2.1 Basic examples

Given an R-module A, we will often deal with submodules of A of the form $\mathfrak {N} \cap A$ for some $\mathfrak {N} \prec (H_\theta ,\in )$ such that $A \in \mathfrak {N}$ , and R is both an element and subset of $\mathfrak {N}$ . This might appear to be a very special collection of submodules of A. But standard set-theoretic facts tell us that almost every submodule of A is of this form! More precisely: fix A, and consider any $H_\theta $ with $A \in H_\theta $ , and let $\kappa $ be any regular uncountable cardinal larger than $|R|$ . Then

$$ \begin{align*} \big\{ B \subset A \ : \ & B = \mathfrak{N} \cap A \text{ for some } \mathfrak{N} \prec (H_\theta,\in) \text{ such that } \\ & \{ A,R \} \cup R \subset \mathfrak{N} \text{ and } |\mathfrak{N}|<\kappa \big\} \end{align*} $$

is a closed and unbounded subset of $\wp _\kappa (A)$ in the sense of Jech [Reference Jech38]. This follows from the “Skolemization trick” of [Reference Foreman, Foreman and Kanamori28, p. 913] (or [Reference Cox11, Lemma 2.3]).

Before covering the applications that combine elementary submodel arguments with the notion of deconstructibility, we present a few lemmas that give the flavor of using elementary submodels in algebra. If F is free and X is a submodule of F, it is not necessarily true that X is freeFootnote ⁶ ; and even if both X and F are free, it does not follow that $F/X$ is free.Footnote ⁷ But “almost all” submodules of free modules behave nicely. The use of elementary submodels in the lemma below is quite an overkill—one could rephrase part (2) in terms of a filter directly on $\wp (F)$ —but thinking in terms of elementary submodels will especially help us later, when dealing with more complicated objects.

Proof The $\Leftarrow $ direction is trivial, since if (2) holds and we take ${\mathfrak {N}:=(H_\theta ,\in )}$ , then $\langle \mathfrak {N} \cap F \rangle = F$ and $\frac {F}{\langle \mathfrak {N} \cap F \rangle } = 0$ . Even if one only assumed (2) to hold for, say, countable $\mathfrak {N}$ , we’d still get freeness of F, since the presence of a single submodule X such that both X and $F/X$ are free implies that F is free (since freeness is closed under extensions).

To prove the $\Rightarrow $ direction, assume F is free. Then there is a free R-basis $B \subset F$ for F, and any such B is an element of $H_\theta $ . The statement “B is an R-basis of F” is easily downward absolute from $(V,\in )$ to $(H_\theta ,\in )$ (it is a $\Sigma _0$ statement in the parameters $R^{<\omega }$ , $B^{<\omega }$ , and F, all of which are elements of $H_\theta $ ). So $(H_\theta ,\in ) \models $ “there exists a free R-basis for F.” Because $R, F \in \mathfrak {N} \prec (H_\theta ,\in )$ , there is such a B with $B \in \mathfrak {N}$ . Trivially, since B is partitioned by $\mathfrak {N} \cap B$ and $B \setminus \mathfrak {N}$ ,

(3) $$ \begin{align} F = \langle \mathfrak{N} \cap B \rangle \oplus \langle B \setminus \mathfrak{N} \rangle, \end{align} $$

which implies that $\frac {F}{\langle \mathfrak {N} \cap B \rangle } \simeq \langle B \setminus \mathfrak {N} \rangle $ . Then

(4) $$ \begin{align} \langle \mathfrak{N} \cap B \rangle \text{ and } \frac{F}{\langle \mathfrak{N} \cap B \rangle} \simeq \langle B \setminus \mathfrak{N} \rangle \text{ are free}, \end{align} $$

because B is linearly independent (which is obviously inherited by subsets of B).

We claim that

(5) $$ \begin{align} \langle \mathfrak{N} \cap B \rangle = \langle \mathfrak{N} \cap F \rangle, \end{align} $$

which, together with (4), will complete the proof. For the nontrivial inclusion of (5), let $x \in \langle \mathfrak {N} \cap F \rangle $ . Then $x = \sum _{i=1}^k r_i x_i$ for some $x_i \in \mathfrak {N} \cap F$ and $r_i \in R$ , but since we are not assuming $R \subset \mathfrak {N}$ , some of the $r_i$ might fail to be in $\mathfrak {N}$ . Now

(6) $$ \begin{align} \text{sprt}_B(x) \subseteq \bigcup_{i=1}^k \text{sprt}_B(x_i) \end{align} $$

because x is in the span of $\{ x_1, \dots , x_k \}$ . And for each $i \in \{ 1, \dots , k \}$ , $\text {sprt}_B(x_i)$ is an element of $\mathfrak {N}$ , because both B and $x_i$ are elements of $\mathfrak {N}$ , and because $\text {sprt}_B(x_i)$ is definable over $(H_\theta ,\in )$ from the parameters B, R, and $x_i$ . So, the right side of (6), which we’ll denote by S, is a finite union of elements of $\mathfrak {N}$ , and hence S is both an element and subset of $\mathfrak {N}$ by Fact 2.1. So the B-support of x is contained in $\mathfrak {N} \cap B$ , showing that $x \in \langle \mathfrak {N} \cap B \rangle $ .

Lemma 2.2 is one of the few places in this article where we do not assume that R is a subset of $\mathfrak {N}$ ; typically we will make that assumption to simplify arguments.

Elementary submodels distribute across quotients, as long as the numerator and denominator are elements of the elementary submodel.

Lemma 2.3. Suppose A is an R-submodule of B and

$$\begin{align*}\{ A,B,R \} \cup R \subset \mathfrak{N} \prec (H_\theta,\in). \end{align*}$$

Then $\mathfrak {N} \cap \frac {B}{A} \simeq \frac {\mathfrak {N} \cap B}{\mathfrak {N} \cap A}$ .

Proof Define

$$\begin{align*}\varphi: \mathfrak{N} \cap B \to \mathfrak{N} \cap \frac{B}{A} \end{align*}$$

by $b \mapsto b + A$ ; note that the output really is an element of $\mathfrak {N} \cap \frac {B}{A}$ , because $\{ b, A,B \} \subset \mathfrak {N} \prec (H_\theta ,\in )$ , so $\mathfrak {N}$ sees the module $B/A$ and the coset $b + A$ . And $\varphi $ is surjective, because for any coset $z \in \mathfrak {N} \cap \frac {B}{A}$ , by elementarity of $\mathfrak {N}$ there is some $b \in \mathfrak {N} \cap B$ such that $z = b + A$ . And clearly $\text {Ker} \ \varphi = \mathfrak {N} \cap A$ . The lemma then follows by the first isomorphism theorem.

If R is $\kappa $ -Noetherian and $\mathfrak {N} \cap \kappa $ is transitive, one can remove the assumption that $R \subset \mathfrak {N}$ from Lemma 2.3, but one has to close each side of the isomorphism in the statement of the lemma under scalar multiplication (see Lemma 3.9 of [Reference Cox8]). We will not use that generalization here.

An R-module B is a pure submodule of C if whenever M is a finite matrix with entries from R, b is a column vector from B, and $Mx = b$ has a solution in C, then it has a solution in B. An embedding $e: B \to C$ is pure if $e[B]$ is pure in C.

Proof For expository reasons we will prove the lemma directly, though we could derive the stronger fact that $\mathfrak {N} \cap A \prec _{\mathcal {L}_R} A$ , where $\mathcal {L}_R$ is the language of R-modules.Footnote ⁸

The fact that R is an element and subset of $\mathfrak {N}$ , together with elementarity of $\mathfrak {N}$ , implies that $\mathfrak {N} \cap A$ is closed under addition and under scalar multiplication from R; so $\mathfrak {N} \cap A$ is a submodule of A. To see that it is a pure submodule, suppose M is an $m \times n$ matrix with entries from R, $b = [ b_1, \dots , b_m]^T$ is a column vector from $\mathfrak {N} \cap A$ , and that $Mx = b$ has a solution in A; so

$$\begin{align*}(V,\in) \models \ \underbrace{\exists a_1 \in A \dots \exists a_n \in A \ \ M [a_1 \dots a_n]^T = b}_{\varphi(A,M,R,b)}. \end{align*}$$

Since $\varphi $ can be expressed a $\Sigma _0$ formula in the language of set theory (in the parameters A, M, b, and R), and

$$\begin{align*}(H_\theta,\in) \prec_{\Sigma_1} (V,\in), \end{align*}$$

it follows that

$$\begin{align*}(H_\theta,\in) \models \varphi(A,M,R,b). \end{align*}$$

By elementarity of $\mathfrak {N}$ , and the fact that M is finite and $R \subset \mathfrak {N} \prec H_\theta $ , it follows from Fact 2.1 part (2a) that M is an element of $\mathfrak {N}$ . Also, b is a finite sequence of elements of $\mathfrak {N}$ , and hence, again by Fact 2.1 part (2a), b is an element of $\mathfrak {N}$ . So all free parameters in the formula $\varphi (A,M,R,b)$ are elements of $\mathfrak {N}$ . So by elementarity,

$$\begin{align*}\mathfrak{N} \models \varphi(A,M,R,b). \end{align*}$$

Let $a_1, \dots , a_n \in \mathfrak {N} \cap A$ witness the existential quantifiers of $\varphi $ . Then $\vec {a}$ is a finite sequence of elements of $\mathfrak {N} \cap A$ that solve $Mx=b$ .

Pure embeddings generalize to $\lambda $ -pure embeddings for any infinite regular $\lambda $ . An inclusion $B \to C$ is $\lambda $ -pure if for any system $\mathcal {S}$ of fewer than $\lambda $ many R-linear equations with parameters from B: if there is a solution for the entire system in C, then there is a solution in B too. This is equivalent to requiring that whenever M is a matrix of entries from R with $<\lambda $ -many rows and (wlog) $<\lambda $ -many columns (and each row finitely supported), b is a vector in B, and $Mx=b$ has a solution in C, then it has a solution in B.

2.2 Deconstructible classes

Given a class $\mathcal {C}$ of R-modules, a $\boldsymbol {\mathcal {C}}$ -filtration is a $\subseteq $ -increasing and $\subseteq $ -continuousFootnote ¹⁰ chain of R-modules

$$\begin{align*}\langle M_i \ : \ i < \zeta \rangle \end{align*}$$

such that:

1. (1) $\zeta $ is an ordinal;

2. (2) $M_0 =0$ Footnote ¹¹ ;

3. (3) for each i such that $i+1 < \zeta $ , $M_{i+1}/M_i$ is isomorphic to some member of $\mathcal {C}$ . Sometimes we will simply write “ $M_{i+1}/M_i \in \mathcal {C}$ ” when we really mean it is isomorphic to some member of $\mathcal {C}$ .

