Differential Graded

We show the left adjoint $F ⊣ G$, the right adjoint $G ⊣ H$ is dual.

The $1:1$ correspondence is clear, as a morphism of DG modules $F(X) → (M,d)$ takes the form $f^∙ (1,d)$. Moreover, $f$ preserves graded action iff $f∘ (1,d)$ does. This proves the adjunction.
The bi-adjoint $G$ preserves all small limits and colimits. To see that $G$ creates small limits and colimits, we show $\varinjlim$ and $\varprojlim$ in $𝐌𝐨𝐝 _{(A,d)}$ admits a termwise computation. Let $\{M_λ\} ⊆ 𝐌𝐨𝐝 _{(A,d)}$ be an inverse system, we show $\varprojlim M_λ$ is the limit. The derivation in $\varprojlim M_λ$ is induced by universal property of limits. For Any cone $N → M_λ$, there is a unique morphism of graded modules $ℓ : N → \varprojlim M_λ$. Since for each $λ$,

$$\begin{equation} d_{M_λ} ∘ p_λ ^n ∘ ℓ ^n = (p_λ ^{n+1} ∘ ℓ ^{n+1}) \circ d_{N}. \end{equation}$$

By universal property of limit, $d_{\varprojlim M_λ } ∘ ℓ ^n = ℓ ^{n+1} ∘ d_N$.

By closed monoidal structure of $C(k)$, $M ⊗ N$ is well-defined. We claim the exact sequence

$$\begin{equation} M ⊗ A ⊗ N \overset φ → M ⊗_k N → M ⊗ _A N → 0;\quad φ (m ⊗ a ⊗ n) = ma ⊗ n - m ⊗ an. \end{equation}$$

Unwinding the second condition (differential), we see from definition that

$$\begin{equation} (M ⊗ _k N, X)_{\text{balanced, graded}} = \ker [(M ⊗_k A ⊗_k N, X) \overset {φ }→ (M ⊗ _k N, X)]. \end{equation}$$

It remains to show that $φ$ preserves $d$. Then $M ⊗ _k N ↠ M ⊗ N$ also preserves $d$, and $\widetilde f : M ⊗ _A N → X$ preserves $d$ by tracking back the preimage.

$$\begin{aligned} d(m ⊗ a ⊗ n) & = (dm) ⊗ a ⊗ n + (-1)^{|m|} m ⊗ (da) ⊗ n + (-1)^{|m| + |a|} m ⊗ a ⊗ (dn) \\ & \xrightarrow φ (d m)a ⊗ n - (dm) ⊗ an + (-1)^{|m|} m da ⊗ n - (-1)^{|m|} m ⊗ (da)n\\ & \qquad + (-1)^{|m| + |a|} ma ⊗ dn - (-1)^{|m| + |a|} m ⊗ a(dn) \\ & = d(ma) ⊗ n + (-1)^{|m| + |a|} ma ⊗ dn - (dm) ⊗ an - (-1)^{|m|} m ⊗ d(an)\\ & = d (ma ⊗ n - m ⊗ an) = d (φ (m ⊗ a ⊗ n)). \end{aligned}$$

We define the right action as $(f ∗ a)(n) = f(an)$.

$$\begin{aligned} (d_{\mathcal{HOM}}(f ∗ a) )(n) &= d (f ∗ a) (n) - (-1)^{|f| + |a|} (f ∗ a)(dn) \\ &= d (f(an)) - (-1)^{|f| + |a|} f (a (dn)) \\ &= d (f(an)) - (-1)^{|f|} f(d(an) - (da)n) \\ &= (d_{\mathcal{HOM}} f)(an) + (-1)^{|f|} f((da)n)\\ &= (d_{\mathcal{HOM}}(f) ∗ a + (-1)^{|f|} f ∗ (da))(n). \end{aligned}$$

Hence, $d_{\mathcal{HOM}}(f ∗ a) = d_{\mathcal{HOM}}(f) ∗ a + (-1)^{|f|} f ∗ (da)$.

