CT2

ZCC
July 25, 2025

Lecture 2
- Abstract
Remarks on E-T Lemma
Acyclic Complexes
- $({}^⟂C_{ac}, C_{ac})$
- $(C_{ac}, C_{ac}^⟂)$
Cotorsion Triple
A Note on Resolutions

Lecture 2

Abstract

We analysis a criterion on a cotorsion pair cogenerated by set, which complete the counter part of Eklof-Trlifaj’s theorem. We introduce DG-projective complexes over the construction of $(^{⟂ }C_{ac}, C_{ac}, C_{ac}^⟂ )$.

Remarks on E-T Lemma

Functorial Approximations

We show in the previous lecture that

(Eklof-Trlifaj lemma). For any object $S ∈ C(R)$, the cotorsion pair cogenerated by $S$ is complete. In paricular, any objects $X$ admits a ses

$$\begin{equation} θ _X : 0 → X → X_F → X_C → 0,\quad (X_F ∈ {}^⟂ S, \ X_C ∈ 𝐅𝐢𝐥 (S)). \end{equation}$$

The construction is functorial as the $|S|$-filtered ordinal $α$ is fixed.

Note that $C(R)$ has enough projectives. Any chain complex $X$ admits a functorial epimorphism $p_X$ from the free complex $𝐅𝐫𝐞𝐞(X)$:

$$\begin{equation} 1_X ∈ (X,X)_{𝐒𝐞𝐭𝐬} ≃ (𝐅𝐫𝐞𝐞(X), X)_{C(R)} ∋ p_X. \end{equation}$$

Show that for $X ∈ 𝐌𝐨𝐝 _R ↪ C(R)$, the above $p_X$ is $R^{(X)} ↠ X, \quad e_x ↦ x$, where $𝐅𝐫𝐞𝐞 (X) = R^{(X)}$. In general,

It is also convenient to give a functorial $τ_X : 0 → X^F → X^C → X → 0$ by

Hence we give a functorial approximations

$$\begin{equation} X ∈ \mathrm{coCone}(S^⟂ , 𝐅𝐢𝐥 (S)) = \mathrm{coCone}(S^⟂ , 𝐅𝐫𝐞𝐞 ∗ 𝐅𝐢𝐥 (S)). \end{equation}$$

In particular, $𝐅𝐢𝐥 (S) ⊆ (𝐅𝐫𝐞𝐞 ∗ 𝐅𝐢𝐥 (S)) ⊆ {}^⟂ (X^⟂)$.

Any $X ∈ {}^⟂ (S^⟂)$ is a summand of some $X' ∈ (𝐅𝐫𝐞𝐞 ∗ 𝐅𝐢𝐥 (S))$.

This follows from the split ses $τ _X$.

$$\begin{equation} (𝐏𝐫𝐨𝐣 ∗ 𝐅𝐢𝐥 (S)) = 𝐅𝐢𝐥 (𝐏𝐫𝐨𝐣 ∪ \{S\}). \end{equation}$$

We show $⊇$. When $M$ is filtrated by either $S$ or some free object $F$ in each step. We denote $M_{α} \overset {\boxed X} ↪ M_{α + 1}$ for the filtration step $(M_{α +1} / M_α) ≃ X$. We construct $M' ↪ M ↠ M''$ by inducting on the following cases:

Note that filtered colimits preserves exactness. Hence,

$$\begin{equation} 0 → \varinjlim{}^{fil} M_γ ' → \varinjlim{}^{fil} M_γ → \varinjlim{}^{fil} M_γ '' → 0 \end{equation}$$

If $X ∈ {}^⟂(S^⟂)$, then $τ _X$ splits. Hence,

$$\begin{aligned} \ &𝐒𝐦𝐝 (𝐏𝐫𝐨𝐣 ∗ 𝐅𝐢𝐥 (S)) = 𝐒𝐦𝐝 (𝐅𝐢𝐥 (𝐏𝐫𝐨𝐣 ∪ \{S\})) \\ = \ &𝐒𝐦𝐝 (𝐅𝐫𝐞𝐞 ∗ 𝐅𝐢𝐥 (S)) = 𝐒𝐦𝐝 (𝐅𝐢𝐥 (𝐅𝐫𝐞𝐞 ∪ \{S\})) \end{aligned}$$

Therefore, one obtains ${}^⟂ (S^⟂ )$ from $S$ via either of the precedures

Reconstruction Lemma

We ask when a cotorsion theory (over $C(R)$) is cogenerated by a set $\mathcal{S}$ (or equivalently, a single element $∐ \mathcal{S}$).

Once $(\mathcal{C}, \mathcal{F})$ is cogenerated by a subset $\mathcal{S}$, we take $κ _0 := |⋃ \mathcal{S}|$. Now $(\mathcal{C}, \mathcal{F})$ is cogenerated by $\{M ∈ \mathcal{C}∣ |M| ≤ κ _0\}$. Indeed,

The isomorphisms classs in $\{M ∈ \mathcal{C}∣ |M| ≤ κ _0\}$ form of a set. Given an infinite set $X$, the number of ordered $ℤ$-partition over $X$ is controlled by $|X|^{ω}$. For each component $L$, the number of $R$-modules structures over over it is controlled by $|L|^{|R| ⋅ |L|}$, as a module is characterised by a morphism of the type $R → \mathrm{End}(L)$.

($κ$-constructable). Say a class $\mathcal{X}$ is $κ$-constructable, provided $$\begin{equation} \mathcal{X} = 𝐅𝐢𝐥 (\mathcal{X}_{≤ κ}) := 𝐅𝐢𝐥 (\{M ∈ \mathcal{X} ∣ |M| ≤ κ\}). \end{equation}$$

(Reconstruction lemma). Let $\mathcal{X} ⊆ C(R)$ be a subclass such that

$\mathcal{X}$ is closed under taking summands and filtrations;
$\mathcal{X}$ consists of all $\{Σ ^k (\overline R)\}_{k ∈ ℕ }$;
$\mathcal{X}$ is $κ$-constructable for some cardinal $κ$.

Then $(\mathcal{X}, \mathcal{X}^⟂ )$ is a cotorsion pair cogenerated by a set.