A module M is said to be $\mathcal {C}$ -filtered if it is the union of some $\mathcal {C}$ -filtration (note that if it is the union of a filtration of successor length, then M is simply the last module in the $\mathcal {C}$ -filtration). The class of $\mathcal {C}$ -filtered modules is denoted $\textit {Filt} {(\mathcal {C})}$ . Clearly $\mathcal {C} \subseteq \text {Filt}(\mathcal {C})$ because each $C \in \mathcal {C}$ can be viewed as the union of the $\mathcal {C}$ -filtration $\langle 0, C \rangle $ of length 2. We say that $\mathcal {C}$ is filtration-closed (or closed under transfinite extensions) if the reverse inclusion holds too; that is, if and only if $\mathcal {C} = \text {Filt}(\mathcal {C})$ . For any class $\mathcal {C}$ , $\text {Filt}(\mathcal {C})$ is closed under filtrations; that is,

(*) $$ \begin{align} \text{Filt}(\text{Filt}(\mathcal{C})) = \text{Filt}(\mathcal{C}). \end{align} $$

To see the nontrivial $\subseteq $ direction, suppose M is the union of some continuous chain $\langle M_i \ : \ i \le \zeta \rangle $ with each $M_{i+1}/M_i \in \text {Filt}(\mathcal {C})$ ; so $M_{i+1}/M_i$ is the union of some $\mathcal {C}$ -filtration $\vec {Z}^i$ . Then by the Correspondence Theorem, we can view $\vec {Z}^i$ as a smooth chain of the form

$$\begin{align*}\left\langle N^i_\xi/M_i \ : \ \xi < \zeta^i \right\rangle, \end{align*}$$

where for each $\xi $ (such that $\xi +1 < \zeta ^i$ ),

$$\begin{align*}\frac{N^i_{\xi+1}/M_i}{N^i_\xi/M_i} \in \mathcal{C}. \end{align*}$$

The quotient above is isomorphic to $N^i_{\xi +1}/N^i_\xi $ ; so $N^i_{\xi +1}/N^i_\xi \in \mathcal {C}$ . Concatenating all the $\vec {N}^i$ ’s together yields a $\mathcal {C}$ -filtration with the same union as the original filtration $\vec {M}$ .

A class $\mathcal {C}$ is called deconstructible if there is a subset $\mathcal {C}_0 \subseteq \mathcal {C}$ —where $\mathcal {C}_0$ is a set, not a proper class!—such that

(7) $$ \begin{align} \mathcal{C} = \text{Filt}(\mathcal{C}_0). \end{align} $$

This definition agrees with, for example, Šťovíček [Reference Šťovíček54], Saorín and Šťovíček [Reference Saorín and Šťovíček48], Šaroch and Trlifaj [Reference Šaroch and Trlifaj51], and most of the newer papers on the topic. In other sources, such as Göbel and Trlifaj [Reference Göbel and Trlifaj33] and the author’s own [Reference Cox8], deconstructiblity was defined by replacing the “ $=$ ” in (7) by “ $\subseteq $ .” We will refer to this notion as weakly deconstructible. For many natural classes, for example, the “left roots of Ext,” the notions are equivalent (by Eklof’s Lemma 3.6). Using (*), it is straightforward to prove the following lemma.

Deconstructibility of $\mathcal {C}$ is equivalent to requiring the existence of a cardinal $\kappa $ such that

$$\begin{align*}\mathcal{C} = \text{Filt}\left( \{ C \in \mathcal{C} \ : \ |C|<\kappa \} \right), \end{align*}$$

and it will be convenient to have a cardinal-specific notion of deconstructibility. Following the prevailing terminology, we will say that $\mathcal {C}$ is $\boldsymbol {\kappa }$ -deconstructible if

$$\begin{align*}\mathcal{C} = \text{Filt}\left( \underbrace{\{C \in \mathcal{C} \ : \ C \text{ is } <\kappa \text{-presented} \}}_{=:\mathcal{C}_\kappa} \right), \end{align*}$$

where “C is $<\kappa $ -presented” means that $C=F/K$ for some free F, where both F and K are generated (as R-modules) by sets of size less than $\kappa $ . Observe that $\mathcal {C}$ is deconstructible if and only if it is $\kappa $ -deconstructible for some $\kappa $ . Similarly, we will say that $\mathcal {C}$ is weakly $\boldsymbol {\kappa }$ -deconstructible if

$$\begin{align*}\mathcal{C} \subseteq \text{Filt}\left( \mathcal{C}_\kappa \right). \end{align*}$$

Clearly, if $\kappa _0 < \kappa _1$ , then $\kappa _0$ -(weak)-deconstructibility implies $\kappa _1$ -(weak)-deconstructibility.

Recall from the introduction that the precovering property of a class is important for relative homological algebra. A class $\mathcal {C}$ is a precovering class if for every module M, there exists some (not necessarily unique) homomorphism $\pi : C \to M$ with $C \in \mathcal {C}$ , such that every morphism from any member $C'$ of $\mathcal {C}$ into M factors through $\pi $ Footnote ¹² :

Such a map $\pi : C \to M$ is called a $\mathcal {C}$ -precover of M.

The key fact about deconstructibility is as follows.

The proof of Theorem 2.9 goes through Quillen’s Small Object Argument from homotopy theory. Theorem 2.9 strengthened the main result of Eklof and Trlifaj [Reference Eklof and Trlifaj19], who had isolated an important special case (for left classes of cotorsion pairs) that wasn’t as obviously connected to the Small Object Argument, but was crucial to the solution of the Flat Cover Conjecture.

There exist filtration-closed classes that are not even precovering. A series of papers (Bazzoni and Šťovíček [Reference Šaroch49], Estrada et al. [Reference Bazzoni and Šťovíček2], Šaroch [Reference Slávik and Trlifaj52], Slávik and Trlifaj [Reference Estrada, Guil Asensio, Prest and Trlifaj24]) culminated in showing that the class of $\aleph _1$ -projective modules—which is the same as the Flat Mittag-Leffler modules (by Herbera and Trlifaj [Reference Herbera and Trlifaj34])—is a precovering class if and only if R is a right perfect ring. In particular, since $\mathbb {Z}$ is not right perfect, the class of $\aleph _1$ -free abelian groups is closed under filtrations, but is not a precovering class.

To head off any possible confusion, we mention a much stronger (and much rarer) property than deconstructibility. A class $\mathcal {C}$ is called decomposable if there is a $\kappa $ such that every member of $\mathcal {C}$ is a direct sum of $<\kappa $ -presented members of $\mathcal {C}$ . Decomposable classes are (weakly) deconstructible, since if

$$\begin{align*}C = \bigoplus_{i < \lambda} C_i \end{align*}$$

with each $C_i$ a $<\kappa $ -presented member of $\mathcal {C}$ , we can view C as the union of the smooth chain

$$\begin{align*}\left\langle \underbrace{\bigoplus_{\xi < i} C_\xi}_{=:D_i} \ : \ i < \lambda \right\rangle, \end{align*}$$

where $D_{i+1}/D_i \simeq C_i$ . So $\mathcal {C}$ is (weakly) $\kappa $ -deconstructible.

Kaplansky’s Theorem [Reference Kaplansky41] states that over every ring, the class of projective modules is decomposable (with $\kappa = \aleph _1$ ). But deconstructibility is strictly weaker than decomposability; for example, the class of flat modules is always deconstructible (see Section 3.1), but for many rings the class of flat modules is not decomposable.

2.3 Top-down characterization of deconstructibility

In [Reference Cox8] we provided a new characterization of deconstructibility of a class of modules, restated in a simpler (but slightly weaker form) as Theorem 2.10 below. The reader is advised to just focus on clauses (I) and (II) for now; clauses (III) and (IV) are technical variants that are only occasionally used.

The (II) $\implies $ (I) direction of Theorem 2.10 provides a “top-down” way to verify that a class is deconstructible, allowing one to avoid having to recursively build filtrations every time. The (I) $\implies $ (II) direction of Theorem 2.10 is a variant of the Hill Lemma ([Reference Göbel and Trlifaj33, Chapter 7]).

Remark 2.11. Theorem 2.10 assumes closure of $\mathcal {C}$ under filtrations. But even without that assumption, one still has the implication

$$\begin{align*}(IV) \implies \mathcal{C} \text{ is } \textit{weakly} \ \kappa \text{-deconstructible}. \end{align*}$$

We want to focus mainly on applications of Theorem 2.10, so we relegate its proof to Appendix A, and only deal with equivalence of (I) with (II) (see [Reference Cox8] for the proof of further equivalence with clauses (III) and (IV), and for how to deal with situations where $\kappa \le |R|$ ).

In Theorem 2.10, it is possible to weaken the assumption that $|R|<\kappa $ to merely assume R is $\kappa $ -Noetherian (all ideals in R are $<\kappa $ -generated), if one also replaces instances of $\mathfrak {N} \cap C$ in the theorem with $\langle \mathfrak {N} \cap C \rangle $ . This is how it was done in Theorem 1.1 of [Reference Cox8], though we avoid doing so here because it considerably complicates the arguments. Though we will not use it here, the $\kappa $ -Noetherian property of a ring can even be characterized in terms of elementary submodels.

3.1 Proof of the Flat Cover Conjecture

The notion of deconstructibility arose out of the following resolution of the Flat Cover Conjecture around the year 2000. We give a new proof of the following theorem.

Long before the proof of Theorem 3.1, Enochs had reduced the problem to proving that the class $\mathcal {F}$ of flat modules is a precovering class. And since $\mathcal {F}$ is closed under extensions and direct limits, it is in particular filtration-closed. So by Theorem 2.9 it suffices to show that $\mathcal {F}$ is weakly deconstructible. We will use the following fact.

Let F be a flat module, and suppose $F \in \mathfrak {N} \prec (H_\theta ,\in , R)$ with $\mathfrak {N} \cap |R|^+$ transitive; then R is both an element and subset of $\mathfrak {N}$ , by Fact 2.1. By Lemma 2.4, $\mathfrak {N} \cap F$ is a pure submodule of F. Then by Fact 3.2, both $\mathfrak {N} \cap F$ and $\frac {F}{\mathfrak {N} \cap F}$ are flat. By the (II) $\implies $ (I) direction of Theorem 2.10, $\mathcal {F}$ is $|R|^+$ -deconstructible, completing the proof of Theorem 3.1.

The argument actually proves the following generalization.

Theorem 3.3 (Bravo, Gillespie, and Hovey [Reference Bravo, Gillespie and Hovey5]).

