Minimal semi-flat-cotorsion replacements and cosupport

Over a commutative noetherian ring $R$ of finite Krull dimension, we show that every complex of flat cotorsion $R$-modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a semi-flat-cotorsion complex as a special type of semi-flat complex, and provide functorial ways to construct a quasi-isomorphism from a semi-flat complex to a semi-flat-cotorsion complex. Consequently, every $R$-complex can be replaced by a minimal semi-flat-cotorsion complex in the derived category over $R$. Furthermore, we describe structure of semi-flat-cotorsion replacements, by which we recover classic theorems for finitistic dimensions. In addition, we improve some results on cosupport and give a cautionary example. We also explain that semi-flat-cotorsion replacements always exist and can be used to describe the derived category over any associative ring.


Introduction
The existence of injective envelopes for modules over any ring yields minimal injective resolutions; dually, in settings where projective covers exist-such as for finitely generated modules over a semi-perfect ring-one can build minimal projective resolutions. These classic forms of minimality are encompassed by the following definition: a complex is minimal if every self homotopy equivalence is an isomorphism; see Avramov and Martsinkovsky [3]. In fact, Avramov, Foxby, and Halperin show [2] that every complex of injective modules decomposes as a direct sum of a minimal complex and a contractible complex, see also Krause [26], thus showing every complex has a minimal semi-injective resolution. A dual statement, considered initially by Eilenberg [12], holds in settings where projective covers exist.
A natural question is whether a complex of flat modules exhibits similar behaviour. Although flat covers do exist for modules over any ring, due to Bican, El Bashir, and Enochs [6], it turns out that minimality is poorly behaved for complexes of flat modules in general: indeed, there exist quasi-isomorphisms between minimal semi-flat complexes that are not isomorphisms of complexes (unlike the case for minimal semi-projective or semi-injective complexes), see for example Christensen and Thompson [10]. We thus restrict our focus to complexes of a special type of flat modules: the flat cotorsion modules.
Let R be a commutative noetherian ring. Enochs shows [13] that flat cotorsion R-modules-i.e., those flat modules that are also right Ext-orthogonal to flat modules-have a unique decomposition, whose structure is akin to that of injective modules over a noetherian ring as shown by Matlis [28]. Further, minimality criteria for complexes of such modules was given by Thompson [43]. One goal of this paper is to show that when R has finite Krull dimension, complexes of flat cotorsion R-modules can be decomposed analogously to complexes of injective modules: Theorem (See 1.8 and 2.4). Assume dim R < ∞. If Y is a complex of flat cotorsion R-modules, then Y = Y ′ ⊕ Y ′′ , where Y ′ is minimal and Y ′′ is contractible.
In Section 3, we give a functorial approach-building on work of Nakamura and Yoshino in [33]-to construct a complex of flat cotorsion R-modules. We also turn to considering semi-flat-cotorsion complexes, that is, semi-flat complexes of flat cotorsion R-modules, as well as replacements by such complexes in the derived category over R; see Appendix A. If F is a semi-flat complex, then Constructions 3.1 and 3.3 yield functorial ways to build a semi-flat-cotorsion complex Y and a quasi-isomorphism F → Y . In particular, we obtain: Theorem (See 3.4). Assume dim R < ∞. Every R-complex has a minimal semiflat-cotorsion replacement in the derived category over R.
Although it is immediate from [43,Theorem 5.2] that every R-module has a minimal semi-flat-cotorsion replacement without the assumption of finite Krull dimension, the assumption here is natural in considering unbounded complexes. One motivation for our approach is that not every R-module admits a surjection from, or injection to, a flat cotorsion R-module-see Example 3.7-and so our method differs from the one for complexes of injective modules given in [26,Appendix B].
In Section 4, we employ the functorial construction of semi-flat-cotorsion replacements from Section 3, along with the Auslander-Buchsbaum formula, to describe the structure of semi-flat-cotorsion replacements; see Lemma 4.1 and Theorem 4.5. In particular, this extends structure of the minimal pure-injective resolution of a flat module described by Enochs [14], and also recovers-see Corollary 4.6-the fact that the finitistic flat dimension of R is at most dim R. In addition, this structure gives a new proof of a classic result of Gruson and Raynaud [36] and Jensen [23]: an R-module of finite flat dimension has projective dimension at most dim R, see Theorem 4.8; in particular, the finitistic projective dimension of R is at most dim R and flat R-modules have projective dimension at most dim R.
In Section 5, we apply our constructions in the context of cosupport. The cosupport of an R-complex X is the set of prime ideals p such that RHom R (κ(p), X) is nontrivial in the derived category over R. As an analogue to work of Chen and Iyengar [8], we give in Example 5.12 an unbounded minimal complex Y of flat cotorsion R-modules such that cosupp R Y is strictly contained in i∈Z cosupp R Y i . This gives a counterexample to [42,Theorem 2.7], unfortunately, and we proceed to give a correction-and improvement-for this result; see Theorem 5.4. In the appendix, we define the notion of semi-flat-cotorsion replacements for any associative ring A, and point to how these complexes can be used to describe the derived category over A. In particular, we note that-due to a result of Gillespie [19]-every A-complex can be replaced by a semi-flat-cotorsion complex in the derived category over A, although minimality remains open. * * * Throughout, let R be a commutative noetherian ring. We use standard cohomological notation for R-complexes (that is, complexes of R-modules), and use H(−) to denote the cohomology functor. Denote by Mod R the category of R-modules, C(R) the category of R-complexes, K(R) the homotopy category of R-complexes, and D(R) the derived category over R. A morphism α : X → Y in C(R) is a quasiisomorphism if H(α) is an isomorphism; an R-complex X is acyclic if H(X) = 0.

