CT4

(← ). Consider

$$\begin{aligned} (x ∈ φ '(M)) ↔ (B' ∣ xA') → (B'Z ∣ xA'Z) ↔ (YB ∣ x(A-XB))\\[6pt] → (B ∣ x(A-XB)) → (B ∣ xA) ↔ (x ∈ φ (M)). \end{aligned}$$

(→). We take $φ' (M) = \{x : B' ∣ xA'\}$ as the projection of the first $k$ coordinates of f.p. module $M: = \{(x,y) : (x,y)\binom{A'}{-B'}=O\} = \operatorname{cok}(A'^T, -B'^T)$. Let $\{z_k\}_{k=1}^n$ denote the image of coordinal basis along $R^n ↠ M$ generating set of $M$, which is a generating set of $M$. By definition, $(z_{[1,k]}, z_{(k,n]})\binom{A'}{-B'} = O$. Hence $(z_{[1,k]}, w)\binom{A}{-B} = O$ for some $w$. We write $w = z_{(k,n]}\binom{X}{Y}$ which recovers the construction.

We write $R^m \overset {C^T} → R^n \overset π → M → 0$, then $\{z_k = π (e_k)\}_{k=1}^n$ generates $M$. Now $z C = 0$. We set $x + z X = 0$ and define

$$\begin{equation} φ_m [x,y] := ∃ y \ ((x,y)\binom{I \ O}{X \ C} = 0). \end{equation}$$

To see $φ _m$ is the least pp condition, we take any $x ∈ φ(M)$ and show that $φ_m ≤ φ$. There is $y'$ s.t. $(x,y')\binom{A}{-B} = 0$. Since $z$ is a generating set, there is $(x,y') = (x,z)\binom{I \ O}{O \ Q}$. Hence,

$$\begin{equation} \binom{I\ O}{O \ Q} ⋅ \binom{A}{-B} = \binom {A}{-QB} ∈ \ker ((x,z) ⋅ ) \overset ⋆= \operatorname{im} (\binom{I \ O}{X \ C}⋅ ). \end{equation}$$

Here $⋆$ is equivalent to $\ker ((0, z)⋅) = \operatorname{im}(\binom{I \ O}{O \ C} ⋅)$, which is more over equivalent to $\ker π = \operatorname{im} (C^T)$.

(←). Clear. (→). Note that $N$ is a filtered colimit of f.p. modules containing $𝐧$. Let $N_0$ be any f.p. module in such system, AB5 shows

$$\begin{equation} \ker [N_0 → N] = ⋃ _{0 → ?} \ker [N_0 → N_?]. \end{equation}$$

Hence, $𝐦 ⊗ 𝐧 = 0$ in some $M ⊗ N_?$. WLOG, we assume $N$ is f.g. at the beginning. We extend $𝐧$ to $(𝐧 \ 𝐥) = \{y_j\}_{j=1}^q$ as a generating set of $N$. Now $(𝐦\ \mathbf0) ⊗ \binom{𝐧 }{𝐤} = 0$ in the image of each $M ⊗ \frac{N}{(\{y_j\}_{j ≠ j_0})}$. By definition, $M ⊗ \frac{R}{(r)} = \frac{M}{Mr}$, there is a vector $𝐯_{j_0}$ s.t.

$$\begin{equation} ∑ (𝐦\ \mathbf0) ⊗ y_{j_0} = ∑ 𝐯 _{j_0} a ⊗ y_{j_0} = ∑ 𝐯 _{j_0} ⊗ a y_{j_0} = 0\quad ∈ M ⊗ N. \end{equation}$$

Combining each $𝐯 _{j_0}$ we obtain $(X \ Y)$.

The morphism $(M, N) → φ (N)$ sends $f ↦ f(z)$ (with diagonal action). To see it is surjective, any $s ∈ φ (N)$ satisfies $\binom{I \ -A}{O \ \ \ B \ } ∣ s I$. Hence, $s$ maps from the generating set $z ⊆ M$. For exactness, we see

$$\begin{equation} \ker [(M, N) \xrightarrow{(-)(z)} φ (N)] = \{f : f(z) = 0\} ≃ (M/(z), N). \end{equation}$$

Note that the latter statement is equivalent to that, f.p. modules lifts $B ↠ C$.
(←). Recall that $φ(-) = \operatorname{cok}((M/(z),-) ↪ (M, -))$. By assumption, both $(M/(z),θ)$ and $(M, θ)$ are exact, thus $φ (θ)$ is exact. We obtain a monomorphis of ses:

By inclusion of cokernels, $\square$ is a pullback. Hence $φ (B) ∩ A^k = φ (A)$.
(→). For any pp condition $φ$, $c ∈ φ (C)$, there is $y$ s.t. $(c,z)H = 0$. Taking the preimage

$$\begin{equation} B^n ↠ C^n,\quad (b,y) ↦ (c , z), \end{equation}$$

we see $(b,y)H ∈ A^m$. By pure embedding, there is some $(a,x) ∈ A^n$ s.t. $(a,x)H = (b,y)H$. Hence, we obtain the induced surjection

$$\begin{equation} π : φ (B) ↠ φ (C),\quad (b-a) ↦ c. \end{equation}$$

Now we show that any morphism $h : K → C$ lifts to $\widetilde h : K → B$. Let $x$ be the generating set of $K$, and take $φ = \min pp^K(x)$. Now $h(x) ∈ φ (C)$. By induced surjection $φ (B) ↠ φ (C)$, there is some $y ∈ φ (B)$ s.t. $π (y) = h(x)$. Since $(K, x)$ is a free realisation of $φ$, there is a morphism $\widetilde h : K → B$ s.t. $\widetilde h (x) = y$. We see that $\widetilde h$ is a lift of $h$.

(←) split monomorphisms and pure. It suffices to show pure morphisms are closed under $\varinjlim^{fil}$. The partial lifting property of pure morphism $f_i$ and f.p. morphism $g$ is equivalent to $\operatorname{im}(2) ⊆ \operatorname{im}(3 ∘ 1)$ in the following diagram:

By exactness of filtered colimit, we obtain $\operatorname{im}(2) ⊆ \operatorname{im}(3 ∘ 1)$ in the colimit diagram.
(→). Any pure morphism $f : X ↪ Y$ is a filtered colimit of the injective system $f_i ⇒ f$. We assign each $i$ a pushout diagram, wherein $e_i$ splits by partial lifting property:

Now $\varinjlim^{fil} Y_i ↪ \varinjlim^{fil} (X ⊔ _{X_i} Y_i) ↪ Y$. Hence the isomorphisms holds.

We show 1 ↔ 2 and 5 ↔ 6, while 3 ↔ 4 ↔ 5 is an exercise. Clearly 2 → 3. It remains to show 5 → 1.
(5 → 1). We shall find the partial lifting map $s$ in the diagram:

Here $\operatorname{cok}(g)$ is f.p.. By 5., there is dashed $t$ making $\circlearrowleft$ commutative. Since $(j - tp) ∈ (π _∗ )$, there is some $s$ s.t. $j - tp= fs$. Now

$$\begin{equation} (sg=i) ↔ (fsg=fi) ↔ ((j - tp)g =jg) ↔ \text{true}. \end{equation}$$

Note that the cardinal of the set of pp conditions is no more than $\max (ω , |R|)$. Set $L_0 := (x)$. For each $x ∈ L_0$ and pp condition $φ$, one can find finitely many bound variables ${}^xS^0_φ ⊆ M$ such that $φ (M) ∩ L_0 ⊆ ({}^xS^0_φ ∪ L_0)$. Take

$$\begin{equation} L_{k+1} := (L_{k} ∪ ⋃ \limits_{x ∈ L_0, φ ∈ \ \text{pp condition}}{}^xS^k_φ),\quad k ∈ ℕ. \end{equation}$$

Hence, for arbitrary $k$ and pp condition $φ$, one has $φ (M) ∩ L_{k} ⊆ L_{k+1}$. Note that $|L_k|≤ \max (ω , |R|)$. We obtain the desired pure submodule $L := ⋃ _{k ∈ ℕ }L_k$.

Summands of $∐ (𝐦𝐨𝐝_R)$ are pure projective, as pure projectives are closed under $∐$. Conversely, any pure porjective $M$ is a filtered colimit of f.p. modules. By definition of pure exactness, the following ses splits:

$$\begin{equation} 0 → K → ∐_{i ∈ I} M_i → \varinjlim{}^{fil} M_i → 0. \end{equation}$$

By the proof of Kaplansky theorem, a summand of coproducts of countably generated modules is also a coproduct of countably generated modules.