Suppose $\mathcal {C}$ is a class of modules closed under filtrations, and such that for any $C \in \mathcal {C}$ and any submodule $K \subset C$ :

$$\begin{align*}K \text{ is pure in } C \ \implies \ \left( K \in \mathcal{C} \text{ and } C/K \in \mathcal{C} \right). \end{align*}$$

Then $\mathcal {C}$ is deconstructible.

The assumptions on $\mathcal {C}$ in Theorem 3.3 closely resemble, but are strictly stronger than, clause (II) of Theorem 2.10. For example, there exist rings for which the projective modules are not closed under pure submodules, so in particular do not satisfy the assumptions of Theorem 3.3.Footnote ¹⁴ But the class of projective modules is always deconstructible, over any ring,Footnote ¹⁵ and hence always satisfy clause (II) of Theorem 2.10.

3.2 The Eklof–Trlifaj extensions of the Flat Cover Conjecture

For a class $\mathcal {B}$ of R-modules, ${}^\perp \mathcal {B}$ refers to the class of A such that $\text {Ext}^1_R(A,B)=0$ . We will use several standard facts about Ext from homological algebra (see, e.g., Chapter 1 of Eklof and Mekler [Reference Eklof and Mekler16]).

Fact 3.5. Suppose

is a short exact sequence of R-modules, and B is an R-module. Then there is an exact sequence

of abelian groups, where $e^*$ and $\pi ^*$ are the results of applying the contravariant functor $\text {Hom}(-,B)$ to e and $\pi $ , respectively.

We will exclusively focus on $\text {Ext}^1$ , so will omit the superscript from now on. Given a class $\mathcal {X}$ of modules,

$$\begin{align*}{}^\perp \mathcal{X} := \bigcap_{X \in \mathcal{X}} \text{Ker} \ \text{Ext}(-,X)= \{ A \ : \ \forall X \in \mathcal{X} \ \text{Ext}(A,X)=0 \} \end{align*}$$

and

$$\begin{align*}\mathcal{X}^\perp := \bigcap_{X \in \mathcal{X}} \text{Ker} \ \text{Ext}(X,-) = \{ B \ : \ \forall X \in \mathcal{X} \ \text{Ext}(X,B)=0 \}. \end{align*}$$

A pair of classes $(\mathcal {A},\mathcal {B})$ is a cotorsion pair if $\mathcal {A}^\perp = \mathcal {B}$ and $\mathcal {A} = {}^\perp \mathcal {B}$ .

Lemma 3.6 (Eklof’s Lemma, see [Reference Göbel and Trlifaj33], Lemma 6.2).

For any class $\mathcal {B}$ , ${}^\perp \mathcal {B}$ is filtration closed. Recall from Section 2.2 this means that

$$\begin{align*}{}^\perp \mathcal{B} = \text{Filt}\left( {}^\perp \mathcal{B} \right). \end{align*}$$

Eklof and Trlifaj generalized Theorem 3.1 as follows. A module C is pure-injective if whenever A is a pure submodule of B, then every morphism from $A \to C$ extends to a morphism from $B \to C$ (i.e., C is “injective with respect to all pure embeddings”). Their generalization statesFootnote ¹⁶ the following theorem.

Proof (of Theorem 3.7) By Eklof’s Lemma 3.6, ${}^\perp \mathcal {B}$ is closed under transfinite extensions, so it suffices to prove that it is weakly deconstructible. Let $\kappa $ be any regular uncountable cardinal larger than R, and consider any $\mathfrak {N} \prec (H_\theta ,\in ,R,\kappa )$ with $\mathfrak {N} \cap \kappa $ transitive. Then R is both an element and subset of $\mathfrak {N}$ by Fact 2.1. Consider any $A \in \mathfrak {N} \cap {}^\perp \mathcal {B}$ ; by Theorem 2.10, it suffices to prove that:

1. (i) $\mathfrak {N} \cap A \in {}^\perp \mathcal {B}$ ;

2. (ii) $\frac {A}{\mathfrak {N} \cap A} \in {}^\perp \mathcal {B}$ .

To prove part (i): since $A \in \mathfrak {N} \prec (H_\theta ,\in ,R)$ , there is a free module F and a submodule K such that $F,K \in \mathfrak {N}$ and $A \simeq F/K$ . Then by Lemma 2.3,

$$\begin{align*}\mathfrak{N} \cap A \simeq \mathfrak{N} \cap \frac{F}{K} \simeq \frac{\mathfrak{N} \cap F}{\mathfrak{N} \cap K}. \end{align*}$$

Since F is free and $F \in \mathfrak {N}$ , $\mathfrak {N} \cap F$ is free by Lemma 2.2. So, in order to prove that $\frac {\mathfrak {N} \cap F}{\mathfrak {N} \cap K}$ is in ${}^\perp \mathcal {B}$ , it suffices by Fact 3.4 to prove that for every $B \in \mathcal {B}$ : every morphism $\varphi : \mathfrak {N} \cap K \to B$ extends to domain $\mathfrak {N} \cap F$ . But by Lemma 2.4, $\mathfrak {N} \cap K$ is a pure submodule of K. Hence, since B is pure-injective, $\varphi $ extends to a homomorphism $\varphi _1: K \to B$ . Since $A=F/K \in {}^\perp \mathcal {B}$ by assumption, Fact 3.4 ensures that $\varphi _1$ extends to a homomorphism $\varphi _2: F \to B$ . So by restricting $\varphi _2$ to $\mathfrak {N} \cap F$ , we have the desired morphism from $\mathfrak {N} \cap F$ into B extending $\varphi $ .

To prove part (ii), fix any $B \in \mathcal {B}$ , and consider the short exact sequence

(8)

where i is inclusion, and the associated long exact sequence (cf. Fact 3.5) gotten by applying $\text {Hom}(-,B)$ to the short exact sequence (8):

(9)

The map $i^*: \text {Hom}(A,B) \to \text {Hom}(\mathfrak {N} \cap A ,B)$ is the restriction map $\varphi \mapsto \varphi \restriction (\mathfrak {N} \cap A)$ . By Lemma 2.4, $\mathfrak {N} \cap A$ is a pure submodule of A, and since B is pure-injective, it follows that the map $i^*$ is a surjection. Then, since $\text {Ext}(A,B)=0$ , exactness of (9) at the lower right entry implies that $\text {Ext} \left ( \frac {A}{\mathfrak {N} \cap A},B \right )$ is zero.

A close reading of the proof, together with Remark 2.6 used in place of Lemma 2.4, yields the next lemma, which will be used later on. Recall the definition of a $\lambda $ -pure embedding, just before Remark 2.6. A module B is $\boldsymbol {\lambda }$ -pure-injective if it is injective with respect to all $\lambda $ -pure embeddings; that is, whenever X is a $\lambda $ -pure submodule of Y, then every morphism from $X \to C$ extends to a morphism from $Y \to C$ .

Lemma 3.9. Suppose R is a ring, $\lambda $ is an infinite regular cardinal, and B is a $\lambda $ -pure-injective R-module. Then for any $\mathfrak {N} \prec (H_\theta ,\in )$ that is $<\lambda $ -closed,Footnote ¹⁷ the following implication holds for any $A \in \mathfrak {N}$ Footnote ¹⁸ :

$$\begin{align*}A \in \mathfrak{N} \cap {}^\perp B \ \implies \left( \mathfrak{N} \cap A \in {}^\perp B \text{ and } \frac{A}{\mathfrak{N} \cap A} \in {}^\perp B \right). \end{align*}$$

3.3 Kaplansky’s Theorem

In this section we show how (a minor variant of) Theorem 2.10 yields the following classic theorem of Kaplansky.

Proof First, notice that if $\langle P_i \ : \ i \le \zeta \rangle $ is a filtration with each $P_{i+1}/P_i$ projective, then each inclusion $P_i \to P_{i+1}$ splits; that is,

$$\begin{align*}P_{i+1} \simeq P_i \oplus \frac{P_{i+1}}{P_i}. \end{align*}$$

So $P_\zeta $ is isomorphic to the direct sum

$$\begin{align*}\bigoplus_{i < \zeta} P_{i+1}/P_i. \end{align*}$$

So it suffices to show that the class of projective modules is $\mathcal {P}_{\aleph _1}$ -filtered, where $\mathcal {P}_{\aleph _1}$ is the class of countably-generated (equivalently, countably-presented) projective modules.

By a minor variant of the proof of the (II) $\implies $ (I) direction of Theorem 2.10 (with $\kappa = \omega _1$ ), it suffices to show that whenever P is projective and

$$\begin{align*}P \in \mathfrak{N} \prec (H_\theta,\in,R) \end{align*}$$

(but where R is not necessarily a subset of $\mathfrak {N}$ ), then both $\langle \mathfrak {N} \cap P \rangle $ and $\frac {P}{\langle \mathfrak {N} \cap P\rangle }$ are projective. Lemma 3.3 of [Reference Cox8] gives a direct proof of this using the dual basis characterization of projectivity, but we can also prove it using Lemma 2.2. By elementarity of $\mathfrak {N}$ , there is some free $F \in \mathfrak {N}$ such that P is a direct summand of F; and again by elementarity we can take the complement to be in $\mathfrak {N}$ . So there is some module $X \in \mathfrak {N}$ such that $F = P \oplus X$ . Using elementarity of $\mathfrak {N}$ (and the fact that $\{ P, F, X \} \subset \mathfrak {N}$ ), it is routine to verify that

(10) $$ \begin{align} \langle \mathfrak{N} \cap F \rangle \simeq \langle \mathfrak{N} \cap P \rangle \oplus \langle \mathfrak{N} \cap X \rangle. \end{align} $$

Since $\langle \mathfrak {N} \cap F \rangle $ is free by Lemma 2.2, this shows that $\langle \mathfrak {N} \cap P \rangle $ is a direct summand of a free module, and is hence projective. And

$$\begin{align*}\frac{F}{\langle \mathfrak{N} \cap F \rangle} \simeq \frac{P \oplus X}{\langle \mathfrak{N} \cap P \rangle \oplus \langle \mathfrak{N} \cap X \rangle} \simeq \frac{P}{\langle \mathfrak{N} \cap P \rangle} \oplus \frac{X}{\langle \mathfrak{N} \cap X \rangle }. \end{align*}$$

By Lemma 2.2, $\frac {F}{\langle \mathfrak {N} \cap F \rangle }$ is free. Hence, $\frac {P}{\langle \mathfrak {N} \cap P \rangle }$ is a direct summand of a free module, and is hence projective.

3.4 Preservation of deconstructibility: Set-sized intersections

Recall the following theorem from the introduction.