$\mathcal{X}_{≤ κ}$ is essentially a small set. Note that $\mathcal{X} ⊆ 𝐅𝐢𝐥 (\mathcal{X}_{≤ κ})$. Taking $𝐅𝐢𝐥$ on both sides yields $𝐅𝐢𝐥 (\mathcal{X}) = 𝐅𝐢𝐥 (\mathcal{X}_{≤ κ})$. Notice that

$$\begin{equation} {}^⟂ ((\mathcal{X}_{≤ κ})^⟂ ) = 𝐒𝐦𝐝 (𝐏𝐫𝐨𝐣 ∗ 𝐅𝐢𝐥 (\mathcal{X}_{≤ κ})) = 𝐒𝐦𝐝 (𝐏𝐫𝐨𝐣 ∗ 𝐅𝐢𝐥 (\mathcal{X})) = \mathcal{X}. \end{equation}$$

By Galois connection, $\mathcal{X}^⟂ = (\mathcal{X}_{≤ κ})^⟂$.

Kaplansky Class

In practice, it is hard to determine whether a class is $κ$-constructable. The main obstackle is to determined whether the transfinite filtration terminates as

$$\begin{equation} M_{γ} = M_{γ + 1} = \cdots \quad ⊊ M. \end{equation}$$

We give a slightly stronger definition, which is called $κ$-Kaplansky class.

($κ$-Kaplansky class). A class $\mathcal{X}$ is called $κ$-Kaplansky, provided for any $x ∈ M ∈ \mathcal{X}$, there is $x ∈ M' ⊆ M$ such that $M' ∈ \mathcal{X}_{≤ κ}$ and $(M/M') ∈ \mathcal{X}$.

Clearly, $M$ in a $κ$-Kaplansky class admits a transfinite $\mathcal{X}_{≤ κ}$-filtration with length $|M|$. In short, a $κ$-Kaplansky class is a $κ$-constructable class where the induction starts anywhere.

Show that $𝐌𝐨𝐝_R$ is an $|R|$-Kaplansky class.

Acyclic Complexes

We denote $C_{ac}(R)$ or simply $C_{ac}$ as the class of acyclic complexes in $C(R)$. $C_{ac}$ is closed under symmetric opetations, e.g. taking extensions, summands, coproducts and products, transfinite filtrations and cofiltrations.

The symmetric behaviours are due to the cotorsion triple $({}^⟂ C_{ac}, C_{ac}, C_{ac}^⟂)$. Trivial examples of cotorsion triples includes $(𝐏𝐫𝐨𝐣 , \mathcal{A}, 𝐈𝐧𝐣)$, and (over Frobenius category) $(\mathcal{A}, 𝐏𝐫𝐨𝐣 , \mathcal{A})$.

For complexes, there are lots of good subcategories containing $𝐏𝐫𝐨𝐣 ∪ 𝐈𝐧𝐣$.

$({}^⟂C_{ac}, C_{ac})$

We show that $({}^⟂C_{ac}, C_{ac})$ is cogenerated by sets with a straightforward construction.

$\mathcal{S} := \{Σ ^k (\underline R)\}_{k ∈ ℤ }$ is cogenerates $({}^⟂C_{ac}, C_{ac})$.

Recall that $Z^d (-) ≃ (Σ ^d (\underline R), -)$ has right derivation $\mathrm R^∙ (Z^d) ≃ H^{k+d}$. Hence $\mathcal{S}^⟂ = C_{ac}$.

$(C_{ac}, C_{ac}^⟂)$

We show that $(C_{ac}, C_{ac}^⟂)$ is cogenerated by sets by showing that $C_{ac}$ is a $\max (ω , |R|)$-Kaplansky class.

For any $(x_n)_{n ∈ ℕ } ∈ E ∈ C_{ac}$, there is a subcomplex $x ∈ E' ∈ (C_{ac})_{≤ κ}$. Here $κ = \max (ω , |R|)$.

We construct first $ω$ filtration of $E$ as follows $$\begin{equation} 0 = E_0 ⊆ E_1 ⊆ E_2 ⊆ \cdots ⊆ E_ω ⊆ E, \end{equation}$$

such that $|E_{n+1} / E_{n}| ≤ κ$ and $E_ω$ is acylic. We take $$\begin{equation} E_1 = [\cdots → ⟨x^k, d(x^{k+1})⟩ → ⟨x^{k+1}, d(x^{k+2})⟩ → \cdots]. \end{equation}$$

We construct $E_{k+1}$ from $E_k$ s.t. $Z(E_{k}) ⊆ B(E_{k+1})$, thus $⋃_{k ∈ ℕ} E_{k}$ is exact. We construct $Q^{p-1} ≃ \mathrm{ker}(d_{E_k}^{p}) × _{Z^p(E)} E^{p-1}$ by a pullback diagram

Here $Q^{p-1}$ reads the preimage of $\mathrm{ker}(d_{E_k}^{p})$ along $d_E^{p-1}$. It suffices to find $M ⊆ Q^{p-1}$ such that $$\begin{equation} |M| ≤ κ , \quad (E_k)^{p-1} ⊆ Q^{p-1},\quad d_E^{p-1} (M) = \ker (d_{E_k}^p). \end{equation}$$ We take any subset $X^{p-1} ⊆ Q^{p-1}$ so that $X^{p-1} \overset ∼ ↠ \ker (d_{E^k}^p)$ is a bijection. Now $|⟨X^{p-1}⟩| ≤ κ ⋅ κ = κ$. Now $M = E_{k+1}^{p-1} := ((E_k)^{p-1} + ⟨X^{p-1}⟩)$ is our desired construction. In particular $Z(E_k) = B(E_{k+1})$.

$(^{⟂}(C_{ac}^⟂ ), C_{ac}^⟂)$ is a complete cotorsion pair cogenerated by a set.

Note that $C_{ac}$ is a $\max (κ , ω)$-Kaplansky class which contains all projective objects, and is closed under taking summands and transfinite filtrations.

One can similarly show that the DG-construction preserves completeness. Moreover, the construction preserves and reflects the hereditary cotorsion pairs (triples).