We apply Herzog’s pairing to the assignment

$$\begin{equation} M ⊗ ∏ _{i ∈ I} N_i → ∏ (M ⊗ N_i),\quad x ⊗ (y_i)_{i ∈ I} ↦ (x ⊗ y_i)_{i ∈ I}. \end{equation}$$

(→). When M is ML, the above is monic. Assume that $pp^M(x)$ is non-principal. We take $N_φ$ be such that $\max pp^N(y_φ) = Dφ$, where $φ$ runs through $pp^M(x)$. Since $x ⊗ (y_φ)_{φ ∈ pp^M(x)} = 0$, there exists pp condition $ψ$ s.t. $x ∈ ψ (M)$ and $$\begin{equation} (y_φ)_{φ ∈ pp^M(x)} ∈ (Dψ) (∏\limits_{φ ∈ pp^M(x)} N_φ), \quad \text{that is,}\quad ⋀\limits_{φ ∈ pp^M(x)} (y_φ ∈ Dψ (N_φ )). \end{equation}$$

Hence, $ψ ∈ pp^M(x)$ is smaller than any $φ ∈ pp^M(x)$, yielding that $x$ is principle. A contradiction.
(←). Assume that $pp^M(x)$ is principal. For any $(x ⊗ y_i)_{i ∈ I} = 0$ in $∏ (M⊗ N_i)$, there is a list of pp conditions $\{φ _i\}_{i ∈ I}$ s.t. $$\begin{equation} x ∈ φ _i (M) ∧ y_i ∈ Dφ _i (N_i),\quad i ∈ I. \end{equation}$$

Taking $ψ = \min pp^M(x)$, we see that

$$\begin{equation} x ∈ ψ (M) ∧ (y_i ∈ Dψ (N_i)),\quad i ∈ I. \end{equation}$$

Since $φ$ preserves $∏$, $ψ$ shows $x ⊗ (y_i)_{i ∈ I} = 0$ in $M ⊗ ∏ _{i ∈ I} N_i$ by Herzog’s pairing.

We show (←), i.e., a countably generated ML module $X$ is pure projective. Let $\{x_i\}_{i ∈ ω }$ be generating set of $X$. For each $n$ we denote $φ _n := \min pp^X_{n+1} (x_0,\ldots, x_n)$. We also define

$$\begin{equation} φ ' _{n+1} [x_0, \ldots, x_n] := ∃ y \ φ _{n+1}[x_0, \ldots, x_n, y]. \end{equation}$$

Let $p : Q ↠ X$ be any pure epimorphism. We shall construct the right inverse of $p$ by assigning $x_i ↦ q_i$. We choose any $q_0$ to be the inverse image of $x_0$ s.t. $(q_0) ∈ φ_0(x_0)$. Suppose that we have $(q_0, \ldots, q_n)$ such that

1. $φ _k [q_0, \ldots, q_k]$ for arbitrary $k ≤ n$, and
2. $x_k ↦ q_k$ for $k ≤ n$.

We shall find $q_{n+1}$ satisfying the inductive conditions. We see that $φ '_{k+1}[q_0, \ldots, q_k]$. Consider the following tuples in $Q^{k+1}$:

1. Let $\widetilde {q_{k+1}}$ be the indeterminant such that $φ_{k+1}[q_0, \ldots, q_k, \widetilde{q_{k+1}}]$.
2. Let $(r_i)_{i ≤ k+1}$ be preimages of each $x_i$.

Hence, $(q_i - r_i)_{i ≤ k} ∈ (\ker p)^{k+1}$. By pure embedding, there is some $λ ∈ (\ker p)^{k+1}$ s.t.

$$\begin{equation} (q_0 - r_0, \ldots , q_k - r_k, λ ) ∈ φ _{k+1} (Q). \end{equation}$$

Set $q_{k+1} := r_{k+1} + λ$, we complete the induction.
Finally, we show $x_i ↦ q_i$ is well-defined. When there is $∑_{i ≤ l} x_i a_i = 0$, we have $φ _l (x_∙) → φ _l (q_∙)$. Hence $∑_{i ≤ l} q_i a_i = 0$.

The functor $X ↦ [\cdots = X = \overset{\deg n} X → 0 = 0 → \cdots → 0]$ has left adjoint $(Y_∙, ψ_∙) ↦ Y_n$. Since an exact left adjoint preserves injective objects, the category of $ω$-towers has enough injectives, i.e.,

$$\begin{equation} (Y_∙, ψ_∙) ↪ I^0(Y_∙) := [\cdots = E(∐ Y_∙)= E(∐ Y_∙)]. \end{equation}$$

We obtain ses of $ω$-towers $0 → Y → I^0 → C → 0$. The morphisms in $I^0$ are surjections, hence $\varprojlim{}^1 I^0 = 0$. By snake lemma, we have the $4$-term ker-coker sequence