We provide a few details that were missing in the sketch in the introduction.

Proof Closure of each $\mathcal {C}_i$ under transfinite extensions implies that ${\mathcal {C}:=\bigcap _{i \in I} \mathcal {C}_i}$ is closed under transfinite extensions. It remains to show that $\mathcal {C}$ is weakly deconstructible.

For each $i \in I$ let $\kappa _i$ witness deconstructibility of $\mathcal {C}_i$ , and fix a regular $\kappa $ with

$$\begin{align*}\kappa> |I| + |R| + \text{sup}_{i \in I} \kappa_i. \end{align*}$$

Then each $\mathcal {C}_i$ is also $\kappa $ -deconstructible (since $\kappa \ge \kappa _i$ ). By the (I) $\implies $ (II) direction of Theorem 2.10, for all sufficiently large regular $\theta> \kappa $ and each $i \in I$ , if

$$\begin{align*}\mathfrak{N} \prec \mathfrak{A}^\theta_i:=\left( H_\theta,\in, R, \mathcal{C}_i \cap H_\theta \right) \end{align*}$$

and $\mathfrak {N} \cap \kappa $ is transitive, then

$$\begin{align*}C \in \mathfrak{N} \cap \mathcal{C}_i \ \implies \ \left( \text{ both } \mathfrak{N} \cap C \text{ and } \frac{C}{\mathfrak{N} \cap C} \text{ are in } \mathcal{C}_i \right). \end{align*}$$

By the (IV) $\implies $ (I) direction of Theorem 2.10, to show that $\mathcal {C}$ is $\kappa $ -deconstructible, it suffices to prove that for any fixed $C \in \mathcal {C}$ , there is a structure in a countable signature (possibly depending on the module C) such that elementary submodels of it have the required reflection properties with respect to the module C. Fix such a C, and let $\theta $ be a regular cardinal with $C \in H_\theta $ (and sufficiently large that $\mathfrak {A}_i^\theta $ is defined for each $i \in I$ ). Set $\Omega := \left ( 2^{|H_\theta |} \right )^+$ and

$$\begin{align*}\mathfrak{B}:=\left( H_{\Omega}, \in, R, I, \kappa, \langle \mathfrak{A}_i^\theta \ : \ i \in I \rangle, C \right). \end{align*}$$

Consider any $\mathfrak {N} \prec \mathfrak {B}$ such that $\mathfrak {N} \cap \kappa $ is transitive. Then I is both an element and subset of $\mathfrak {N}$ , and $\mathfrak {A}_i^\theta \in \mathfrak {N}$ for all $i \in I$ . It follows that

$$\begin{align*}\forall i \in I \ \ \mathfrak{N} \cap H_\theta \prec \mathfrak{A}^\theta_i, \end{align*}$$

and hence, since $C \in \mathfrak {N} \cap H_\theta $ , we have

$$\begin{align*}\forall i \in I \ \mathfrak{N} \cap C \in \mathcal{C}_i \text{ and } \frac{C}{\mathfrak{N} \cap C} \in \mathcal{C}_i. \end{align*}$$

That is, both $\mathfrak {N} \cap C$ and $\frac {C}{\mathfrak {N} \cap C}$ are in $\mathcal {C}$ .

3.5 Preservation of deconstructibility: Direct summands

Theorem 3.12 (Šťovíček and Trlifaj [Reference Šťovíček and Trlifaj55]).

If $\mathcal {C}$ is a decontructible class of modules, then so is the class

$$\begin{align*}\mathcal{C}_\oplus:= \{ M \ : \ \exists C \in \mathcal{C} \ \ M \text{ is a direct summand of } C \}. \end{align*}$$

Proof Suppose $\mathcal {C}$ is $\kappa $ -deconstructible. Fix any $M \in \mathcal {C}_{\oplus }$ ; by clause (IV) of Theorem 2.10, it suffices to find a first-order structure (possibly depending on M) whose elementary submodels reflect in the desired way.

By definition of $\mathcal {C}_{\oplus }$ , there is some $C \in \mathcal {C}$ such that $M \oplus X = C$ for some X. Fix a regular $\theta $ such that $M,X,C \in H_\theta $ ,Footnote ¹⁹ and set

$$\begin{align*}\mathfrak{A}:= (H_\theta,\in,R,\kappa, X,C,M, \mathcal{C} \cap H_\theta). \end{align*}$$

We claim that $\mathfrak {A}$ satisfies clause (IV) of Theorem 2.10, for the class $\mathcal {C}_{\oplus }$ . Consider any $\mathfrak {N} \prec \mathfrak {A}$ such that $\mathfrak {N} \cap \kappa $ is transitive. By equivance of (I) with (II) for the class $\mathcal {C}$ , both $\mathfrak {N} \cap C$ and $\frac {C}{\mathfrak {N} \cap C}$ are in $\mathcal {C}$ . And, similarly as in our proof of Kaplansky’s Theorem (Theorem 3.10),

$$\begin{align*}\mathfrak{N} \cap C = (\mathfrak{N} \cap M) \oplus (\mathfrak{N}\cap X) \end{align*}$$

and

$$\begin{align*}\frac{C}{\mathfrak{N} \cap C} = \frac{M \oplus X}{\mathfrak{N} \cap (M \oplus X)} = \frac{M \oplus X}{(\mathfrak{N} \cap M) \oplus (\mathfrak{N} \cap X)} \simeq \frac{M}{\mathfrak{N} \cap M} \oplus \ \frac{X}{\mathfrak{N} \cap X}. \end{align*}$$

In particular, $\mathfrak {N} \cap M$ is a direct summand of $\mathfrak {N} \cap C$ , and $\frac {M}{\mathfrak {N} \cap M}$ is a direct summand of $\frac {C}{\mathfrak {N} \cap C}$ . Hence, both $\mathfrak {N} \cap M$ and $\frac {M}{\mathfrak {N} \cap M}$ are in $\mathcal {C}_\oplus $ .

We omit the proof that $\mathcal {C}_\oplus $ is closed under transfinite extensions; it follows from the fact that $\mathcal {C}$ is closed under transfinite extensions, and has nothing to do with deconstructibility or elementary submodels.

5.1 Resolutions by deconstructible classes

Homological algebra uses projective resolutions to approximate non-projective modules. More generally, given an R-module M and a class $\mathcal {F}$ of R-modules, an $\boldsymbol {\mathcal {F}}$ -resolution of $\boldsymbol {M}$ is a long exact sequence of the form

where each $F_i$ is either 0 or an element of $\mathcal {F}$ . A module M has $\boldsymbol {\mathcal {F}}$ -dimension at most $\boldsymbol {n}$ if it has an $\mathcal {F}$ -resolution such that $F_i = 0$ for all $i> n$ . M has finite $\mathcal {F}$ -dimension if it has $\mathcal {F}$ -dimension at most n, for some natural number n.

Kaplansky’s Theorem 3.10 implies that the class of modules with projective dimension 0—that is, the projective modules—is deconstructible. Theorem 3.1 (proof of the Flat Cover Conjecture) says the same is true for the class of flat modules. These facts were generalized by Aldrich et al. [Reference Aldrich, Enochs, Jenda and Oyonarte1] to the classes of modules of projective or flat dimension at most n, and this was further strengthened by Slávik and Trlifaj [Reference Slávik and Trlifaj52], restated as Theorem 5.1 below. It says that if $\langle \mathcal {C}_n \ : \ n < \omega \rangle $ is a sequence of $\kappa $ -deconstructible classes of modules, then the class of modules possessing some “ $\vec {\mathcal {C}}$ -resolution”—that is, those modules M for which there exists an exact sequence

with each $C_n \in \mathcal {C}_n$ —is $\kappa $ -deconstructible.Footnote ²² More precisely, we have the following theorem.

We use Theorem 2.10 to provide a new, short proof of Theorem 5.1 (but under the simplifying assumption $|R|<\kappa $ ; the full version of Theorem 5.1 can be proved using the version of Theorem 2.10 from Cox [Reference Cox8]).

Proof First we check weak $\kappa $ -deconstructibility. Suppose $D \in \mathcal {D}$ , as witnessed by some $\vec {\mathcal {C}}$ -resolution

This can be viewed as a (truncated) acyclic complex, which we will denote by $X_\bullet $ .

Fix a regular $\theta $ with all these parameters in $H_\theta $ , and consider any

$$\begin{align*}\mathfrak{N} \prec \mathfrak{A}:=(H_\theta,\in,R,\kappa, \langle S_n \ : \ n < \omega \rangle) \end{align*}$$

such that $X_\bullet \in \mathfrak {N}$ and $\mathfrak {N} \cap \kappa $ is transitive. By the (IV) $\implies $ (I) direction of Theorem 2.10, it suffices to prove that both $\mathfrak {N} \cap D$ and $\frac {D}{\mathfrak {N} \cap D}$ have $\vec {\mathcal {C}}$ -resolutions. By Lemma 4.2, since $X_\bullet $ is acyclic, then

is a short exact sequence of acyclic complexes, that is, the following has all exact rows and columns:

(12)

Since $C_n \in \mathcal {C}_n \cap \mathfrak {N}$ for each $n \in \omega $ and $\mathcal {C}_n$ was $\kappa $ -deconstructible, the (I) $\implies $ (II) direction of Theorem 2.10 tells us that

$$\begin{align*}\forall n \in \omega \ \ \mathfrak{N} \cap C_n \in \mathcal{C}_n \text{ and } \frac{C_n}{\mathfrak{N} \cap C_n} \in \mathcal{C}_n. \end{align*}$$

So the row of (12) corresponding to $X_\bullet \restriction \mathfrak {N}$ is a $\vec {\mathcal {C}}$ -resolution of $\mathfrak {N} \cap D$ , and the row corresponding to $X_\bullet / \mathfrak {N}$ is a $\vec {\mathcal {C}}$ -resolution of $\frac {D}{\mathfrak {N} \cap D}$ .

This completes the proof that $\mathcal {D}$ is weakly $\kappa $ -deconstructible. We omit the proof that $\mathcal {D}$ is filtration-closed; this is a purely algebraic proof, using the fact that each $\mathcal {C}_n$ is a precovering class (by Theorem 2.9).

5.2 Almost everywhere closure under quotients

In [Reference Cox8] we isolated the following property, after observing that classes of complexes associated with Gorenstein Homological Algebra typically possess it. It is a very weak version of closure of a class under quotients.

Definition 5.2 [Reference Cox8].