Clearly $E ∈ C(k)$, and $\deg 1_E = 0$. The multiplication if given by composition of morphisms. Hence $E^p E^q ⊆ E^{p+q}$. For homogeneous $f$,

$$\begin{aligned} d_{\mathcal{HOM}}(gf) & = gfd - (-1)^{|g|+|f|}dfg \\ & = gfd - (-1)^{|f|}gdf + (-1)^{|f|} gdf - (-1)^{|g| + |f|}dgf\\ & = d_{\mathcal{HOM}}(g)f + (-1)^{|g|} g d_{\mathcal{HOM}}(f). \end{aligned}$$

By definition, $θ : E ⊗ M → M, \quad f ⊗ m ↦ f(m)$ maps $E^p ⊗ M^q$ to $M^{p+q}$. To show such it is compatible with $d$, we take homogeneous $f ∈ E^p$ and $m ∈ M^q$,

$$\begin{aligned} d_{E ⊗ M} (f ⊗ m) & = d_E(f) ⊗ m + (-1)^{|f|} f ⊗ d_M(m) \\ & \xrightarrow θ d_{\mathcal{HOM}}(f)m + (-1)^{|f|}fd_M(m) \\ & = d_M (fm) - (-1)^{|f|} f (d_M m) + (-1)^{|f|}fd_M(m) \\ & = d_M(fm)\quad = f_M ( θ (f ⊗ m) ). \end{aligned}$$

Inflations (deflations) are precisely dws monomorphisms (epimorphisms).

1. (EX0). Trivial ses are dws.
2. (EX1, EX1’). Inflations and deflations are closed under compositions.
3. (EX2, EX2’). Split monomorphisms (epimorphisms) are closed under pushout (pullback).
   • Recall that the forgetful functor $G: 𝐌𝐨𝐝 _{(A,d)} → 𝐆𝐫𝐌𝐨𝐝 _A$ creates all limits and colimits, pullbacks and pushouts in $𝐌𝐨𝐝 _{(A,d)}$ are computated degreewise.

We show enough projective and injectives. Recall the adjoint $∞$-ple $\cdots ⊣ F ⊣ G ⊣ H ⊣ \cdots$, as a direct computation shows $F ≃ Σ H$. We claim that $\operatorname{im}F = \operatorname{im}H$ are precisely projectives and injectives.

• $FX$ lifts dws epimorphism $p$, since $X$ lifts split epimorphism $Gp$. By the same analysis, $HX$ lifts dws monomorphisms.
• If $P$ is projective, then $FGP ↠ P$ splits. It suffices to show that $\operatorname{im} F$ is closed under summands. For $N ⊕ N' = FX$, we see $F\mathrm{coim}(N) ⊕ F\mathrm{coim}(N') = F\mathrm{coim}(FX) ≃ FX$. By functoriality of $? ↦ F\mathrm{coim}(?)$, $N ≃ F\mathrm{coim}(N)$.
• There are enough projectives, which coincides injectives.

Let $Q_1$ and $Q_2$ be quotient functors. We show there are induced $α$ and $β$ s.t. $α ∘ Q_1 = Q_2$ and $β ∘ Q_2 = Q_1$. Therefore $α ∘ β$ and $β ∘ α$ are identical.

Consider the dws ses $0 → X \xrightarrow {\binom 10} HGX \xrightarrow {(0 \ 1)} ΣX → 0$, where $HGX$ is injective.

Note that $\mathrm{Cone}(\binom 01)$ is a chain complex

Now we show the following chain maps are homotopy inverses:

$$\begin{equation} \left(\substack{-f\\1\\0}\right) :\quad \mathrm{Cone}\left(\substack {1\\0}\right) ⇆ Σ X \quad : (0 \ 1 \ 0). \end{equation}$$

It suffices to show $1 - \left(\substack{-f\\1\\0}\right) ⋅ (0 \ 1 \ 0)$ is null-homotopic. Consider

$$\begin{equation} \begin{pmatrix}1&f&0\\0&0&0\\0&0&1\end{pmatrix} = \begin{pmatrix}-d&0&0\\0&-d&0\\1&f&d\end{pmatrix}\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}+\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}\begin{pmatrix}-d&0&0\\0&-d&0\\1&f&d\end{pmatrix}. \end{equation}$$

