(1 → 2 → 3) is clear. For (3 → 1), we take any such $q$ with $q ∘ α = p$. Since $p$ is a $\mathcal{P}$-precover, there is and $q = p ∘ β$. We see $(β ∘ α)$ is monic. Hence $α$ is monic. (4 → 2) is clear.
We show (3 → 4). If not, then there exists a stricly injection $i : P → P$ such that $p ∘ i = p$. We shall construct a transfinite composition which terminates at $κ ≫ |P|$:

$$\begin{equation} P_0 \xrightarrow {f_{1,0}} P_1 \xrightarrow {f_{2,1}} P_2 → \cdots P_γ \xrightarrow {f_{γ +1,γ }} \cdots P_{κ}. \end{equation}$$

For any $\mathcal{P}$-precover $p_γ : P_γ → M$, we define $P_{γ + 1} := P$ and $p_{γ + 1} = p$. In particular,

when $P_γ = P$, we take $f_{γ + 1, γ } = i$,
when $γ$ is a limit ordinal, then $p_{γ}$ is induced by universal property of colimit. By closure property of $\mathcal{P}$, we see $P_γ ∈ \mathcal{P}$.

Clearly, $p_{γ} = p$ when $γ$ is not a limit ordinal. We take the subsystem $S := \{P_γ ∣ p_γ = p\}$ consisting of strict monomorphisms. As a result, the subobjects of $P$ has cardinality $κ$. Since $κ$ is arbitrary large, a contradiction.

There is a small gap in the proof. Let $α$ be any infinite ordinal. Show that

$$\begin{equation} |\{γ ∈ α ∣ ∃ β \ \text{s.t.} \ γ = β^+\}| = |α|. \end{equation}$$

Therefore, in the over-category $\mathcal{P} → M$, the $\mathcal{P}$-cover dominates the injections. It is convenient to define the following.

The collection of admissible kernels of $q : Q → M$, a morphism from $\mathcal{P}$ to $M$, is defined as

$$\begin{equation} \mathrm{KER}(q) := \{\ker α ⊆ Q ∣ ∃ \ \text{precover} \ p, \ \text{and} \ p ∘ α =q\}. \end{equation}$$

Clearly, for any $\mathcal{P}$-precover $p$, $\mathrm{KER}(p)$ is a singleton $\{0\}$ iff $p : P → M$ is a $\mathcal{P}$-cover.

Now we show every $M$ has a $\mathcal{P}$-cover when there is a $\mathcal{P}$-precover.

Again by Grothendieck’s insight, we firstly show the existence of relative covering.

(Base change). Let $q = p ∘ α$ be a commutative diagram where

$p$ is a $\mathcal{P}$-precover from $P$ to $M$,
$α : P → Q$ is a morphism from $\mathcal{P}$ to $\mathcal{P}$.

Then there is a induced map

$$\begin{equation} α ^♯ : \mathrm{KER}(p) → \mathrm{KER}(q), \quad \ker (?) ↦ \ker (? ∘ α). \end{equation}$$

We say $α ^♯$ is a relative cover, provided $\operatorname{im}(α ^♯)$ is a singleton $\{\ker α \}$.

Now we show the existence of relative $\mathcal{P}$-covering.

For any morphism $q : Q → M$, there is some $α : Q → P$ ($p = q ∘ α$, and $p$ is a $\mathcal{P}$-precover) such that $α ^♯$ is a relative cover.

If not, then there is an transfinite composition

$$\begin{equation} (Q; q) := (P_0; p_0) \xrightarrow {f_{1,0}} (P_1; p_1) \xrightarrow {f_{2,1}} \cdots (P_γ; p_γ ) \xrightarrow {f_{γ +1,γ }} \cdots (P_{κ}; p_κ), \end{equation}$$

such that

$p_{γ + 1 } ∘ f_{γ +1, γ } = p_γ$ for each $γ$;
if $γ$ is a successor ordinal, $p_γ$ is a $\mathcal{P}$-precover;
if $γ$ is a limit ordinal, then $p_γ$ is induced by colimit;
$\{\ker f_{γ , 0}\}$ strictly ascends when composing $f_{β +2, β +1}$.