Fix a ring R, a class $\mathcal {C} \subseteq R$ -Mod, and a regular uncountable cardinal $\kappa $ . We say $\mathcal {C}$ is $\boldsymbol {\kappa }$ -a.e. closed under quotients if whenever

$$\begin{align*}\mathfrak{N} \prec \mathfrak{N}' \prec (H_\theta,\in,R,\mathcal{C} \cap H_\theta) \end{align*}$$

and both $\mathfrak {N} \cap \kappa $ and $\mathfrak {N}' \cap \kappa $ are transitive, then for every $C \in \mathfrak {N}$ the following implication holds:

$$ \begin{align*} \left( C \in \mathcal{C} \text{ and } \mathfrak{N} \cap C \in \mathcal{C} \text{ and } \mathfrak{N}' \cap C \in \mathcal{C} \right) \ \implies \ \frac{\mathfrak{N}' \cap C}{\mathfrak{N} \cap C} \in \mathcal{C}. \end{align*} $$

We also define this notion for classes $\mathcal {C} \subseteq \text {Ch}(R)$ of complexes, where for $C_\bullet \in \mathfrak {N}$ the implication above is replaced by

$$ \begin{align*} \left( C_\bullet \in \mathcal{C} \text{ and } C_\bullet \restriction \mathfrak{N} \in \mathcal{C} \text{ and } C_\bullet \restriction \mathfrak{N}' \in \mathcal{C} \right) \ \implies \ \frac{C_\bullet \restriction \mathfrak{N}'}{C_\bullet \restriction \mathfrak{N}} \in \mathcal{C}. \end{align*} $$

We say $\mathcal {C}$ is eventually a.e. closed under quotients if it is $\kappa $ -a.e. closed under quotients for all sufficiently large regular $\kappa $ .

If $\mathcal {C}$ is eventually a.e. closed under quotients, and you’re trying to prove that it is deconstructible, then the task of trying to verify clause (II) (or clause (III)) in the characterization of deconstructibility (Theorem 2.10) becomes easier, because one only has to verify the “ $\mathfrak {N} \cap C \in \mathcal {C}$ ” part of that clause; the “ $\frac {C}{\mathfrak {N} \cap C} \in \mathcal {C}$ ” part is then taken care of by eventual a.e. closure of $\mathcal {C}$ under quotients, by taking $\mathfrak {N}':= H_\theta $ .

But in fact, eventual a.e. closure under quotients makes the task even easier for us: it allows us to turn a stationary set of reflections to a closed unbounded one (assuming filtration-closure of the class), and then to extend further to larger elementary submodels.

Theorem 5.4 (essentially Theorem 6.3 of [Reference Cox8]).

Suppose $\mathcal {C}$ is a class of R-modules, $\kappa $ is a regular uncountable cardinal larger than $|R|$ , and:

1. (a) $\mathcal {C}$ is filtration-closed;

2. (b) $\mathcal {C}$ is $\kappa $ -a.e. closed under quotients;

3. (c) For every $C \in \mathcal {C}$ , the collection

   $$\begin{align*}\mathcal{C} \cap \wp_\kappa(C) \end{align*}$$

   is stationary in $\wp _\kappa (C)$ .

Then $\mathcal {C}$ is $\kappa $ -deconstructible.

Note: If $\mathcal {C}$ is a class of complexes instead of modules, the same theorem holds, with clause (c) replaced by the assumption: For every $C_\bullet \in \mathcal {C}$ , the collectionFootnote ²³

$$\begin{align*}\big\{ \mathfrak{N} \in \wp_\kappa(H_\theta) \ : \ C_\bullet \in \mathfrak{N} \text{ and } C_\bullet \restriction \mathfrak{N} \in \mathcal{C} \big\} \end{align*}$$

is stationary in $\wp _\kappa (H_\theta )$ for all sufficiently large $\theta $ .

Proof (Sketch) Assumptions (a) and (b) ensure that whenever

$$\begin{align*}\vec{\mathfrak{N}} = \langle \mathfrak{N}_\alpha \ : \ \alpha < \zeta \rangle \end{align*}$$

is a $\subseteq $ -continuous (but not necessarily $\in $ -increasing) sequence of elementary submodels of $(H_\theta ,\in ,R,\kappa , \mathcal {C} \cap H_\theta )$ with each $\mathfrak {N}_\alpha \cap \kappa $ transitive, then the following implication holds for all $C \in \mathfrak {N}_0 \cap \mathcal {C}$ , where $\mathfrak {N}:= \bigcup _{\alpha < \zeta } \mathfrak {N}_\alpha $ :

(13) $$ \begin{align} \left( \mathfrak{N}_0 \cap C \in \mathcal{C} \text{ and } \forall \alpha < \zeta \ \frac{\mathfrak{N}_{\alpha+1} \cap C}{\mathfrak{N}_\alpha \cap C} \in \mathcal{C} \right) \implies \ \left( \mathfrak{N} \cap C \in \mathcal{C} \text{ and } \frac{C}{\mathfrak{N} \cap C } \in \mathcal{C} \right). \end{align} $$

To see why this implication holds, assume its hypothesis; then $\mathfrak {N} \cap C \in \mathcal {C}$ because of assumption (a). Now set $\mathfrak {N}':= H_\theta $ ; it follows from assumption (b) that $\frac {\mathfrak {N}' \cap C}{\mathfrak {N} \cap C} = \frac {C}{\mathfrak {N} \cap C}$ is an element of $\mathcal {C}$ .

Then one can prove, by induction on $|\mathfrak {N}|$ and using (13), that clause (II) of Theorem 2.10 holds. Assumption (c) is only used to deal with the base case $|\mathfrak {N}|<\kappa $ .

5.3 Further extensions of the Flat Cover Conjecture

Recall that the Eklof–Trlifaj Theorem 3.7 was a generalization of the Flat Cover Conjecture (Theorem). We consider strengthenings of Theorem 3.7, the first due to Cortés-Izurdiaga and Šaroch (Theorem 5.5), and a new one due to the author (Theorem 5.6). In all three theorems, one assumes $\lambda $ is an infinite regular cardinal, $\mathcal {B}$ is a class of $\lambda $ -pure-injective modules, and one wants to prove (possibly with an additional assumption) that ${}^\perp \mathcal {B}$ is deconstructible. Since ${}^\perp \mathcal {B}$ is filtration-closed by Eklof’s Lemma, the problem reduces (by Theorem 2.10) to showing that for some sufficiently large $\kappa $ , the implication

(14) $$ \begin{align} A \in {}^\perp \mathcal{B} \cap \mathfrak{N} \ \implies \left( \mathfrak{N} \cap A \in {}^\perp \mathcal{B} \text{ and } \frac{A}{\mathfrak{N} \cap A} \in {}^\perp \mathcal{B} \right) \end{align} $$

holds for almost every $\mathfrak {N} \prec (H_\theta ,\in ,\dots )$ whose intersection with $\kappa $ is transitive.

In our proof of the Eklof–Trlifaj Theorem 3.7, where $\lambda = \omega $ , the implication (14) held automatically for any $\mathfrak {N} \prec (H_\theta ,\in )$ , as long as $R \in \mathfrak {N}$ and $\mathfrak {N} \cap |R|^+$ was transitive. Lemma 2.4 was the key there, and that lemma was ultimately due to the fact that every elementary submodel of $(H_\theta ,\in )$ is closed under finite subsets (Fact 2.1 part (2a)). That is, $\omega $ -purity reflects to every $\mathfrak {N} \prec (H_\theta ,\in )$ , as long as R is both an element and subset of $\mathfrak {N}$ .

Remark 2.6 says that if $\mathfrak {N} \prec (H_\theta ,\in )$ is closed under $<\lambda $ sequences, then its trace on any $A \in \mathfrak {N}$ is $\lambda $ -pure in A. If $\kappa ^{<\lambda } = \kappa $ , the collection of such $\mathfrak {N}$ is stationary in $\wp _\kappa (H_\theta )$ , but it is also co-stationary if $\lambda \ge \omega _1$ .Footnote ²⁴ In this situation, the implication (14) is guaranteed to hold for $<\lambda $ -closed ${\mathfrak {N} \in \wp _\kappa (H_\theta )}$ ,Footnote ²⁵ which at least ensures that ${}^\perp \mathcal {B}$ is a Kaplansky Class. But being a Kaplansky class is weaker, in general, than deconstructibility (and does not abstractly imply the class is precovering). To achieve deconstructibility, one also needs the implication (14) to hold even for those $\mathfrak {N}$ that are not $<\lambda $ -closed. Theorems 5.5 and 5.6 employ different assumptions to achieve this; the former assumes closure of ${}^\perp \mathcal {B}$ under direct limits, while the latter assumes ${}^\perp \mathcal {B}$ is eventually a.e. closed under quotients.

Theorem 5.5 (Cortés-Izurdiaga and Šaroch [Reference Cortés-Izurdiaga and Šaroch7, Theorem 4.6 and Corollary 4.7]).

Let $\lambda $ be an infinite regular cardinal. Suppose $\mathcal {B}$ is a class of $\lambda $ -pure-injective modules, and that

(†) $$ \begin{align} {}^\perp \mathcal{B} \text{ is closed under direct limits.} \end{align} $$

Then ${}^\perp \mathcal {B}$ is deconstructible.

One could re-prove Theorem 5.5 in a way similar to our proof of the Eklof–Trlifaj Theorem 3.7. The argument from the proof of Theorem 3.7, together with Remark 2.6, would imply that (under the assumptions of Theorem 5.5)

$$\begin{align*}A \in \mathfrak{N} \cap {}^\perp \mathcal{B} \ \implies \ \left( \mathfrak{N} \cap A \in {}^\perp \mathcal{B} \text{ and } \frac{A}{\mathfrak{N} \cap A} \in {}^\perp \mathcal{B} \right) \end{align*}$$

holds whenever $\mathfrak {N}$ is closed under sequences of length $<\lambda $ . Then one could use assumption (†) to achieve the same implication for almost every $\mathfrak {N}$ of any size, by induction on $|\mathfrak {N}|$ and using filtrations of $\mathfrak {N}$ .Footnote ²⁶

We strengthen Theorem 5.5.

Theorem 5.6. Theorem 5.5 holds when the assumption (†) is replaced by

(††) $$ \begin{align} {}^\perp \mathcal{B} \text{ is eventually a.e. closed under quotients}. \end{align} $$

Remark 5.7. We can also weaken the assumption on $\mathcal {B}$ , to only require injectivity with respect to inclusions of the form

$$\begin{align*}\mathfrak{N} \cap A \to A, \end{align*}$$

where $A \in \mathfrak {N} \prec (H_\theta ,\in )$ and $\mathfrak {N}$ is closed under $<\lambda $ sequences (all such inclusions are $\lambda $ -pure, by Remark 2.6).