Note that the forgotful functor $𝔘 : 𝐆𝐫𝐌𝐨𝐝 _A → 𝐆𝐫𝐒𝐞𝐭𝐬$ admits a free functor sending

$$\begin{equation} 𝔉: 𝐆𝐫𝐒𝐞𝐭𝐬 → 𝐆𝐫𝐌𝐨𝐝 _A, \quad ⨆ \limits_{n ∈ ℤ } S^n ↦ ∐\limits _{n ∈ ℤ } Σ^n ∐\limits _{S^n}A. \end{equation}$$

On one hand, $𝔉𝔘 X$ is projective. Recall that $𝔘$ preserves epimorphisms. Since $(𝔘 X, -)_{𝐆𝐫𝐒𝐞𝐭𝐬}$ lifts epimorphisms, and so is $(𝔘 X, 𝔘 (-))_{𝐆𝐫𝐒𝐞𝐭𝐬} ≃ (𝔉 𝔘 X, -)_{𝐆𝐫𝐌𝐨𝐝 _A}$. If $X$ is projective, then $𝔉 𝔘 X ↠ X$ is is dws epic. Hence, projective modules are summands of $\operatorname{im} 𝔉$.

Note that $FUM$ is a projective object in $𝐌𝐨𝐝 _{(A,d)}$, we find dws ses

$$\begin{equation} θ _M := 0 → Σ^{-1} M \xrightarrow {\binom 01} FGM \xrightarrow {(1 \ 0)} M → 0. \end{equation}$$

The morphism $(Σ^{-1} M, -) → \mathrm{Ext}^1(M, -),\quad f ↦ f_∗ θ _M$ is surjective, as $FGM$ lifts epimorphisms. The kernel consists of morphisms passing through projective objects. Hence $\mathrm{Ext}(M, -) ≃ (Σ ^{-1} M, -)_K$. Therefore,

$$\begin{equation} \mathrm{Ext}^1_{dw}(M,-) = (Σ ^{-1} M, -)_K ≃ \mathrm{Ext}^1(M, -). \end{equation}$$

A calculation shows that

$$\begin{aligned} \mathrm{Ext}^1(M, -) & = \mathrm{Ext}_{dw} ^1 (M, -) = H^0\mathcal{HOM}(Σ^{-1} M, -) \\ & = H^0\mathcal{HOM}(∐ _{n ∈ ℤ } Σ ^{n-1} A^{(I_d)}, -) \\ & = ∏ _{n ∈ ℤ } Σ ^{1-n} (H^0\mathcal{HOM}(A, -))^{I_d} \\ & = ∏ _{n ∈ ℤ } (H^{1-n} (-))^{I_d} \\ \end{aligned}$$

We explain the second equality with $M = A$.

$$\begin{equation} \mathrm{Ext}^1(A,-) = H^0\mathcal{HOM}(Σ^{-1} A,-) ≃ Σ H^0(-) ≃ \mathcal{HOM}(Σ^{-1} A, H^0(-)). \end{equation}$$

Note that the $A$-module structure of $H^0(M)$ is given by $$\begin{equation} H^0(M) A^0 = \frac{Z^0(M) A^0}{B^0(M)A^0} ⊆ H^0(M);\quad H^0(M)A^{≠ 0} = 0. \end{equation}$$

Clearly $\mathcal{P} ⊆ \mathrm{Loc}(\{A\})$. We show $\mathcal{P} = \mathrm{Loc}(\mathcal{P})$. Note that the coproduct and (de)suspension of a projective filtration is also a well-defined projective filtration. By Cantor’s diagonal argument, $\mathrm{Loc}(\mathcal{P})$ is closed under countable filtrations.
We show that $\mathrm{Loc}(\mathcal{P})$ is closed under cones. For any $X = ⋃ P_∙$, $Y = ⋃ Q_∙$ and a morphism $f : X → Y$, $⋃ \mathrm{Cone}(f_∙)$ is a filtration of $\mathrm{Cone}(f)$, wherein $f_k : P_k ↪ X \xrightarrow f Y$. It remains to show that each $\mathrm{Cone}(f_k)$ is a filtration. WLOG, we assume $P_k = ∐_{i ∈ I} A$. The underlying morphism of $f_k : A^{(I)} → Y$ in $𝐆𝐫𝐌𝐨𝐝 _A$ identifies a map between graded set $\widetilde {f_k} : ∐ S → 𝔘Y$. Hence, $\mathrm{Cone}(f_k)$ has a projective filtration with underlying graded set

$$\begin{equation} A^{(I_0)} ⊕ Q_0 ⊆ A^{(I_1)} ⊕ Q_1 ⊆ \cdots ⊆ A^{(I_n)} ⊕ Q_n ⊆ \cdots;\quad I_k = \widetilde {f_k}^{-1}(𝔘 Q_k). \end{equation}$$