Cotorsion Triple

We are curious about the cotorsion triple $(^⟂ (\mathcal{S}^⟂), \mathcal{S}^⟂ , (\mathcal{S}^⟂ )^⟂)$, where $\mathcal{S}^⟂ = C_{ac}$. In fact, there are much more examples

$(𝐏𝐫𝐨𝐣 , \mathcal{A}, 𝐈𝐧𝐣)$, the most elementary one. It is complete and hereditary.
$(\mathcal{G}\mathcal{P}, \mathcal{W}, \mathcal{G}\mathcal{P})$, aka the Gorenstein cotorsion triple, where $\mathcal{W} = \{M ∣ p.d. < ∞\}$. The triple is complete and hereditary under some good conditions (e.g., $R$ is Iwanaga-Gorenstein).
$(\mathcal{D}\mathcal{P}, \mathcal{W}, \mathcal{D}\mathcal{I})$, aka the 丁 cotorsion triple, where $\mathcal{W} = \{M ∣ f.d. < ∞\}$. The triple is complete and hereditary over 丁-陈 ring.
…

Relative DG

By a note on projectives in the end, we know that $^⟂ (\mathcal{S}^⟂)$ is the collection of DG-projective complexes with another two equivalent definitions

The complex lifts surjective quasi-isomorphisms;
The $K$-projective complex with projective terms.

Give equivalent definition of $(\mathcal{S}^⟂ )^⟂$, (called the DG-injective complexes).

J. Gillespie introduce introduce a procedure which construct cotorsion pairs in $C(R)$ from a cotorsion pair in $𝐌𝐨𝐝 _R$.

Let $(\mathcal{C}, \mathcal{F})$ be a cotorsion pair over an Abelian category.

($\widetilde {\mathcal{C}}$). $\mathcal{C}$ complexes are those acyclic with boundaries in $\mathcal{C}$;
($\text{DG-}\widetilde {\mathcal{F}}$). DG-$\mathcal{C}$ complexes consist of $X ∈ C(\mathcal{C})$ such that $\mathcal{HOM}(X, F)$ is acyclic for all $\mathcal{F}$ complexes.

Notice that the cotorsion pair $(𝐏𝐫𝐨𝐣 , \mathcal{A})$ creates two cotorsion theories:

$(\widetilde {𝐏𝐫𝐨𝐣}, \text{DG-}\widetilde {\mathcal{A}})$, which is $(𝐏𝐫𝐨𝐣 (C(\mathcal{A})), \mathcal{A})$, and
$(\text{DG-}\widetilde {𝐏𝐫𝐨𝐣}, \widetilde {\mathcal{A}})$, which is $({}^⟂ C_{ac}, C_{ac})$.

Show that the $\mathcal{F}$-complexes are precisely right othogonal to any of the following classes:

all stalk complexes $\{Σ ^d (\underline C) ∣ C ∈ \mathcal{C}, d ∈ ℤ\}$,
all bounded complexes taking values in $\mathcal{C}$,
all bounded below complexes taking values in $\mathcal{C}$,
all DG-$\mathcal{C}$ complexes ($\text{DG-}\widetilde {C}$).

Moreover, show that $(\text{DG-}\widetilde {C}, \widetilde {\mathcal{F}})$ is a cotorsion pair.
Hint: the above generates the same cotorsion classes ($(\text{DG-}\widetilde {C})^⟂$). By 4., and $\underline {(-)} ⊣ Z^0 (-)$, we see $X ∈ \text{DG-}\widetilde {C}$ has cycles in $\mathcal{F}$. Deriving $Z^k (-)$, $X$ is acyclic. Clearly $(\text{DG-}\widetilde {C})^⟂ = \widetilde {\mathcal{F}}$. Conversely, ${}^⟂(\widetilde {\mathcal{F}})$ consists of complexes lifting epimorphisms with kernel in $\widetilde {\mathcal{F}}$. The analysis is similar to that in the note below.

C. H. Cotorsion Pair

(Hereditary). Say a cotorsion pair $(\mathcal{C},\mathcal{F})$ is hereditary, provided

$\mathcal{C}$ is a resolving subcategory, i.e., contains projectives, closed under extensions, and closed under kernel of epimorphisms, and
$\mathcal{F}$ is coresolving subcategory, i.e., contains injectives, closed under cokernels of monomorphisms, and closed under cokernel of monomorphisms.

Let $P$ be any property of cotorsion pair. Say the cotorsion triple $(\mathcal{A},\mathcal{B},\mathcal{C})$ is $P$, provided $(\mathcal{A}, \mathcal{B})$ is $P$ and $(\mathcal{B},\mathcal{C})$ is $P$.

Show that $(𝐏𝐫𝐨𝐣 , \mathcal{A}, 𝐈𝐧𝐣)$ is complete and hereditary.

Although the terminology hereditary originates from the vanishing of $\mathrm{Ext}^2$; it has nothing to do with the global dimension of $\mathcal{A}$.

There is a $1$-sided criterion for a complete cotorsion pair to be hereditary.

Let $(\mathcal{C}, \mathcal{F})$ be a complete cotorsion pair. Then $\mathcal{C}$ is resolving iff $\mathcal{F}$ is coresolving.

We show → only. Let $0 → F_1 → F_2 → X → 0$ be ses. We show that any ses $0 → X → A → C → 0$ splits. We complete the Noether square from the the compostion $A^C ↠ A ↠ C$. $C' ∈ \mathcal{C}$ by assumption. Then we denote $δ ∈ \mathrm{Ext}^1(C,C')$ and $q_∗ δ ∈ \mathrm{Ext}^1(C,X)$.

We notice that $(C', F_2) → (C', X) → \mathrm{Ext}^1(C', F_1) = 0$. Hence $q$ we can write $q : [X' \xrightarrow a F_2 \xrightarrow b X]$. Clearly $a_∗ δ ∈ \mathrm{Ext}^1 (C, F_2)= 0$. We obtain $q_∗ δ = 0$.