Proof (of Theorem 5.6) Fix a sufficiently large regular $\kappa $ such that:

• • ${}^\perp \mathcal {B}$ is $\kappa $ -a.e. closed under quotients;

• • $\alpha ^{<\lambda } <\kappa $ for all $\alpha < \kappa $ .

Then by Fact 2.1, for any $\theta> \kappa $ the set

$$\begin{align*}S_{\kappa,\lambda}:=\{ \mathfrak{N} \prec (H_\theta,\in) \ : \ |\mathfrak{N}|<\kappa, \ \mathfrak{N} \cap \kappa \in \kappa, \text{ and } {}^{<\lambda} \mathfrak{N} \subset \mathfrak{N} \} \end{align*}$$

is stationary in $\wp _\kappa (H_\theta )$ . By Lemma 3.9, for every $\mathfrak {N} \in S_{\kappa ,\lambda }$ ,

$$ \begin{align*} A \in \mathfrak{N} \cap {}^\perp \mathcal{B} \ \implies \ \mathfrak{N} \cap A \in {}^\perp \mathcal{B}. \end{align*} $$

So

$$\begin{align*}\{ \mathfrak{N} \cap A \ : \ A \in \mathfrak{N} \in S_{\kappa,\lambda} \} \end{align*}$$

is a stationary subset of ${}^\perp \mathcal {B} \cap \wp _\kappa (A)$ . By Theorem 5.4, ${}^\perp \mathcal {B}$ is $\kappa $ -deconstructible.

The assumptions of Theorem 5.6 follow from those of Theorem 5.5: suppose $\mathcal {B}$ satisfies the assumptions of Theorem 5.5. Then, by that theorem, ${}^\perp \mathcal {B}$ is deconstructible, and hence by Lemma 5.3, ${}^\perp \mathcal {B}$ is eventually a.e. closed under quotients.

However, under “smallness” assumptions on the universe—in particular, if $0^\sharp $ does not exist—then the assumptions of Theorems 3.7, 5.5, and 5.6 are equivalent. This follows from a theorem of Cortés-Izurdiaga and Šaroch [Reference Cortés-Izurdiaga and Šaroch7], which states that in the presence of enough nonreflecting stationary sets (in particular, if $0^\sharp $ does not exist), $\lambda $ -pure-injectivity is equivalent to ordinary $\aleph _0$ -pure injectivity, for any $\lambda $ and over any ring. In such a model, if $\mathcal {B}$ is a class of $\lambda $ -pure-injectives, then they are pure-injective, and it follows that ${}^\perp \mathcal {B}$ is closed under direct limits by a result of Angeleri, Mantese, Tonolo, and Trlifaj (cf. proof of Lemma 9 of [Reference Eklof and Trlifaj18]). And by Eklof–Trlifaj’s Theorem 3.7, ${}^\perp \mathcal {B}$ is deconstructible.

5.4 Gorenstein homological algebra

As mentioned in the introduction, one goal of approximation theory is to do homological algebra relative to certain classes, in place of the classical ones (e.g., projective and injective modules). Some of the most commonly-encountered classes in the recent literature are the Gorenstein Projective ( $\mathcal {GP}$ ), Gorenstein Flat ( $\mathcal {GF}$ ), and Gorenstein Injective ( $\mathcal {GI}$ ) modules (Enochs and Jenda [Reference Enochs and Jenda23], Iacob [Reference Iacob36]); these are introduced after Definition 5.11 below. Early on, the question was raised of whether analogues of Kaplansky’s Theorem 3.10 and/or the Flat Cover Theorem 3.1 hold in the Gorenstein setting. In particular, the following question appears many times in the literature.Footnote ²⁷

For $\mathcal {GF}$ the answer is affirmative, over ZFC alone, for all rings, as shown by Yang and Liang (see Theorem 5.13 below). For $\mathcal {GP}$ , the answer is affirmative in ZFC for certain kinds of rings, and for all rings if there are large cardinals [Reference Cortés-Izurdiaga and Šaroch7, Reference Cox8], but it is still open in general. Section 5.4.1 focuses on what’s known to be provable in ZFC alone about Question 5.8, while Section 5.4.2 discusses what can be done with large cardinal assumptions.

5.4.1 ZFC results

We will use Theorems 2.10 and 4.3 to re-prove the following theorems (see below for the meaning of $\mathcal {X}$ - $\mathcal {GF}$ and $\mathcal {X}$ - $\mathcal {GP}$ ).

Theorem 5.9 (Estrada, Iacob, and Pérez [Reference Estrada, Iacob, Pérez, Srivastava, Leroy, Herzog and Guil Asensio26]).

Any class of the form $\mathcal {X}$ - $\mathcal {GF}$ is a precovering class. In particular, $\mathcal {GF}$ is always a precovering class (that $\mathcal {GF}$ is precovering—that is, that $\mathcal {X}$ - $\mathcal {GF}$ is precovering when $\mathcal {X}$ is the class of injectives—was first proved by Yang and Liang [Reference Yang and Liang58]).

Theorem 5.10. For any infinite regular $\lambda $ and any class $\mathcal {X}$ of $\lambda $ -pure-injective modules, the class $\mathcal {X}$ - $\mathcal {GP}$ is deconstructible.

Theorem 5.10 modestly strengthens a result from Cortés-Izurdiaga and Šaroch [Reference Cortés-Izurdiaga and Šaroch7], who proved it under the additional assumption that $\mathcal {X}$ contains all the projective modules. In particular, their result showed that $\mathcal {GP}$ is always deconstructible over left $\Sigma $ -pure-injective rings (which includes, e.g., rings that are simultaneously right coherent and left perfect; see [Reference Cortés-Izurdiaga and Šaroch7]).

Definition 5.11. If $\mathcal {C}$ is a class of acyclic complexes in $\text {Ch}(R)$ and $\mathcal {F}$ is a (possibly class-indexed)Footnote ²⁸ collection of functors from $\text {Ch}(R) \to \text {Ch}(\mathbb {Z})$ , set

$$\begin{align*}\text{Ac}(\mathcal{C},\mathcal{F}):= \Big\{ C_\bullet \in \mathcal{C} \ : \ \forall F \in \mathcal{F} \ F(C_\bullet) \text{ is acyclic} \Big\}. \end{align*}$$

Definition 5.12. If $\mathcal {D}$ is a class of complexes of R-modules,

$$\begin{align*}\text{Cyc}(\mathcal{D}):=\big\{ C \in R \text{-Mod} \ : \ \exists D_\bullet \in \mathcal{D} \ C = \text{ker}(D_0 \to D_1) \big\}. \end{align*}$$

All classes in the literature on Gorenstein homological algebra known to the author are of the form $\text {Cyc} \left ( \text {Ac}(\mathcal {C},\mathcal {F}) \right )$ for some class $\mathcal {C}$ of complexes and some collection $\mathcal {F}$ of functors. For example:

1. (A) For a fixed class $\mathcal {X}$ of modules: $\mathcal {X}$ - $\mathcal {GP}$ is $\text {Cyc}\left ( \text {Ac}(\mathcal {C}_{\text {proj}}, \text {Hom}(-,\mathcal {X}) )\right )$ , where $\mathcal {C}_{\text {proj}}$ is the class of acyclic complexes with projective entries, and $\text {Hom}(-,\mathcal {X})$ is the collection of (contravariant) functors of the form

   $$\begin{align*}\text{Hom}(-,X): \text{Ch}(R) \to \text{Ch}(\mathbb{Z}), \end{align*}$$

   where X ranges over all members of $\mathcal {X}$ . The (ordinary) Gorenstein projective modules $\mathcal {GP}$ is the special case where $\mathcal {X}$ is the class of projective modules.

2. (B) For a fixed class $\mathcal {X}$ of (right) modules: $\mathcal {X}$ - $\mathcal {GF}$ is $\text {Cyc}\left (\text {Ac}(\mathcal {C}_{\text {flat}}, \mathcal {X} \otimes - )\right )$ , where $\mathcal {C}_{\text {flat}}$ is the class of acyclic complexes with flat entries, and $\mathcal {X} \otimes -$ is the collection of (covariant) functors of the form

   $$\begin{align*}X \otimes -: \text{Ch}(R) \to \text{Ch}(\mathbb{Z}), \end{align*}$$

   where X ranges over all members of $\mathcal {X}$ . The (ordinary) Gorenstein flat modules $\mathcal {GF}$ is the special case where $\mathcal {X}$ is the class of (right) injective modules.

To re-prove Theorem 5.9 we will use the following theorem.

Theorem 5.13 (Yang and Liang [Reference Yang and Liang58, pp. 3083–3084]).

Suppose $\mathcal {D}$ is a precovering class in $\text {Ch}(R)$ , and each member of $\mathcal {D}$ is acyclic. Then $\text {Cyc}(\mathcal {D})$ is a precovering class in $\text {Ch}(R)$ .

The proof of Theorem 5.13 is a purely algebraic proof, showing how given any module M, a $\mathcal {D}$ -precover of the complex $\ldots \longrightarrow 0 \longrightarrow M \longrightarrow 0 \longrightarrow \ldots \ \text{in} \ \text {Ch}(R)$ can be used to define a $\text {Cyc}(\mathcal {D})$ -precover of M in R-Mod (see [Reference Yang and Liang58] for details).

Now we prove the Estrada–Iacob–Pérez Theorem.

Proof (of Theorem 5.9) By definition, $\mathcal {X} \text {-} \mathcal {GF} = \text {Cyc}\Big ( \text {Ac}(\mathcal {C},\mathcal {F}) \Big )$ , where:

• • $\mathcal {C}$ are the acyclic complexes with flat entries;

• • $\mathcal {F}$ is the collection of functors of the form $X \otimes -$ , for $X \in \mathcal {X}$ .

We will prove that $\text {Ac}(\mathcal {C},\mathcal {F})$ is deconstructible; then by Theorem 2.9 it will be precovering (in $\text {Ch}(R)$ ), and finally by Theorem 5.13, $\text {Cyc}\big ( \text {Ac}(\mathcal {C},\mathcal {F}) \big )$ (= $\mathcal {X}$ - $\mathcal {GF}$ ) will be precovering in R-Mod.