By injective resolution $Y \overset ι ↪ I^0 \overset {g_0}→ I^1\overset {g_1}→ \cdots$,

$$\begin{equation} (\mathrm R^1 \varprojlim) Y = \frac{\ker [\varprojlim (g_1)]}{\operatorname{im} [\varprojlim (g_0)]} ≃ \frac{\varprojlim \ker (g_1)}{\operatorname{im} [\varprojlim (g_0)]}≃ \frac{\varprojlim C}{\operatorname{im} [\varprojlim I^0 → \varprojlim C]} ≃ \varprojlim {}^1 C \end{equation}$$

We show (1 → 2 → 3 → 1). We show in previous subsection shows (1 ↔ 2). Recall that in the proof of (2 → 1), the collection $N_φ$ s.t. $φ = \min pp^{N_φ }(y_φ )$ is chosen to be f.p. modules. Hence, (3 → 1) holds.
We show (6 → 5 → 4 → 6).
(6 → 5). We take any $e_i : M_i → M$ and $g : M_i → X$ s.t. the pushouts are pure embeddings. Note that $\operatorname{cok}[M_j → M_j ⊔ _{M_i} X]$ is f.p. (pure projective), $e_i$ factors through $g$ via some $s :X → M$. By smallness of compact objects, there is $M_j$ making the following diagram commute:

To see that the pushout morphisms of $e_i$ and $f_{j,i}$ are pure embeddings, we observe $$\begin{equation} \ker (f_{j,i} ⊗ -) ⊆ \ker (e_i ⊗ -) = \ker (g ⊗ -) ⊆ \ker (f_{j,i} ⊗ -). \end{equation}$$ The equality holds.
(5 → 4). Note that for any $i$, there is $i → j$ s.t. $\ker (e_i ⊗ -) = \ker (f_{j,i} ⊗ -)$. We obtain the pure embedings:

Note that pure embedding splits when cokernel is pure projective. Hence $f_{j,i}$ factors through $f_{k,i}$.
(4 → 6). We take arbitrary $f : X → M$. There is $e_i$ where $f$ passes through. By ML condition, there is $j$ s.t. for any $j → k$, $f_{j,i}$ factors through $f_{k,i}$.

For each $j → k$ we have the left pushout square. $M_j → M_j \sqcup _{X}M_k$ is split monic by factoring morphism. Taking filtered colimit we obtain the right pushout square. The morphism $X_j → M_j \sqcup _X\varinjlim ^{fil}M_k$ is a pure embedding since it is a filtered colimit of split monomorphisms. By cofinality, $\varinjlim ^{fil}_{k \ (j → k)}M_k ≃ M$. The morphism $X → M$ factors through $e_j$, thus the pushout of $X → M_j$ also splits.
We finally show (1,2,3 ↔ 4,5,6). Note that the canonical morphism in 2. is

$$\begin{equation} \varinjlim{}^{fil}_i ∏_λ (M_i ⊗ N_λ) → ∏_λ \varinjlim{}^{fil}_i (M_i ⊗ N_λ). \end{equation}$$

(3 → 6). For any $f : X → M$, we take index set $$\begin{equation} Λ := ⨆ \limits_{K ∈ 𝐦𝐨𝐝 _R} \ker (f ⊗ K) =\{(λ, K_λ)\}. \end{equation}$$

Explicitly, $λ := (λ, K_λ)$ is an element in $Λ$, $λ ∈ \ker (f ⊗ K_λ)$. $K_λ$ is a representative of the isomorphism class of $𝐦𝐨𝐝 _R$. The above is well-defined since

• $𝐦𝐨𝐝 _R$ is essentially small, and
• $X ⊗ ∏ K_λ ≃ ∏ (X ⊗ K_λ)$ as $X$ is f.p..

Let $x := (λ)_{λ ∈ Λ } ∈ ∏ (X ⊗ K_λ)$ be the universal element. By Herzog’s tensor pairing, any $x ∈ \ker (f ⊗ ∏ _{λ ∈ Λ} K_λ)$ lies in some $\ker (g ⊗ ∏ _{λ ∈ Λ} K_λ)$, where $g : P → Y$ is a morphism between f.p. modules.