Clearly $\mathrm{Loc}(A) = \mathrm{Loc}(\mathcal{P})$. We show that for any collection of morphisms $f_i : X_i → Y_i$ such that $(A, f_i)_K$ is epic, $(A, ∐ f_i)_K$ is also epic. Note that

$$\begin{equation} (A, ∐ f_i)_K ≃ \mathcal{HOM}(A, H^0 (∐ f_i)) ≃ \mathcal{HOM}(A, ∐ H^0 (f_i)). \end{equation}$$

Taking underlying set of graded modules, we see $\mathcal{HOM}(A, -)$ preserves and reflects epimorphisms. Hence $(A, ∐ f_i)_K$ is epic iff $∐ H^0 (f_i)$ is epic, iff each $H^0(f_i)$ is epic, iff each $(A, f_i)_K$ is epic, thus iff $∐ (A, f_i)_K$ is epic. This shows that $A$ is a perfect generator of $\mathrm{Loc}(\mathcal{P})$.
By Brown’s representability, the functor

$$\begin{equation} (-, M)_K: \mathrm{Loc}(\mathcal{P}) → 𝐌𝐨𝐝 _k,\quad P ↦ (P, M)_K \end{equation}$$

is cohomological and sends coproducts to products. By Brown’s representability, the functor is representable by some $(-, ρ M)_{\mathrm{Loc}(\mathcal{P})}$. Let $c : ρM → M$ be the morphism $$\begin{equation} c ∈ (ρ M, M)_K ≃ (ρ M, ρ M)_{\mathrm{Loc}(\mathcal{P})} ∋ 1_{ρ M}. \end{equation}$$ This indicates that any $P → M$ factors through $c : ρ M → M$. By functoriality, $H^∙ (c)$ is an isomorphism.

(1 → 2). Suppose there exists a free filtration $M = ⋃ F_∙ = \varinjlim ^{fil} F_∙$. A compotation of Grothendieck’s spectal sequence shows ($∀ p ≥ 0$):

$$\begin{equation} 0 → \varprojlim{}^1 \mathrm{Ext}^{p} (F_∙ , -) → \mathrm{Ext}^{p+1} (\varinjlim F_∙, -) → \varprojlim\mathrm{Ext}^{p+1} (F_∙, -) → 0. \end{equation}$$

For $p = 0$, we show $(F_∙ , -)$ is Mittag-Leffler. It suffices to show in $𝐆𝐫𝐌𝐨𝐝 _A$, as the $G$ preserves image. WLOG we assume $G(F_∙) = 𝔉S_∙$, where $S_0 = ∅$ and $S_k ⊆ S_{k+1}$. By adjunction and $𝐏𝐫𝐨𝐣 (𝐆𝐫𝐒𝐞𝐭𝐬 ) = 𝐆𝐫𝐒𝐞𝐭𝐬$, $\{(𝔉S_k, -)\} ≃ \{(S_k, 𝔘 (-))\}$ is an epic tower. We claim $\mathrm{Ext}^{1} (F_∙, -)$ vanished on acyclic complexes, since

$$\begin{equation} \mathrm{Ext}^{1} (F_∙ , -) = \mathrm{Ext}^1_{dw}(F_∙ , -) ≃ (F_∙ , H^0(-))_K. \end{equation}$$