Suppose $F_\bullet \in \mathcal {C}$ ; that is, $F_\bullet $ is an acyclic complex of flat modules. For each $n \in \mathbb {Z}$ let $C_n:= \text {ker}(F_n \to F_{n+1})$ (the n-th cycle). Then the following equivalences hold, via purely homological arguments, where $X^+$ denotes the character dual $\text {Hom}_{\mathbb {Z}}(X,\mathbb {Q}/\mathbb {Z})$ Footnote ²⁹ :

$$ \begin{align*} & F_\bullet \in \text{Ac}(\mathcal{C},\mathcal{F}) & \\ \iff & \forall X \in \mathcal{X} \ X \otimes F_\bullet \text{ is acyclic} & \text{(by definition)} \\ \iff & \forall n \in \mathbb{Z} \ \forall X \in \mathcal{X} \ \text{Tor}\left( X, C_n \right)=0 & \text{(using flatness of the } F_n \text{'s)} \\ \iff & \forall n \in \mathbb{Z} \ \forall X \in \mathcal{X} \ \text{Ext}(C_n,X^+)=0 & \text{(by Ext-Tor relations, }[33] \text{ Lemma 2.16)} \\ \iff & \forall n \in \mathbb{Z} \ C_n \in {}^\perp \underbrace{\{ X^+ \ : \ X \in \mathcal{X} \}}_{=: \mathcal{Y}} & \text{(by definition)}. \end{align*} $$

By Eklof’s Lemma 3.6, the class ${}^\perp \mathcal {Y}$ is filtration closed in R-Mod, and this easily implies (via the equivalences above) that $\text {Ac}(\mathcal {C},\mathcal {F})$ is filtration-closed in $\text {Ch}(R)$ . So it remains to show that $\text {Ac}(\mathcal {C},\mathcal {F})$ is weakly $|R|^+$ -deconstructible in $\text {Ch}(R)$ . Since character duals are pure-injective (Enochs and Jenda [Reference Enochs and Jenda23, Proposition 5.3.7]), the Eklof–Trlifaj Theorem 3.7 implies that

(15) $$ \begin{align} {}^\perp \mathcal{Y} \text{ is } |R|^+ \text{-deconstructible in } R\text{-Mod}. \end{align} $$

Consider any $F_\bullet \in \text {Ac}(\mathcal {C},\mathcal {F})$ and assume

$$\begin{align*}F_\bullet \in \mathfrak{N} \prec (H_\theta,\in, \mathcal{Y} \cap H_\theta), \end{align*}$$

where $\mathfrak {N} \cap |R|^+$ is transitive. Now each $C_n=\text {ker} (F_n \to F_{n+1})$ is an element of $\mathfrak {N} \cap {}^\perp \mathcal {Y}$ , so by deconstructibility of ${}^\perp \mathcal {Y}$ and Theorem 2.10,

$$\begin{align*}\forall n \in \mathbb{Z} \ \mathfrak{N} \cap C_n \in {}^\perp \mathcal{Y} \text{ and } \frac{C_n}{\mathfrak{N} \cap C_n} \in {}^\perp \mathcal{Y}. \end{align*}$$

Moreover, the n-th cycles of $F_\bullet \restriction \mathfrak {N}$ and $F_\bullet / \mathfrak {N}$ are, respectively, $\mathfrak {N} \cap C_n$ and $\frac {C_n}{\mathfrak {N} \cap C_n}$ . So all cycles of $F_\bullet \restriction \mathfrak {N}$ and $F_\bullet / \mathfrak {N}$ lie in ${}^\perp \mathcal {Y}$ , so by the equivalences above, both complexes are in $\text {Ac}(\mathcal {C},\mathcal {F})$ .

The next theorem, in conjunction with Theorem 5.4, will be convenient when trying to show that various classes in Gorenstein homological algebra are deconstructible.

Theorem 5.14. Suppose $\mathcal {C}$ is a class of acyclic complexes in $\text {Ch}(R)$ , $\mathcal {F}$ is a collection of functors from $\text {Ch}(R) \to \text {Ch}(\mathbb {Z})$ , and:

1. (1) $\mathcal {C}$ is eventually a.e. closed under quotients in $\text {Ch}(R)$ (recall this holds if, e.g., $\mathcal {C}$ is deconstructible).

2. (2) For every short exact sequence of the form,

   

   with $\{ C_\bullet , R \} \cup R \subset \mathfrak {N} \prec \mathfrak {N}' \prec (H_\theta ,\in )$ and both $\mathfrak {N} \cap \kappa $ and $\mathfrak {N}' \cap \kappa $ transitive: if all three inner entries of $\mathcal {E}$ are in $\mathcal {C}$ , then each $F \in \mathcal {F}$ preserves the exactness of $\mathcal {E}$ ; that is,

   

   —with the direction of the arrows depending on whether F is covariant or contravariant—is a short exact sequence in $\text {Ch}(\mathbb {Z})$ .

Then $\text {Ac}(\mathcal {C},\mathcal {F})$ is eventually a.e. closed under quotients in $\text {Ch}(R)$ .

Proof This is essentially due to the 3-by-3 lemma. Fix any regular $\kappa> |R|$ witnessing eventual a.e. closure of $\mathcal {C}$ under quotients. Suppose

$$\begin{align*}C_\bullet \in \mathfrak{N} \prec \mathfrak{N}' \prec (H_\theta,\in,\kappa,R, \mathcal{C} \cap H_\theta ), \end{align*}$$

$\mathfrak {N}$ and $\mathfrak {N}'$ both have transitive intersection with $\kappa $ , and that

(16) $$ \begin{align} \Big\{ C_\bullet, \ C_\bullet \restriction \mathfrak{N}, \ C_\bullet \restriction \mathfrak{N}' \Big\} \subset \text{Ac}(\mathcal{C},\mathcal{F}). \end{align} $$

We need to prove that $\frac {C_\bullet \restriction \mathfrak {N}'}{C_\bullet \restriction \mathfrak {N} } \in \text {Ac}(\mathcal {C},\mathcal {F})$ . Since $\mathcal {C}$ is $\kappa $ -a.e. closed under quotients, and $\text {Ac}(\mathcal {C},\mathcal {F}) \subseteq \mathcal {C}$ by definition, (16) implies that the complexes $C_\bullet \restriction \mathfrak {N}$ , $C_\bullet \restriction \mathfrak {N}'$ , and $\frac {C_\bullet \restriction \mathfrak {N}'}{C_\bullet \restriction \mathfrak {N} }$ are all in $\mathcal {C}$ . So it only remains to prove that $\frac {C_\bullet \restriction \mathfrak {N}'}{C_\bullet \restriction \mathfrak {N} }$ remains acyclic after applying any functor $F \in \mathcal {F}$ . Consider the short exact sequence

with i the inclusion. Since each of the three inner terms are, by assumption, in $\text {Ac}(\mathcal {C},\mathcal {F})$ and in particular in $\mathcal {C}$ , assumption (2) ensures that

is a short exact sequence in $\text {Ch}(\mathbb {Z})$ (with the direction of the arrows depending on whether F is contravariant or covariant).

By assumption (16), both $C_\bullet \restriction \mathfrak {N}$ and $C_\bullet \restriction \mathfrak {N}'$ are in $\text {Ac}(\mathcal {C},\mathcal {F})$ , so both $F(C_\bullet \restriction \mathfrak {N})$ and $F(C_\bullet \restriction \mathfrak {N}')$ are acyclic. Then by the 3-by-3 lemma and exactness of $F(\mathcal {E})$ , it follows that $F \left ( \frac {C_\bullet \restriction \mathfrak {N}'}{C_\bullet \restriction \mathfrak {N}}\right )$ is acyclic.

Lemma 5.15. The classes of complexes $\text {Ac}(\mathcal {C}_{\text {proj}}, \text {Hom}(-,\mathcal {X}) )$ and $\text {Ac}(\mathcal {C}_{\text {flat}}, \mathcal {X} \otimes - )$ —that is, those used to define $\mathcal {X}$ - $\mathcal {GP}$ and $\mathcal {X}$ - $\mathcal {GF}$ (resp.) in (A) and (B) on page 26—satisfy the assumptions of Theorem 5.14.

Proof Since the flat modules and projective modules are deconstructible in R-Mod, a routine use of Theorems 4.3 and 2.10 implies that the classes $\mathcal {C}_{\text {proj}}$ and $\mathcal {C}_{\text {flat}}$ are deconstructible in $\text {Ch}(R)$ . So by Lemma 5.3, they are eventually a.e. closed under quotients. So both classes satisfy clause (1) of the assumptions of Theorem 5.14.

To see that $\text {Ac}(\mathcal {C}_{\text {proj}}, \text {Hom}(-,\mathcal {X}) )$ satisfies clause (2) of Theorem 5.14, suppose $\kappa> |R|$ , $P_\bullet $ is an acyclic complex of projective modules, $\{ C_\bullet , R \} \subset \mathfrak {N} \prec \mathfrak {N}' \prec (H_\theta ,\in )$ , and $\mathfrak {N}$ and $\mathfrak {N}'$ have transitive intersection with $\kappa $ . Consider the short exact sequence

with i the inclusion. By Theorem 3.10, for each $n \in \mathbb {Z}$ , the module $\frac {\mathfrak {N}' \cap C_n}{\mathfrak {N} \cap C_n}$ is projective; so $\mathcal {E}$ is “degreewise” split. Then for any X and any $n \in \mathbb {Z}$ , $\text {Hom}(-,X)$ preserves exactness of $\mathcal {E}$ at degree n. Hence, $\text {Hom}(\mathcal {E},X)$ is a short exact sequence in $\text {Ch}(\mathbb {Z})$ .

That $\text {Ac}(\mathcal {C}_{\text {flat}}, \mathcal {X} \otimes - )$ satisfies clause (2) of Theorem 5.14 follows from a similar argument, except one uses instead that $\mathcal {E}$ is pure in each degree (by Lemma 2.4), so that tensoring with any module preserves exactness.Footnote ³⁰

To prove Theorem 5.10, we will need a variant of the Yang–Liang Theorem 5.13.

Theorem 5.16. Suppose $\mathcal {D}$ is a deconstructible class in $\text {Ch}(R)$ . Then $\text {Cyc}(\mathcal {D})$ is weakly deconstructible in R-Mod (but possibly not filtration-closed).

Proof Say $\kappa> |R|$ witnesses deconstructibility of $\mathcal {D}$ . Then by Theorem 4.3,

$$\begin{align*}D_\bullet \in \mathfrak{N} \cap \mathcal{D} \ \implies \left( D_\bullet \restriction \mathfrak{N} \in \mathcal{D} \text{ and } D_\bullet / \mathfrak{N} \in \mathcal{D} \right) \end{align*}$$

whenever $\mathfrak {N} \prec (H_\theta ,\in ,\kappa , \mathcal {D} \cap H_\theta )$ and $\mathfrak {N} \cap \kappa $ is transitive.