Recall that $\mathrm{Hom}_{\mathcal{A}/ω }(\mathcal{C}, \mathcal{F}) = 0$. For hereditary cases,

$$\begin{equation} \ker \mathrm{Hom}_{\mathcal{A}/ω }(\mathcal{C}, - ) = \mathcal{F}, \quad \ker \mathrm{Hom}_{\mathcal{A}/ω }(-, \mathcal{F}) = \mathcal{C}. \end{equation}$$

We show $⊆$ part of the second identity only. Suppose that any $X → F$ factors through $ω$. We consider the induced dashed ses

$Q ∈ \mathcal{F}$ as $(\mathcal{C},\mathcal{F})$ is hereditary. By assumption, $q$ factors through $W ∈ ω$. Since $W ∈ \mathcal{C}$, $q$ factors through $p$. Now $X ∈ 𝐒𝐦𝐝 (X^C) ⊆ \mathcal{C}$.

Under some good conditions, $(\mathcal{C} , \mathcal{F})$ is a torsion theory in $\mathcal{A} / ω$.

C. H. Cotorsion Triple

Suppose that $(\mathcal{A}, \mathcal{B}, \mathcal{C})$ is a complete cotorsion triple. Then,

it is hereditary iff $\mathcal{B}$ is closed thick subcategory of $\mathcal{A}$, i.e., closed under taking summands, two-out-three property of ses (extensions, kernel of epimorphisms, and cokernel of monomorphisms);
$\mathcal{A} ∩ \mathcal{B} = 𝐏𝐫𝐨𝐣$, and $\mathcal{B} ∩ \mathcal{C} = 𝐈𝐧𝐣$.

We mentioned that, there is no need to assume $\mathcal{A}$ to have enough $𝐏𝐫𝐨𝐣$ or $𝐈𝐧𝐣$.

$1$ is clear by one-sided definition of complete hereditary cotorsion pair.
For $2$., we show any $T ∈ \mathcal{A} ∩ \mathcal{B}$ is projective. Any ses $0 → M → N → T → 0$ admits a diagram

Since $\mathcal{B}$ is thick, $E ∈ \mathrm{coCone}(\mathcal{B},\mathcal{B}) = \mathcal{B}$. Since $\mathrm{Ext}^1(T,E) = 0$, pushing out $q$ yields the original split ses.

There are some connections between $\mathcal{A}$ and $\mathcal{C}$. We commence with the definition of left approximations, or preenvelopes.

(Left approximation = preenvelope). Let $\mathcal{E}$ be a class (closed under isomorphisms). For any $E ∈ \mathcal{E}$ and arbitrary $X$, a morphism $φ : X → E$ is called a left $\mathcal{E}$-approximation, or an $\mathcal{E}$-preenvelope, provided it the following equivalent conditions:

any $φ' : X → E'$ extends along $φ$,
the natural transformation $φ ^∗ : (E, -)_\mathcal{E} ↠ (X, (-)|_{\mathcal{E}})_\mathcal{A}$ is a projective precover.

Show that for complete cotorsion pair $(\mathcal{C}, \mathcal{F})$, any $X$ admits

a left $\mathcal{F}$-approximation ($X → X_F \ \color{grey}{⇢ X_C}$), and
a right $\mathcal{C}$-approximation (${\color{grey}{X^F ⇢}} \ X^C → X$).

Such approximations are special due to their (co)kernels. Moreover, show that

any left $\mathcal{F}$-approximation $φ : M → F$ has a summand $i _F : M ↪ M_F$,
any right $\mathcal{C}$-approximation $ψ : C → N$ has a summand $p_C : N^C ↠ N$.

Now we characterise $\mathcal{A}$ with the lifting property of left $\mathcal{C}$-approximations, and vise versa.

Let $(\mathcal{A}, \mathcal{B}, \mathcal{C})$ be a complete hereditary cotorsion triple. Then $X ∈ \mathcal{A}$ iff

for any monic left $\mathcal{C}$ approximation $φ : M ↪ C$, any morphism $f : X → \operatorname{cok}(φ)$ lifts along $C ↠ \operatorname{cok}(φ)$.

(→). Consider the commutative diagram of ses

Since $X ∈ \mathcal{A}$, the morphism $X → \operatorname{cok}(i) → M_B$ factors through some $P ∈ \mathcal{A} ∩ \mathcal{B}$. The lifting property yields $P → M_C$. A pullback diagram yields $X → C$.
(←). We show that for any such $X$, ses $0 → B → E → X → 0$ splits. Notice that $\mathcal{B} ∩ \mathcal{C} = 𝐈𝐧𝐣$, we have the commutative diagram of ses：

Notice that $i$ is induced by lifting property of $𝐈𝐧𝐣$, $j$ is induced by cokernel. The right square is both a pullback and pushout. By property of $X$, there is a dashed arrow $s$. Hence $0 → B → E → X → 0$ splits.

Throughout, we make no assumption that the category has enough projectives or injectives. However, for a complete hereditary cotorsion triple $(\mathcal{A},\mathcal{B},\mathcal{C})$, there are enough projectives for $\mathcal{A}$, and enough injectives for $\mathcal{C}$.

A Note on Resolutions

The objects in ${}^⟂C_{ac}$ has the name dg-projective, Dold projective, homotopically projective objects, etc. We summarise projective related definitions in this note.

The Definitions

(Projective complex). $C(𝐏𝐫𝐨𝐣(\mathcal{A}))$, the complex with projective terms.

($𝐏𝐫𝐨𝐣 (C(\mathcal{A}))$). The projective objects in $C(\mathcal{A})$ are precisely the split acyclic projective complexes.

$⊇$ is clear. For $⊆$, the split ses $0 → P → \mathrm{Cone}(1_P) → Σ P → 0$ shows $P ≃ 0$ in $K(\mathcal{A})$. An termwise verification yields $P$ is has projective terms.

Show the complex $\cdots \xrightarrow {× 2}ℤ / 4ℤ \xrightarrow {× 2}ℤ / 4ℤ → \cdots$ in $C(ℤ / 4ℤ)$ is acyclic with projective terms, but it is not a projective object.