Therefore, $\mathrm{Ext}^{1} (\varinjlim F_∙, -)$ vanishes.
(2 ↔ 3). Note that $\mathrm{Ext}^1(A,B) = 0$ iff any ses $0 → B → ? → A → 0$ splits.
(2 → 4(1)). By $\mathrm{Ext}^1(M, -) = H^0\mathcal{HOM}(M, -)$.
(4(1) → 4(2)). Clear.
(4(2) → 4(3)). Note that $\mathcal{HOM}(M,-)$ is homological and preserves cones.
(4(3) → 4(4)). Clear.
(4(4) → 1). Brown’s representability shows the existence of quasi-isomorphism $ρ M → M$, where $ρ M$ has a projective filtration. We take $k = 0$ and see that $(M, ρ M)_K \overset {c_∗ }≃ (M, M)_K$. Hence $M$ is homotopy equivalent to a summand of $ρ M$. Since $M$ is a projective module, there exists some projective objects $P$ and $Q$ s.t. $M ⊕ P ≃ (\text{summand of } \ ρ M) ⊕ Q$. Clearly $M$ has a projective filtration.

It suffices to show that $\mathcal I ⊆ K(𝐌𝐨𝐝 _{(A,d)})^{\mathrm{op}}$ is closed under (de)suspension, cones, and products. The proof is the same as the projective case except the final step. To show that $\mathrm{Cone}(Y \overset φ → ∏ A^+)$ admits a tower, we consider

$$\begin{equation} (Y,∏ A^+)_{(A,d)} ≃ (∐ A, Y^+)_{(A,d)^{\mathrm{op}}} \xrightarrow {\text{G}} (∐ A, Y^+)_{𝐆𝐫𝐌𝐨𝐝 _A} ≃ (S, 𝔘 (Y^+))_{𝐆𝐫𝐒𝐞𝐭𝐬 _A}. \end{equation}$$

We take $\widetilde φ ∈ (S, 𝔘 (Y^+))_{𝐆𝐫𝐒𝐞𝐭𝐬 _A}$ as the image of $φ : Y → ∏ A^+$. The tower of $Y = \varprojlim J_∙$ induces a filtration $Y^+ = ⋃ J_∙ ^+$. We take $∏ \limits _{(\widetilde φ)^{-1}(𝔘(J_k))}A^+ ↪ ∏ A^+$ as the $k$-th entry in the tower of $∏ A^+$. This establishes a filtration of $\mathrm{Cone}(φ )$.

We show that $\mathrm{Loc}(A^+) = \mathrm{Loc}(\mathcal{I})$. Note that the dual category opposite the monomorphisms and epimorphisms. It remains to show that for any collection of monomorphisms $f_i : X_i ↪ Y_i$ such that $(f_i, A^+)_{K(A,d)} ≃ (A, f_i ^+)_{K((A,d)^{\mathrm{op}})}$ is epic, $(∏ f_i, A^+)_{K(A,d)} ≃ (A, (∏ f_i) ^+)_{K((A,d)^{\mathrm{op}})}$ is also epic. A calculation shows that

$$\begin{aligned} ↔\ \ & (A, (∏ f_i) ^+)_{K((A,d)^{\mathrm{op}})} &&\text{is epic} \\ ↔\ \ & H^0 ((∏ f_i) ^+) &&\text{is epic} \\ ↔\ \ & ((∏ H^0 (f_i)) ^+) &&\text{is epic} \\ ↔\ \ & H^0 (f_i) &&\text{is monic for all} \ i \\ ↔\ \ & (A, (f_i)^+)_{K((A,d)^{\mathrm{op}})} &&\text{is epic for all} \ i. \end{aligned}$$

Since $(-, M)_{K^{\mathrm{op}}}$ is cohomological and sends coproducts to products, by Brown’s representability, the functor is representable by some $(-, ι M)_{\mathrm{Loc}(\mathcal{I})}$. Let $d : M → ι M$ be the morphism, then any $M → J$ factors through $d : M → ι M$. By functoriality, $H^∙ (d)$ is an isomorphism.

We show (1 → 2) only. The rests are the same as the DG-projective case. We take $Y = \varprojlim ∏ A_∙ ^+$ as the limit of free-injective tower. A computation of Grothendieck’s spectral sequence shows

Clearly $\varprojlim \mathrm{Ext}^1(X, \prod A_\bullet ^+) = 0$. It suffices to show that $(X,\prod A_\bullet ^+)$ is Mittag-Leffler. Note that

$$\begin{equation} (X,∏ A_∙ ^+) ≃ ((∐ A)_∙ , X^+) \xrightarrow G (𝔉 S_∙ , (X^+))_{𝐆𝐫𝐌𝐨𝐝 _A} ≃ (S_∙ , 𝔘 (X^+))_{𝐆𝐫𝐒𝐞𝐭𝐬 _A}. \end{equation}$$

This is clearly an epic tower.