To prove weak deconstructibility of $\text {Cyc}(\mathcal {D})$ , we use the (IV) $\implies $ (I) direction of Theorem 2.10 (which, by Remark 2.11, does not need filtration-closure). Consider any $C \in \text {Cyc}(\mathcal {D})$ , and fix a $D_\bullet \in \mathcal {D}$ such that $C = \text {Ker} (D_0 \to D_1)$ . Fix a $\theta $ such that $C,D_\bullet \in H_\theta $ . Set

$$\begin{align*}\mathfrak{A}:=(H_\theta,\in,\kappa,C,D_\bullet, \mathcal{D} \cap H_\theta) \end{align*}$$

and consider any $\mathfrak {N} \prec \mathfrak {A}$ whose intersection with $\kappa $ is transitive. Then $D_\bullet \restriction \mathfrak {N}$ and $D_\bullet / \mathfrak {N}$ are both in $\mathcal {D}$ , and hence their respective 0-th cycles are in $\text {Cyc}(\mathcal {D})$ . The 0-th cycle of $D_\bullet \restriction \mathfrak {N}$ is $\mathfrak {N} \cap C$ , and the 0-th cycle of $D_\bullet / \mathfrak {N}$ is $\frac {C}{\mathfrak {N} \cap C}$ .

Finally, we prove the following proof of theorem.

Proof (of Theorem 5.10) Assume that $\lambda $ is an infinite regular cardinal and $\mathcal {X}$ is a class of $\lambda $ -pure-injective modules. By definition,

$$\begin{align*}\mathcal{X} \text{-} \mathcal{GP} = \text{Cyc}\Big( \text{Ac}(\mathcal{C},\mathcal{F}) \Big), \end{align*}$$

where:

• • $\mathcal {C}$ are the acyclic complexes with projective entries;

• • $\mathcal {F}$ is the collection of functors of the form $\text {Hom}(-, X)$ , for $X \in \mathcal {X}$ .

The class $\mathcal {X}$ - $\mathcal {GP}$ is known to be filtration-closed; the homological argument from Enochs, Iacob, and Overtoun [Reference Enochs, Iacob and Jenda21, Theorem 3.2], for the case $\mathcal {X} = \text {Proj}$ , works for any $\mathcal {X}$ . So it remains to show that $\mathcal {X}$ - $\mathcal {GP}$ ( $= \text {Cyc}\big ( \text {Ac}(\mathcal {C},\mathcal {F} \big )$ ) is weakly deconstructible, and by Theorem 5.16 it suffices to prove that $\text {Ac}(\mathcal {C},\mathcal {F})$ is deconstructible in $\text {Ch}(R)$ . Filtration-closure of $\text {Ac}(\mathcal {C},\mathcal {F})$ follows from the proof of Theorem 2.6 of Enochs, Iacob, and Overtoun [Reference Enochs, Iacob and Jenda21] argument, so it only remains to show weak deconstructibility of $\text {Ac}(\mathcal {C},\mathcal {F})$ in $\text {Ch}(R)$ .

By Theorem 5.14 and Lemma 5.15, $\text {Ac}(\mathcal {C},\mathcal {F})$ is eventually a.e. closed under quotients. Fix a regular $\kappa $ such that $\text {Ac}(\mathcal {C},\mathcal {F})$ is $\kappa $ -a.e. closed under quotients and $\alpha ^{<\lambda } < \kappa $ for all $\alpha < \kappa $ . So by Theorem 5.4, it suffices to find a stationary set of $\mathfrak {N} \in \wp _\kappa (H_\theta )$ such that $\mathfrak {N}$ reflects membership in $\text {Ac}(\mathcal {C},\mathcal {F})$ ; that is, such that

$$\begin{align*}P_\bullet \in \mathfrak{N} \cap \text{Ac}\big(\mathcal{C},\mathcal{F} \big) \ \implies \ P_\bullet \restriction \mathfrak{N} \in \text{Ac}\big(\mathcal{C},\mathcal{F} \big). \end{align*}$$

We will show this holds for any $<\lambda $ -closed $\mathfrak {N}$ , the collection of which constitutes a stationary set by Fact 2.1.

For any acyclic complex $P_\bullet $ of projective modules with cycles $C_n:= \text {ker}(P_n \to P_{n+1})$ , we have the following equivalences:

$$ \begin{align*} & P_\bullet \in \text{Ac}(\mathcal{C},\mathcal{F}) & \\ \iff & \forall X \in \mathcal{X} \ \text{Hom}(P_\bullet,X) \text{ is acyclic} & \text{(by definition)} \\ \iff & \forall X \in \mathcal{X} \ \forall n \in \mathbb{Z} \ C_n \in {}^\perp X & \text{(using projectivity of the } P_n \text{'s)} \\ \iff & \forall n \in \mathbb{Z} \ C_n \in {}^\perp \mathcal{X} & \text{(by definition)}. \end{align*} $$

Consider any $<\lambda $ -closed $\mathfrak {N} \prec (H_\theta ,\in ,\mathcal {X} \cap H_\theta )$ whose intersection with $\kappa $ is transitive, and suppose $P_\bullet \in \mathfrak {N}$ is a member of $\text {Ac}(\mathcal {C},\mathcal {F})$ . Then ${C_n \in \mathfrak {N} \cap {}^\perp \mathcal {X}}$ for all $n \in \mathbb {Z}$ . Since $\mathcal {X}$ consists of $\lambda $ -pure-injective modules, and by $<\lambda $ -closure of $\mathfrak {N}$ , Lemma 3.9 ensures that

$$\begin{align*}\forall n \in \mathbb{Z} \ \underbrace{\mathfrak{N} \cap C_n}_{\text{ the } n \text{-th cycle of } P_\bullet \restriction \mathfrak{N}} \in {}^\perp \mathcal{X} \end{align*}$$

(Lemma 3.9 also tells us $\frac {C_n}{\mathfrak {N} \cap C_n} \in {}^\perp \mathcal {X}$ , but we don’t need that here, because we only need to check the assumptions of Theorem 5.4). Also, $P_\bullet \restriction \mathfrak {N}$ consists of projective modules, by Kaplansky’s Theorem 3.10. So by the equivalences above, $P_\bullet \restriction \mathfrak {N} \in \text {Ac}(\mathcal {C},\mathcal {F})$ .

Remark 5.17. The proof does not actually need projectivity of the complexes. The argument works as long as $\mathcal {C}$ is a deconstructible class of complexes with entries from ${}^\perp \mathcal {X}$ .

5.4.2 Maximum econstructibility and Gorenstein homological algebra

Theorem 5.18 (Cortés-Izurdiaga and Šaroch [Reference Cortés-Izurdiaga and Šaroch7]).

If there is a proper class of strongly compact cardinals, then over every ring, the class of Gorenstein projective modules is deconstructible.

The author independently proved the same conclusion, but under the stronger assumption of a proper class of supercompact cardinals (Cox [Reference Cox8]). The proofs from [Reference Cortés-Izurdiaga and Šaroch7, Reference Cox8] are very different, and generalize in different ways (see the introduction of [Reference Cox8] for a discussion). The argument from [Reference Cox8] uses (of course) elementary submodel methods, and generalizes to show the following theorem.

Theorem 5.19 [Reference Cox8].

Vopěnka’s Principle implies that all classes of the form $\mathcal {X}$ - $\mathcal {GP}$ are deconstructible.

Theorem 5.19 follows from an even more powerful theorem. Recall by Lemma 5.3 that for any class $\mathcal {C}$ of modules or complexes that is filtration-closed,

$$ \begin{align*} \mathcal{C} \text{ is deconstructible } \implies \ \mathcal{C} \text{ is eventually a.e. closed under quotients.} \end{align*} $$

Under Vopěnka’s Principle, the converse holds the following theorem.

Theorem 5.20 (Cox [Reference Cox8]).

Vopěnka’s Principle implies the Maximum Deconstructibility principle, which asserts that for any class $\mathcal {C}$ of modules or complexes: if $\mathcal {C}$ is filtration-closed and eventually a.e. closed under quotients, then it is deconstructible.

We refer to [Reference Cox8] for the proof, but roughly: given any (isomorphism-closed and filtration-closed) class $\mathcal {C}$ , Vopěnka’s Principle provides many regular $\kappa $ such that for a proper class of $\theta $ there are stationarily many $\mathfrak {N} \in \wp ^*_\kappa (H_\theta )$ such that for any C,

$$\begin{align*}C \in \mathfrak{N} \cap \mathcal{C} \ \implies \ \mathfrak{N} \cap C \in \mathcal{C}. \end{align*}$$

But this doesn’t quite satisfy clause (II) of Theorem 2.10,Footnote ³¹ because we only have it for stationarily many $\mathfrak {N} \in \wp _\kappa (H_\theta )$ , and we are still missing the requirement that the quotient $\frac {C}{\mathfrak {N} \cap C}$ is also in $\mathcal {C}$ . And we must assume something about $\mathcal {C}$ if we want to derive its deconstructibility, since there are ZFC examples of filtration-closed $\mathcal {C}$ that are not deconstructible.Footnote ³² But the assumption that $\mathcal {C}$ is eventually a.e. closed under quotients provides a way, via Theorem 5.4 from page 22, to “convert” the mere stationary set of $\mathfrak {N} \in \wp ^*_\kappa (H_\theta )$ into a club, and moreover obtain the desired quotient behavior.

To see that Theorem 5.20 implies Theorem 5.19, fix any ring R and any class $\mathcal {X}$ of R-modules. Using the notation of Section 5.4,

(17) $$ \begin{align} \mathcal{X} \text{-} \mathcal{GP} = \text{Cyc}\left( \text{Ac}(\mathcal{C},\mathcal{F}) \right), \end{align} $$

where $\mathcal {C}$ is the class of acyclic complexes with projective entries, and $\mathcal {F}$ is the collection of functors of the form $\text {Hom}(-,X)$ , where X ranges over all members of the class $\mathcal {X}$ . By Theorem 5.14 and Lemma 5.15, $\text {Ac}(\mathcal {C},\mathcal {F})$ is eventually a.e. closed under quotients in $\text {Ch}(R)$ . The class $\text {Ac}(\mathcal {C},\mathcal {F})$ is also (via purely homological arguments) closed under filtrations. So by the Maximum Deconstructibility Principle, $\text {Ac}(\mathcal {C},\mathcal {F})$ is deconstructible in $\text {Ch}(R)$ . By Theorem 5.16, the class (17) is weakly decontructible in R-Mod. But it is also closed under filtrations (by the same argument as Enochs, Iacob, and Overtoun [Reference Enochs, Iacob and Jenda21] did when $\mathcal {X} = \text {Proj}$ ). So $\mathcal {X}$ - $\mathcal {GP}$ is deconstructible.