Show that $\mathcal{HOM}(X,-) : K(\mathcal{A}) → K(𝐀𝐛)$ is well-defined. Hint: show that

$$\begin{equation} C(\mathcal{A}) \xrightarrow{\mathcal{HOM}(X,-)} C(𝐀𝐛) ↠ K(𝐀𝐛),\quad C_{\text{null homo}} ↦ 0. \end{equation}$$

This shows that $(K(\mathcal{A}), ⊗ ,\mathcal{HOM})$ is again closed monoidal.

($K$-projective). $K$-projective complexes are defined in $K(\mathcal{A})$ with the following equivalent conditions:

$\mathcal{HOM}(P, - ): K(\mathcal{A}) → K(𝐀𝐛 )$ preserves acyclic complexes;
$\mathcal{HOM}(P, - ): K(\mathcal{A}) → K(𝐀𝐛 )$ preserves quasi-isomorphisms;
$H^k (\mathcal{HOM}(P, - )) : K(\mathcal{A}) → 𝐀𝐛$ sends quasi-isomorphisms to isomorphisms for each/some $k ∈ ℤ$;
$H^k (\mathcal{HOM}(P, -)) : K(\mathcal{A}) → 𝐀𝐛$ annihilates acyclic complexes for each/some $k ∈ ℤ$.

We use either each/some in the last two statemtents, as $Σ^±$ preserves and reflects quasi-isomorphisms and acyclic complexes.

(1 → 2). Note that any quasi-isomorphsm $f$ fits into a distinguished triangle $⋅ \xrightarrow f ⋅ → \mathrm{Cone}(f)$. Applying $\mathcal{HOM}(P, - )$ yields the long exact sequence. Since $\mathcal{HOM}(P, \mathrm{Cone}(f))$ is acyclic, $\mathcal{HOM}(P, f)$ is a quasi-isomorphism.
(2 → 3). Taking $H^k$ kills the statement.
(3 → 4). Notice that $E$ is acyclic iff $0 → E$ is an quasi-isomorphism.
(4 → 1). For any acyclic complex $E$, $H^∙ \mathcal{HOM}(P, E) ≡ 0$. Hence $\mathcal{HOM}(P, E)$ is acyclic.

($π$-projective). A complex $P ∈ C(\mathcal{A})$ is called $π$-projective, provided it is $K$-projective in $K(\mathcal{A})$.

Note that $P$ is $π$-projective iff $\mathcal{HOM}(P, -) : C(\mathcal{A}) → C(𝐀𝐛)$ preserves either all quasi-isomorphisms or all acyclic complexes.

(Easy). Find a $π$-projective complex with no projective terms.

(DG-projective). A complex $P$ is called DG-projective, provided the following equivalent conditions hold:

$(P, -)_{C(\mathcal{A})} : C(\mathcal{A}) → 𝐀𝐛$ lifts epic quasi-isomorphisms;
$P$ is $π$-projective with projective terms;
$\mathrm{Ext}^1(P, E) = 0$ for acyclic complexes $E$.

(1 → 3). Every ses in $\mathrm{Ext}^1(P, E)$ splits.
(3 → 2). We show $P$ has projective terms. If $P^1$ is not projective, then there is a non-split ses $0 → E → T → P^1 → 0$. Hence we have

Here $T \times _{P^1}P^0$ is pullback, and the dashed arrow $P^{-1} → T \times _{P^1}P^0$ is induced by $d_P^{-1}$ and $0_{T,P^{-1}}$.
We also show $P$ is $π$-projective, i.e., $(P, C)_{K(\mathcal{A})} = 0$ for any acyclic complex $C$. We pullback the partial projective resolution $0 → Ω (C) → P(C) → C → 0$ along any $φ : P → C$. Since $Ω (C)$ is acyclic, the pullback ses splits. Thus $φ$ factors through $P(C)$, which is the zero object in $K(\mathcal{A})$.
(2 → 1). For any epic quasi-isomorphism $p : A ↠ B$ and $f : P → B$, we have the pullback ses $0 → \ker p → P × _BA → P → 0$ which

split termwise ($(P^i)$’s are projective), and
$P × _BA → P$ is a homotopy equivalence ($P$ is $K$-projective in $K(\mathcal{A})$).

By homotopy invariance of ses, it splits.

Resolutions ($K^-(𝐩𝐫𝐨𝐣)$)

We begin with resolution of chain complex bounded above.

Let $X ∈ C^-(\mathcal{A})$ be a complex bounded above. Then $X$ has a projective resolution.

By Eilenburg-Cartan resolution, we obtain a locally finite double complex with projective terms and exact columns. The total complex ($∐$) is again exact and quasi-isomorphic to $X$.

There is ususall no easy constructions for unbounded complexes. We enumerate some special cases.

If an unbounded chain complex has uniform projective/injective dimensions, then Eilenberg-Cartan resolution yields a projective/injective resolution, as the total complex is locally finite.
By Chase’s theorem, $R$ is right perfect and left coherent if and only if projective right modules are closed under $∏$. In this case, we take Eilenberg-Cartan resolution of an unbounded chain complex $P^{∙ , ∙ } ↠ X^∙$, then $\mathrm{Tot}^∏ (P) ↠ X$ is a projective resolution.
By a well-known characterisation of Noetherian rings, we take Eilenberg-Cartan coresolution of an unbounded chain complex $X^∙ ↪ I^{∙ , ∙ }$, then $X ↪ \mathrm{Tot}^{∐ }(I)$ is an injective resolution.

For acyclic complex $C$ and projective complex $P$, if either one is bounded above, then $(P,C)_{K(\mathcal{A})} = 0$.

Let $f : P → C$ be such chain map. Say $f$ is ok for $k$, when there is $s^{≥ k}$ s.t. $f^l = ds^{l}+s^{l+1}d$ for all $l ≥ k$. By assumption $f$ is ok for large $k$. We show $f$ is ok for $k-1$ when it is ok for $k$.

We construct $s^{k-1} : P^{k-1} → C^{k-2}$ as follows. The morphism $Δ := (f^{k-1} - s^k d_P^{k-1})$ belongs to $\ker d_X^{k-1}$, thus $Δ$ factors through $B^{k-1}(C)$. We take $s^{k-1} : P^{k-1} → C^{k-2}$ by lifting property of projectives.

We denote $K^-(𝐏𝐫𝐨𝐣)$ the subcategory of $K(\mathcal{A})$ consisting of projective complexes bounded above.