We show DG-projective case. Let $φ : P → Q$ be a quasi-isomorphism between DG-projective modules. Since $(Q, -)_K$ preserves quasi-isomorphisms, we find a right inverse of $φ$ in the homotopy category. The left inverse of $φ$ exists by applying $(-, P)_K$. Hence $φ$ is a homotopy equivalence. The injective case is similar. A non-example of flat case is

$$\begin{equation} [\cdots → 0 → ℤ → ℚ → 0 \cdots \cdots] ↠ [\cdots → 0 → 0 → ℚ / ℤ → 0 → \cdots ]. \end{equation}$$

We show FR’s and only, the verification of LFR’s are dual. Note that for any distinguished triangle $⋅ \xrightarrow f ⋅ → K$, $f$ is quasi-isomorphism iff $K$ is acyclic.
(FR1). $1_X ∈ S$ and $S ∘ S = S$.
(FR2). Given the quasi-isomorphism $s : X ⇒ Y$ and a morphism $Z → Y$, we construct distinguished triangle $W ⇒ Z → \mathrm{Cone}(s)$. By TR1, there is an induced $W → X$ making the square $\binom {W \ Z}{X \ Y}$ commutative.
(FR3). When $0 = f ∘ s$ for some $s ∈ S$, then $f$ factors through $\mathrm{Cone}(s)$ which is acyclic. Assume $f : X → Y$, then there is a dintinguished triangle $\mathrm{Cone}(s) \overset i → Y \overset t⇒ Z$. Since $f ∈ \operatorname{im}(i ∘ -)$, we see $t ∘ f = 0$.
(FR4). $Σ^± S = S$.
(FR5). For morphisms $(α , β , γ ) : \triangle → \triangle '$ where $α, β ∈ S$, we see $γ ∈ S$ by homological long exact sequence.
(Saturated). If $fg, gh ∈ S$, then $H^∙ (g)$ has left and right inverses. Hence $g ∈ S$.

The induced morphism $Ι$ is well-defined, since

$$\begin{equation} K (𝐌𝐨𝐝 _{(A,d)}^κ) ↪ K (𝐌𝐨𝐝 _{(A,d)}^{κ '}) → (S^{κ '})^{-1} K (𝐌𝐨𝐝 _{(A,d)}^{κ '}),\quad (S^{κ})^{-1} → \mathrm{Iso}. \end{equation}$$

We assume $|(-)| ≤ κ$ and $|(-)'| ≤ κ '$. To see $Ι$ is full, we see any $X ⇐ W' → Y$ takes the form $X ⇐ ρ X → Y$. To see $Ι$ is faithful, we assume $X ⇐ M → Y$ is zero in $(S^{κ '})^{-1} K (𝐌𝐨𝐝 _{(A,d)}^{κ '})$. Then there exists a zero composition $W' ⇒ M → Y$. The induced composition $ρ W' → ρ M → ρ Y$ is zero in $K(𝐌𝐨𝐝 _{(A,d)})$. Since $ρ W' → ρ M$ is a homotopy equivalence, we see $ρ M → ρ Y$ is null-homotopic. Hence $M → Y$ is zero.

We show the first isomorphism. Clearly $d_∗ : (M, N)_D ≃ (M, ι N)_D$. Note that for any quasi-isomorphism $s : M ⇒ ι N$, $(s, ι N)_K$ is an isomorphism. Hence $s$ is a homotopy equivalence. This shows that $(M, ι N)_K ≃ (M, ι N)_D$.

The assignment $C(A) → C(A) → D(A)$, $X ↦ \mathcal{HOM}(ρM, X)$ is uniquely defined.
To see this passing through the localisation $C(A) → D(A)$, any quasi-isomorphism $f$ yields $\mathcal{HOM}(ρM, f)$, which is again a quasi-isomorphism as $ρ M$ is DG-projective.
$\mathcal{HOM}(ρ M, -)$ is a triangle functor, since it preserves $\mathrm{Cone}$.