Again, let $κ ≫ |P|$, we see the contradiction.

Let $\mathcal{P}$ be a class of modules or complexes closed under summands and filtered colimits. When $X$ has a $\mathcal{P}$-precover, then it has a $\mathcal{P}$-cover.

We construct a transfinite composition of relative covers:

We explain the followings ($γ$ is a limit ordinal).

$\varinjlim^{fil}_{\beta < \gamma }(P_\beta) → M$, induced by colimit, is just a morphism from $\mathcal{P}$, which not necessary a precover. We take $\overline {p_γ} : \overline {P_γ} → M$ as arbitrary precover it factors through. Then there is a relative cover $\overline {P_γ } → P_γ$.
We claim $α _1 |_{\mathrm{im}(α _0)} : \mathrm{im}(α _0) ↪ \mathrm{im}(α _1)$ is an injection. This is due to $\ker (α _1 ∘ α _0) = \ker (α _0)$.
The monomorphism $\varinjlim^{fil}_{\beta < \gamma }(\alpha _\beta) ↪ \mathrm{im}(\alpha _\gamma)$ is induced by the monomorphisms $\mathrm{im}(\alpha _\beta) ↪ \mathrm{im}(\alpha _\gamma)$. Notice that the coposition $P_β → P_γ$ is a relative cover.

If we fix a certian precover $\overline {p_γ} : \overline {P_γ} → M$ for all limit ordinals $γ$, then the inclusion chain for $\operatorname{im}(α _β)$’s stablises for $|β| ≫ |P_γ|$. Now such $\mathrm{im}(α _β)$ belongs to $𝐒𝐦𝐝 (\mathcal{P}) = \mathcal{P}$.
We denote $q : S → M$ as such summand of $p_{β +1} : P_{β +1} → M$. It remains to show that $q$ is a cover. Note that $q$ is a precover. It suffices to show that

for any precover $r : R → M$ and $r ∘ δ = q$, $δ$ is monic.

Consider the following diagram.

By definition of relative covering ($⋆$),

$$\begin{equation} \ker (π _β) = \ker (α _β ) \overset ⋆= \ker ((δ ∘ π _{β +1}) ∘ α _β) = \ker (δ ∘ π _β ). \end{equation}$$

Hence $δ$ is monic. Now we see that $q$ is a $\mathcal{P}$-cover.

Show the (partly) equivalent definition of (right) perfect rings. All modules are right modules.

$𝐅𝐥𝐚𝐭 = 𝐏𝐫𝐨𝐣$,
$𝐅𝐥𝐚𝐭$-precovers are $𝐏𝐫𝐨𝐣$-precovers,
$𝐏𝐫𝐨𝐣$-cover exists.

Hint: (1 → 2) is clear. (2 → 3) is due to the analysis above, as $𝐏𝐫𝐨𝐣$-precover always exists. (3 → 1) is due to the fact that flat module with a projective cover is projective (see above section).

Special Preenvelope ⇒ Special Envelope

We show a criterion that, for any subclass of modules or complexes $\mathcal{C}$ closed under summand, extensions and filtered colimits, if any $M$ admits a special $\mathcal{C}^⟂$-preenvelope (monic pre-envelope with cokernel in $\mathcal{C}$), then there is a special $\mathcal{C}^⟂$-envelope.

Recall the procedure for showing the existence of $\mathcal{P}$-cover when there is a $\mathcal{P}$-precover.

For a general morphism $q$ from $Q ∈ \mathcal{P}$ to $M$, one can find $α$ from $q$ to some precover $p$ such that $α ^♯$ is a relative cover.
We start from any precover a transfinite composition of relative covers. We see the ascending chain of images. Filling every gap at the limit ordinals with a fixed precover, we obtain a stabilised summand $s : S → M$, which is a precover.
For any morphism of precovers from $s$, e.g., $α : s → p$, we show $α$ is monic. Hence $s$ must be a cover for set-theoretic issues.

Anaglously, we substitute

Morphisms of the type $\mathcal{P} → M$, with ses of the type $0 → M → ? → \mathcal{C} → 0$;
$\mathcal{P}$-precover, with those ses with lifting property;
$\mathcal{P}$-cover, with minimal ses with lifting property.