The minus $(⋅ )^-$ means $X ∈ K^- (𝐏𝐫𝐨𝐣)$ almost lies in the negative part.

For Abelian category $\mathcal{A}$ with enough projectives,

Quasi-isomorphisms in $K^-(𝐏𝐫𝐨𝐣)$ are precisely homotopy equivalences;
Every $X ∈ K^-(\mathcal{A})$ admits a $K^- (𝐏𝐫𝐨𝐣)$-replacement $ρ X$ with epic quasi-isomorphism $c_X : ρ X ↠ X$;
It is possible to let $ρ$ be a triangulated functor right adjoint to the inclusion $K^-(𝐏𝐫𝐨𝐣 ) ↪ K^-(\mathcal{A})$;
$K^-(𝐏𝐫𝐨𝐣)$ are exactly the DG-projective complexes bounded above.

(1). Let $f : P → Q$ be a quasi-isomorphism in $K^-(𝐏𝐫𝐨𝐣)$. Since $Q ↪ \mathrm{Cone}(f)$ is null-homotopic, $f$ is a homotopy equivalence.
(2). We take Cartan-Eilenburg resolution $c_X : ρX ↠ X$, which is assumed to be functorial by axiom of choice. In particular, $c_P : ρP ↠ P$ is a canonical isomorphim. The replacement means the natural isomorphism $(c_X)_∗ : (P, ρ X)_{K^-(𝐏𝐫𝐨𝐣 )} ≃ (P, X)_{K^-(\mathcal{A})}$. This is due to $(P, \mathrm{Cone}(c_X)) = 0$.
(3). We show that $ρ$ is an additive functor right adjoint to the inclusion. The inclusion is triangle functor. By adjoint functor theorem, $ρ$ is also triangulated.
(4). We show that $P ∈ K^- (𝐏𝐫𝐨𝐣)$ is $K$-injective with projective terms. Hence it is DG-projective.

The following exercises requires some techniques of diagram chasing.

We know that $P ∈ C^- (𝐏𝐫𝐨𝐣)$ lifts epic quesiisomorphisms.

For general $Q ∈ C (𝐏𝐫𝐨𝐣)$ and $Y ∈ C^- (\mathcal{A})$, show that any epic quasi-isomorphism $p : X ↠ Y$ is lifted by $P$.
Show the natural isomorphism: $$\begin{equation} \{f ∈ (X,Y) ∣ \text{epic quasi-iso.}\} \overset{1:1}↔ \{\widetilde {f} ∈ (X, (\operatorname{cok} (X)) ×_{\operatorname{cok} (Y)} Y) ∣ \text{epic}\}. \end{equation}$$

Resolutions via Brown’s Rep

We discuss the projective resolutions of general Abelian category with exact arbitrary coproducts (AB3, AB3*) and a projective generator $U$ (e.g. $R$ for $𝐌𝐨𝐝 _R$).

Show the following properties of projective generators.

$(U, -)$ preserves and reflect exactness (e.g. monomorphisms and epimorphisms),
Moreover, if $ι : L ↪ X$ is monic and $(U, ι)$ is an isomorphism, then $ι$ is an isomorphism. Hint: consider the pair of morphisms $\binom π 0 : X ⇉ X / L$, where $ι ∘ π = ι ∘ 0$.
Show that the subobjects of $X$ form a set. Hint: $(U, -) : 𝐒𝐮𝐛 (X) → 𝐒𝐮𝐛 ((U,X))$ is a strict inclusion. The latter is a set.

The latter two have nothing to do with projective.

The discussion are based on Brown’s representability.

Unwinding its existence, we denote $K_{\mathrm{h \ proj}}$ as the class of projective resolutions of $K(\mathcal{A})$. We wish there is still a functor $ρ : K(\mathcal{A}) → K_{\mathrm{h \ proj}}$ right adjoint to the inclusion $K_{\mathrm{h \ proj}} ↪ K(\mathcal{A})$. That is,

$$\begin{equation} (i(-), Y) : K_{\mathrm{h \ proj}} → 𝐀𝐛 \end{equation}$$

is represented by some $ρ Y$.

$\underline U$ is the perfect generator of the additive full subcategory $\mathrm{Loc}(\underline U)$.

$\mathrm{Loc}(\underline U)$ is the smallest triangulated category generated by $U$ under $Σ ^±$, $∐$ and $∗$.
$U$ is perfect in $\mathcal{T}$, provided $\mathrm{Loc}(\underline U) = \mathcal{T}$, and $(\underline U, ∐ _{i ∈ I} f_i)$ is surjection when every $(\underline U, f_i)$ is surjective.

(PG1). We show for any set of morphisms $f_i : A_i → B_i$ with $(\underline U,f_i)$ surjective, the morphism $(\underline U, ∐ f_i)$ is again surjective. Note that

$$\begin{aligned} (\underline U, ∐ f_i)_{K(\mathcal{A})} &≃ ( U, H^0(∐ (f_i)))_\mathcal{A} \overset {\text{Ab3*}}≃ (U, ∐ H^0(f_i))_\mathcal{A}\quad \text{is surj.}\\[6pt] & ↔ ∐ H^0(f_i) \quad \text{is surj.}\\[6pt] & ↔ \{H^0(f_i)\}_{i ∈ I} \quad \text{are surj.}\\[6pt] & ↔ \{(U, H^0(f_i))_\mathcal{A}\}_{i ∈ I} \quad \text{are surj.}\\[6pt] & ↔ ∐ (U, H^0(f_i))_\mathcal{A} ≃ ∐ (\underline U, f_i)_{K(\mathcal{A})} \quad \text{is surj.}. \end{aligned}$$

(PG2). $\mathrm{Loc}(\underline U) = \mathrm{Loc}(\underline U)$ is clear.

We set $K_{\mathrm{h \ proj}} := \mathrm{Loc}(\underline U)$.

$i : K_{\mathrm{h \ proj}} ↪ K(\mathcal{A})$ admits a right adjoint $ρ : K(\mathcal{A}) → K_{\mathrm{h \ proj}}$, which is triangulated.