It suffices to consider in the right column, that is, $x = (λ) ∈ ∏ X ⊗ K_λ$ belongs to $∏ \ker (f⊗ 1)$. By structure of $λ$, $\ker (g ⊗ K) ⊆ \ker (f ⊗ K)$ for any $K ∈ 𝐦𝐨𝐝 _K$. Let $f'$ and $g'$ denote the pushout morphisms of $f$ and $g$. Therefore, $g'$ is pure monic and $f'$ splits monic.
(5 → 2). When $(M_∙ , X)$ is ML system, we show $\ker (M ⊗ ∏ N_∙ → ∏ (M ⊗ N_∙ )) = 0$. Equivalently, for any $i$ there is $i → j$ s.t. any

$$\begin{equation} x ∈ \ker \left(M_i ⊗ ∏ N_∙ → ∏ (M ⊗ N_∙ )\right) ≃ ∏_{λ ∈ Λ } \ker (e_i ⊗ N_λ ) ∋ (x_λ ) \end{equation}$$

maps to $0$ under $f_{j,i} ⊗ 1$. Note that there is $j$ s.t. $\ker (e_i ⊗ -) = \ker (f_{j,i} ⊗ -)$ by ML system.

Recall that $M ↪ M^{++}$, $M ↪ M^{cc}$, and $∐ _{λ ∈ Λ } M ↪ ∏ _{λ ∈ Λ } M$ are pure embeddings. Hence, (1 ↔ 2 → 3 ∧ 5). is clear.
(3 → 4). It suffices to show all scalar-compact modules are algebraically compact. We take arbitrary system $r_{∙,∙} ∈ ∏ _I ∐ _J R$. Note that for each finite $I_0 ⊆ I$, $M^{I_0}$ is zero of a continuous map, which is closed. Hence the solution set (with $I_0$ linear equations) is a closed subset of $M^I$, thus it is compact. By finite intersection argument of compact sets, $M$ is algebraically compact.
(4 → 1). For any pure submodule $X ↪ Y$, we show $h : X → M$ extends to $Y → M$. We take infinite turple $f ∈ M^Y$ with the following set of equations

$$\begin{equation} \{f_{y}r+f_{y'}r'-f_{yr+y'r'}=0\}_{y,y' ∈ Y, \text{and} \ r,r' ∈ R} ∪ \{f_y = h(y)\}_{y ∈ X ⊆ Y}. \end{equation}$$

When these equality holds, we obtain a homomorphism $f : Y → M$ with $f|_X = h$. Any finite subset of the above equations are of the form $𝐦 ⋅ 𝐑 = 𝐦 _0$, which has a solution in the pushout $M ⊔ _A B$. Since $M → M ⊔ _A B$ pure embedding, any such finite set of equation has a solution in $M$. By assumption of algebraically compactness, the global solution exists.
(5 → 4). A linear equation of the form $𝐱_{I_k} ⋅ 𝐑 ^{n × m} = 𝐱_0$ determines a composition of morphism

$$\begin{equation} ∐ _{I_k} M \xrightarrow{⋅ 𝐑 \ \text{termwise}} ∐ _{I_k}∐ _{J_k} M↪∐ _{I}∐ _{J_k} M \xrightarrow {∐ _{J_k}Σ} ∐ _{J_k}M. \end{equation}$$

We assume $J_∙$ to be the juxtaposition of constant terms, i.e., for $i ≠ j$, we take $J_i ∩ J_j = ∅$. By lifting property of $Σ$, we obtain

Since $\coprod_{J_0}\Sigma$ is surjective, $\coprod_{J_i}\widetilde {\Sigma}$ is surjective. The colimit functor of $\varinjlim _{J_i}$ preserves surjection, thus $\coprod_{J}\widetilde {\Sigma}$ is surjective. We denote $𝐲$ as the disjoint union of the constant part of the equation, taking $I$-projection of any $(\coprod_{J}\widetilde {\Sigma})^{-1}(𝐲)$ yields the solution.

Let $F$ be such a functor. Let $\widetilde {Σ} : ∏ _I X → X$ denote any lifting map of $Σ : ∐ _I X→ X$. Now $∐ _I (FX) \xrightarrow Σ FX$ has a lifting map

$$\begin{equation} ∏_I FX ≃ F∏_I X \xrightarrow {F\widetilde {Σ}} FX. \end{equation}$$

(1 → 2, 3). Clear.
(3 → 1). Assume $M ↪ I$, is pure embedding, then any monomorphism $M ↪ X$ passes through $X → I$. Recall that

• if $gf$ is pure embedding, then so is $f$.

Hence $M ↪ X$ is a pure embedding.
(2 → 3). Let $M ↪ I$ be any injective preenvelope. Since $\mathrm{Ext}^1(𝐦𝐨𝐝 _R, M) = 0$, for any $K ∈ 𝐦𝐨𝐝 _R$ the morphism $K → I/M$ lifts up to $K → I$. Hence $M → I → I/M$ is pure exact.