Recall that the morphisms for $\mathcal{P} → M$ are

$$\begin{equation} \text{General Morphism} → \text{Precover} → \text{Cover}. \end{equation}$$

Now analogously take the morphisms of extensions in $\mathrm{Ext}^1(\mathcal{C}, M)$ as

$$\begin{equation} \text{General Extension} → \text{Preenvelift} → \text{Envelift}. \end{equation}$$

We analysis the admissible kernels at $P$ and $F$, respectively:

The proof starts here. We define the so-called envelift of ses among the those of the type $\mathrm{Ext}^1(\mathcal{C}, M)$.

Say $[θ] ∈ \mathrm{Ext}^1(\mathcal{C}, M)$ is a $\mathcal{C}$-preenvelift, provided for any ses $[τ] ∈ \mathrm{Ext}^1(\mathcal{C}, M)$, there is a morphism of extensions $θ → τ$. Say a pre-envelift $θ$ is a $\mathcal{C}$-envelift, provided for any endomorphism of extension $α : θ → θ$, $α$ is an automorphism.

We use the morphism in the middle $f$ term to stand for the morphism of extensions $(1_M, f, f|_{\mathrm{coker}})$.

For a general extension $[δ] ∈ \mathrm{Ext}^1(\mathcal{C}, M)$, we denote

Moreover, when there is a morphism $α : δ → θ$ from $δ$ to a $\mathcal{C}$-preenvelift $θ$, we have the base change

$$\begin{equation} α ^♯ := \mathrm{KER}(θ) → \mathrm{KER}(δ), \quad \ker (?) ↦ \ker (? ∘ α). \end{equation}$$

Say $α$ is a relative envelift, provided $\operatorname{im}(α ^♯)$ is a singleton $\{\ker α\}$.

We show firstly the existence of relative envelift.

If $δ$ has a preenvelift, then $δ$ has a relative envelift.

If not, then there is a transfinite composition of morphisms of extensions:

$$\begin{equation} δ = θ _0 \xrightarrow {f_{1,0}} θ _1 \xrightarrow {f_{2,1}} \cdots θ _γ \xrightarrow {f_{γ +1,γ }} \cdots θ _{κ}, \end{equation}$$

such that

$θ _{γ + 1} ∘ f_{γ +1, γ } = θ _γ$ for each $γ$;
if $γ$ is a successor ordinal, $θ _γ$ is a $\mathcal{C}$-preenvelift;
if $γ$ is a limit ordinal, then $θ _γ$ is induced by colimit;
- Remark: $0 → M → \varinjlim^{fil}X_∙ → \varinjlim^{fil}C_∙ → 0$ is again exact. Meanwhile, $\varinjlim^{fil}C_∙ ∈ \mathcal{C}$ by assumption.
$\{\ker f_{γ , 0}\}$ strictly ascends when composing $f_{β +2, β +1}$.

Let $κ ≫ |θ|$, we see the contradiction.

When $δ$ (of the type $\mathrm{Ext}^1(\mathcal{C}, M)$) admits a $\mathcal{C}$-preenvelift $θ$, then it has a $\mathcal{C}$-envelift.

We construct a transfinite composition of relative envelifts:

$$\begin{equation} θ = θ _0 \xrightarrow {α _0} θ _1 \xrightarrow {α _1} \cdots θ _γ \xrightarrow {α _γ} \cdots θ _{κ}. \end{equation}$$

For every limit ordinal $γ$, we firstly take $\overline {θ _γ }$ by colimit, then take $\overline {θ } → θ$. The stablised image $ε$ sequence is a summand of some preenvelift, thus $ε$ is a preenvelift.