For any $X ∈ K(\mathcal{A})$, the functor $(i(-), X)_{K(\mathcal{A})} : K_{\mathrm{h \ proj}}^{\mathrm{op}} → 𝐀𝐛$ is cohomological, and sends $∐$ to $∏$. By functorial construction of Brown’s representability, there is a natural isomorphism

$$\begin{equation} (-, ρ X)_{K_{\mathrm{h \ proj}}} ≃ (i(-), X)_{K_{\mathrm{h \ proj}}} \end{equation}$$

Semi-(DG-)free Complexes

We have shown the existence and some functorial properties of DG complexes. A DG-projective complex $P$ has the following equivalent definitions:

For any epic quasi-isomorphism $f$, then $(P, f)$ is a surjection.
$\mathrm{Ext}^1(P,E) = \mathrm{Ext}^1_{dw}(P,E) = (P, Σ E)_{K(\mathcal{A})} = 0$ for any acyclic $E$.
$\mathcal{HOM}(P, -)$ preserves acyclic complexes, and quasi-isomorphisms.
$\mathcal{HOM}(P, -)$ preserves exact sequences, and quasi-isomorphisms.
Any surjective quasi-isomorphism (aka resolution) $f : P → X$ is a homotopy equivalence, and $P$ has projective terms.
…

In early study of DG-algebra, the prefix DG- is replaced by semi-.

(Revised definition). Let $𝔉$ denote the collection of stalk complexes with free terms. Say $X$ is semi-free, provided $X ∈ 𝐅𝐢𝐥 (𝔉)$.

Semi-free complexes are DG-projective complexes with free terms.

Clearly $𝐅𝐢𝐥 (𝔉) ⊆ {}^⟂C_{ac}$. It remains to show semi-free modules has free terms. By termwise analysis, we show a transfinite filtration of free modules are free. For any free filtration

$$\begin{equation} 0 = F_0 ⊆ F_1 ⊆ \cdots ⊆ F_γ ⊆ \cdots F_α \end{equation}$$

We show the statement $P_γ : [F_{γ} = ∐ _{β < γ}\frac{F_{β + 1}}{F_β }]$ holds for all ordinals $γ ≤ α$.
(Initial). Clearly $F_1 = F_1 / 0$.
(Successor). Assume $P_γ$ holds, then split ses shows $F_{γ + 1} = F_{γ} ⊕ \frac{F_{γ +1}}{F_{γ}}$. Hence $P_{γ + 1}$ holds.
(Limit). Assume $P_γ$ holds for all $γ < β$ wherein $β$ is a limit ordinal. We see

$$\begin{equation} F_β = \varinjlim_{γ < β } \left[\frac{F_1}{0} \xrightarrow {\binom 10} \frac{F_1}{0} ∐ \frac{F_2}{F_1} \xrightarrow {\binom 10} \frac{F_1}{0} ∐ \frac{F_2}{F_1} ∐ \frac{F_2}{F_2} \xrightarrow {\binom 10} \cdots \right]. \end{equation}$$

Clearly, $∐ _{γ <β } \frac{F_{γ + 1}}{F_γ }$ is also the colimit of this system.

Show that the above filtration can be deformed into a countable filtration.

Corollary, DG-projective complexes are exactly the summand of semi-free complexes. It suffices to use countable cardinal to characterise DG-projective complexes in sense of homotopy colimit.

We show some examples of DG-projective complexes.

All projective complexes bounded above are DG-projective.
The tensor product of two DG-projective complexes is DG-projective.
Dold’s well-known example $\cdots → ℤ / 4ℤ \xrightarrow {× 2} ℤ / 4ℤ→ \cdots$ is NOT DG-projective.

DG-Flat

(DG-Flat). Say $F$ is a DG-flat complex, provided the following equivalent consitions.

$F ⊗ -$ preserves monic quasi-isomorphisms;
$F ⊗ -$ preserves exact sequences, and quasi-isomorphisms.

Show the following properties of DG-flat complexes.

Any complex in $C^- (𝐅𝐥𝐚𝐭 )$ is DG-flat.
The tensor product of two DG-flat complexes is again DG-flat.
$F$ is DG-flat iff $F^+$ is DG-injective.

For the subcategory of DG-projective (resp. DG-injective) complexes, quasi-isomorphisms are precisely homotopy equivalence. This no longer holds for DG-flat complexes.

Let $φ : F → F'$ be a quasi-isomorphisms between DG-flat complexes. Then

$φ ^+$ is a homotopy equivalence, i.e., $s : φ^+ ∼ 0$.
Note that $(X ⊗ φ)^+ = (X, φ ^+)$ is a homotopy equivalence. Both $(X ⊗ φ )$ and $(φ ⊗ Y)$ are quasi-isomorphisms.
$φ$ is not necessary a homotopy equivalence, since $s$ does not take the form $t^+$ in general.

For example, we take bounded DG-flat complexes

$X = [\cdots → 0 → ℤ ↪ ℚ → 0 → \cdots ]$, and
$Y = [\cdots → 0 → ℚ / ℤ → 0 → \cdots ]$.

The quasi-isomorphism $X ↠ Y$ is not a homotopy equivalence.

Derived Category and Its Closed Monoidal Structure

This is an easy approach to derived category, as X.W. Chen mentioned that

$$\begin{equation} \boxed{\text{resolutions make life easier, but not really easier}} \end{equation}$$

We explain derived functors $\mathrm R^∙ \mathcal{HOM}(-, ?)$ and $\mathrm L_∙ (- ⊗ ?)$ in DG-complexes. We write $𝐑 \mathrm{Hom}$ and $⊗^𝐋$ for simplicity.

The functors are defined for general Abelian categories with enough projectives and injectives. We talk about $C(R)$ ($R$ commutative) only.

The right derived Hom-functor is defined as the endofunctor $$\begin{equation} 𝐑 \mathrm{Hom}(M, -) := \mathcal{HOM}(ρM, -) : D(R) → D(R). \end{equation}$$

where $ρM ↠ M$ is chosen as a functorial resolution of $M$.

The above $𝐑 \mathrm{Hom}(M, -)$ is a well-defined triangle functor.

The assignment $C(R) → C(R) → D(R)$, $X ↦ \mathcal{HOM}(ρM, X)$ is uniquely defined.
To see this passing through the localisation $C(R) → D(R)$, 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 $\mathrm{Cone}$.

We also define $𝐑 \mathrm{Hom}$ from injective coresolution. $𝐑 \mathrm{Hom}(X, ?)(Y) ≃ 𝐑 \mathrm{Hom}(-, Y)(X)$ comes from the functorial quasi-isomorphisms in $C(R)$:

$$\begin{equation} \mathcal {HOM}(\rho M, N) → \mathcal {HOM}(\rho M, \iota N) ← \mathcal{HOM}(M,\iota N). \end{equation}$$

Define and show the balance of derived tensor product

$$\begin{equation} (- ⊗ ^𝐋 Y)(X) = X ⊗ ρ Y \xrightarrow[∼ ]{⋆} ρ X ⊗ Y \xrightarrow[∼ ]{⋆^{-1}} (X ⊗ ^𝐋 ?)(Y) \end{equation}$$

One can use Yoneda lemma to see $(X ⊗ \mathrm{Cone}(f), -) ≃ (\mathrm{Cone}(X ⊗ f), -)$.

Explain flat-resolution fails in $⋆$, while the projective resolution works well.

One can also conclude some natural isomorphisms, inhererited from the closed monoidal structures of $C(R)$.

(Attempt). Once you attempt to show the monoidal structure of $D(R)$, you fails to show the identity $ρ (X ⊗ ^L Y) = ρ X ⊗ ρ Y$. This makes $(D(R), ⊗ ^L, \underline R)$ a lax monoidal category.

Thanks to MacLane, there is a way (possibly by a complicated conjugation) to reform the strict structure from the lax one.

We seek for a direct ways to obtain the strict monoidal structure of $D(R)$. Consider the quasi-isomorphisms:

$$\begin{equation} ρ (X ⊗ ^𝐋 Y) = ρ (X ⊗ ρ Y) ← ρ (ρ X ⊗ ρ Y) \overset {c} → ρX ⊗ ρY. \end{equation}$$

Note that $ρX ⊗ ρY$ is again DG-projective.

(The symmetric derived tensor). We redefine the assignment

$$\begin{equation} X ⊗^L Y := ρ X ⊗ ρ Y\quad (\xrightarrow[∼ ] {c} ρ (ρ X ⊗ ρ Y)). \end{equation}$$

Set $α ^ρ = (c ⊗ 1)^{-1} ∘ α ∘ (1 ⊗ c)$. The rule of associativity writes

(The pentagon rule). Consider the identity in $C(R)$:

$$\begin{equation} (α _{X,Y,Z} ⊗ W) ∘ (α _{X, Y ⊗ Z, W}) ∘ (X ⊗ α _{Y, Z, W}) = (α _{X⊗ Y, Z,W}) ∘ (α _{X,Y,Z ⊗ W}). \end{equation}$$

We apply $(-)^ρ$ on both sides, and show

$$\begin{equation} (α^ρ _{X,Y,Z} ⊗^𝐋 W) ∘ (α^ρ _{X, Y ⊗^𝐋 Z, W}) ∘ (X ⊗^𝐋 α^ρ _{Y, Z, W}) = (α^ρ _{X ⊗^𝐋 Y, Z,W}) ∘ (α^ρ _{X,Y,Z ⊗^𝐋 W}). \end{equation}$$

(The unit). $ρ (\underline R)$ is the unit in $D(R)$ with the (left) identity rule

$$\begin{equation} l_X^ρ : \underline R ⊗^𝐋 X = ρ \underline R ⊗ ρ X \xrightarrow [∼ ]{c ⊗ 1} \underline R ⊗ ρ X\xrightarrow {λ} ρX \xrightarrow [∼ ]{c} X. \end{equation}$$

Now feel free to verify $(D(R), ⊗ ^𝐋 , \underline R)$ is closed monoidal. For instance, we verify the pentagon diagram of left unit,

with the commutative diagram

Here all parallel line with the same colour yields gives the commutative square by functoriality. $\blacktriangle$ is by definition, $★$ is given by the monoidal structure of $C(R)$.

(Lax monoidal functor). Recall that we define the functor of resolution $ρ : D(R) → K(R)$ with axiom of choice of the class. This extends to a lax monoidal functor

$$\begin{equation} ρ : (D(R), ⊗ ^𝐋 , \underline R) → (K(R), ⊗ , \underline R) \end{equation}$$

consisting of

a functorial assignment $X ↦ ρ X$,
a natural transformation $ε : ρX → X$,
a natural transformation $μ : ρ (X ⊗^𝐋 Y) → ρ X ⊗ ρ Y$,

which preserves the associativity and unit rules.

The natural transformation $ε$ and $μ$ are isomorphisms in $D(R)$.
The word lax means not strict. It has nothing to do with Peter Lax.

We then proceed to the symmetric structure.

Note the isomorphism

$$\begin{equation} τ ^ρ : X ⊗ ^𝐋 Y = ρ (ρ X ⊗ ρ Y) \xrightarrow {ρ τ _{ρ X, ρ Y}} ρ (ρ Y ⊗ ρ X) = Y ⊗ ^𝐋 X. \end{equation}$$

One can verify the hexagon rule and the unit rule are satisfied.

$𝐑 \mathrm{Hom}$ is precisely the internal Hom of the symmetric monoidal category $(D(R), ⊗ ^𝐋 , \underline R)$.

Recall the definiton $𝐑 \mathrm{Hom} (-, ?) := \mathcal{HOM}(ρ (-), ?)$. A direct computation shows

$$\begin{aligned} & 𝐑 \mathrm{Hom}(X ⊗^𝐋 Y, Z) = \mathcal{HOM}(ρ X ⊗ ρ Y, Z) \\[6pt] ≃ \ & \mathcal{HOM}(ρ X ,\mathcal{HOM}(ρ Y, Z)) = 𝐑 \mathrm{Hom}(X, 𝐑 \mathrm{Hom}(Y,Z)). \end{aligned}$$

Moreover, there is also finiteness conditions as we discussed in Lec0.