In this case, any endomorphism $α : ε → ε$ is injective. Since all objects are well-powered, $α$ is an automorphism. Once we show that the middle term of $ε$ ($ε _2$) lies in $C^⟂$, we complete the proof. We take Noether’s isomorphism:

$C' ∈ 𝐅𝐢𝐥 (\mathcal{C}) = \mathcal{C}$. Hence $q : ε → ε '$ is a morphism of extensions of type $\mathrm{Ext}^1(M, \mathcal{C})$. Since $ε$ is a $\mathcal{C}$-preenvelift, there is another morphism $s : ε ' → ε$. Since $s ∘ q$ is an automorphism, $ε_2 ∈ 𝐒𝐦𝐝 (\mathcal{C}^⟂) = \mathcal{C}^⟂$.

Envelope ⇒ Preenvelope

Suppose $\mathcal{E}$ is a class of modules closed under summand, along with the following concession.

(Concession). If there is a preenvelope $i : M → E$, then any morphism from $M$ to the cofiltered limit $\varprojlim^{cof} E_∙$ factors through $i$.

In particular, the concession holds if $\mathcal{E}$ is closed under cofiltered limits.

We denote $\widetilde {\mathcal{E}}$ as the $\varprojlim ^{cof}$-completion of $\mathcal{E}$, and introduce the relative cokernel (or relative image for convenience).

(Image). For a general morphism of the type $j : M → \widetilde {\mathcal{E}}$, we define

$$\begin{equation} \mathrm{IM}(j) := \{\operatorname{im}(α ) ∣ α : i → j,\ i \ \text{ is a preenvelope}\}. \end{equation}$$

We also have the cobase change. Say $α ∈ \mathrm{IM}(j)$ is a relative envelope, provided

$$\begin{equation} α _♯ : \mathrm{IM}(i) → \mathrm{IM}(j), \quad \operatorname{im}(?) ↦ \operatorname{im}(α ∘ ?),\ \text{such that} \ \operatorname{im}(α _♯ ) = \{\operatorname{im}(α )\}. \end{equation}$$

We show the existence of relative envelope.

Any morphism of the type $j : M → F ∈ \widetilde {\mathcal{E}}$ has a relative envelope.

If not, then there is a transfinite composition of towers

$$\begin{equation} \cdots \overset {f_{γ +1, γ+0}}→ (E _{γ +1}, i _{γ +1}) \overset {f_{γ,γ+1}} → (E_γ, i _γ ) \cdots \overset {f_{1,2}}→ (E_1, i _1) \overset {f_{0,1}}→ (E_0, i _0) = (F, j), \end{equation}$$

such that

for any $β$, $i_β ∘ α _β = i_{β +1}$;
if $β$ is a successor ordinal, then $i_β$ is a preenvelope;
if $β$ is a limit ordinal, then $i_β$ is induced by limit;
$\{\operatorname{im}(f_{0, γ})\}$ strictly descends when composing $f_{β +1, β +2}$.

Let $κ ≫ |F|$, we see the contradiction.

We leave the dual procedure to readers.

Show that

Any preenvelope $i : M → E$ admits a trasnfinite tower of relative envelopes;
The stablised image $ε$ is a preenvelope;
Any endomorphism $α : ε → ε$ is surjective, hence bijective.

Special Precover ⇒ Special Cover

Suppose $\mathcal{F}$ is a class of modules closed under summands, extensions, and cofiltered limits. A special ${}^⟂\mathcal{F}$-(pre)cover is an epic ${}^⟂ \mathcal{F}$-(pre)cover with kernel in $\mathcal{F}$.

For $M$ admits a special ${}^⟂\mathcal{F}$-precover, $M$ there has a special ${}^⟂ \mathcal{F}$-cover.

Recall in the subsection Special Preenvelope ⇒ Special Envelope, we did not use AB5 property.

Remarks

We are interested in cogenerating construction for set-theoritic issuse. For instance,

(Shelah). It is independent of ZFC that every whitehead group ($^⟂ℤ$) is a free group. A minicourse.

Completion (with ideal-metric) is a special case of cofiltered limit. We might see that in the study of (Enoch’s) cotorsion modules, etc.

One can similarly define $\mathrm{Tor}$-cotorsion theory. Note that the left and right class of modules are closed under extensions, summands, coproducts, and filtered colimits. In this case, any $\varinjlim^{fil}$-closed $\mathcal{C}$-class corresponds to a $\mathrm{Tor}$-torsion theory.