Decomposing complexes of flat cotorsion modules
For a complex P of finitely generated free modules over a local ring (R, m, k), there exists a decomposition P = P ′ ⊕ P ′′ such that k ⊗ R P ′ has zero differential and P ′′ is contractible; this was shown in [2]. Although such a phenomenon does not extend to all complexes of infinitely generated projective modules (see Example 1.5), we are able to find a similar decomposition if we take m-adic completions of complexes of free modules. In this section, we explain this fact and extend it to the case of complexes of flat cotorsion modules.
We start with the following elementary lemma.
is a local ring. Let T and T ′ be m-adic completions of free R-modules and let ϕ : (1) There exists a homomorphism ϕ : (2) If ϕ is an isomorphism, then any such ϕ : T → T ′ is an isomorphism. (1). For (2), since the maximal ideal of R/m n+1 is nilpotent, an artinian version of Nakayama's lemma implies that φ n is an isomorphism for every n ≥ 1, hence ϕ is an isomorphism.
Remark 1.2. The argument in the proof of Lemma 1.1 is essentially the same as the proof of [16,Lemma 6.7.4], which shows that the m-adic completion of any flat R-module is isomorphic to the m-adic completion of a free R-module.
For an index set A and an R-module M , we denote by M (A) or A M the direct sum of A-copies of M . If (R, m, k) is local, then we write M for the m-adic completion of M . Lemma 1.1 is also a consequence of the fact that the canonical map R (A) → k (A) is a flat cover; see [13,Example,p. 181]. Lemma 1.3. Assume (R, m, k) is a local ring. Let A and A ′ be some index sets, and let ∂ : Proof. There are isomorphisms k ⊗ R R (A) ∼ = k (A) and k ⊗ R R (A ′ ) ∼ = k (A ′ ) , hence we may view k ⊗ R ∂ as a linear transformation of k-vector spaces. Since ker(k ⊗ R ∂) and im(k ⊗ R ∂) are subspaces (and hence direct summands), we may find disjoint partitions A = B ⊔ C and A ′ = B ⊔ C ′ such that the following diagram commutes: The maps α and β lift, by Lemma 1.1, to isomorphisms α : We thus obtain a commutative diagram: Thus Lemma 1.1 implies that i is an isomorphism; the conditions on f , g, and h allow for an elementary translation of the diagram into the desired one.
We aim to apply Lemma 1.3 to complexes of completions of free modules. Towards this end, assuming (R, m, k) is local and D and D ′ are some other index sets, consider an R-complex of the form By Lemma 1.3, it may be identified with the following one: where k ⊗ R ∂ ′ = 0 and 0 → R (B) = − → R (B) → 0 is a contractible direct summand. Moreover, we can again apply Lemma 1.3 to a : R (D) → R (C) and b : R (C ′ ) → R (D ′ ) to find contractible direct summands. This observation can be used to show that the following lemma holds.
Inductively work outward, using Lemma 1.3 to build contractible summands Y ′′ (n) of Y /Y ′′ (n − 1) for each n ≥ 1, such that the differential of Y /Y ′′ (n − 1) in degrees −n through n is zero upon application of k ⊗ R −. Induction yields a contractible direct summand Y ′′ of Y such that the differential of The next example exhibits the necessity of taking completions to obtain a suitable decomposition.
Example 1.5. Let (R, m, k) be a local ring with dim R ≥ 1. Let x ∈ m be an element that is not nilpotent. The localization R x is therefore nonzero and has a projective resolution of the form P is exact, thus P has no nonzero direct summand P ′ such that k ⊗ R P ′ has zero differenatial. However, P is not contractible since R x is nonzero.
The goal of this section is to extend Lemma 1.4 above to the case of complexes of flat cotorsion modules, and so we begin with some basic facts about these. Here we return to the setting of any commutative noetherian ring R.
An R-module M is flat cotorsion if it is both flat and cotorsion, that is, M is flat and Ext 1 R (F, M ) = 0 for every flat R-module F . Enochs shows in [13] that an Rmodule M is flat cotorsion if and only if M ∼ = p∈Spec R T p , where T p is the p-adic completion of a free R p -module. For an ideal a of R, let Λ a = lim ← −n≥1 (R/a n ⊗ R −) denote the a-adic completion functor; for an R-module M , also write Λ a M = M ∧ a . A motivation for studying complexes of flat cotorsion R-modules is their relationship to cosupport, as defined by Benson, Iyengar, and Krause [5]. For an R-complex X, the cosupport of X is defined as: where κ(p) stands for the residue field R p /pR p . This is dual to the notion of support defined by Foxby [17]; the support of X is: From this and tensor-hom adjunction, we see that cosupp( A R p ) ∧ p ⊆ {p}. Consequently it follows that a flat cotorsion R-module M has cosupport contained in a subset W of Spec R if and only if M ∼ = p∈W T p , where T p is the p-adic completion of a free R p -module. We can therefore translate Lemma 1.4 to: such that the complex κ(p) ⊗ R Y ′ has zero differential and Y ′′ is contractible.
Proof. Reduce to a local ring (R, m, k); this is just a restatement of Lemma 1.4.
For a subset W of Spec R, we define dim W as the supremum of the lengths of chains of prime ideals in W . As is standard, dim(Spec R) is denoted by dim R; this is the Krull dimension of R. The next theorem is the main result of this section. In its proof, we use several basic facts about complexes of flat cotorsion R-modules; they are summarized at the end of this section.
such that the complex κ(p) ⊗ R Hom R (R p , Y ′ ) has zero differential for every p ∈ W and Y ′′ is contractible.
Proof. We proceed by induction on dim W . First suppose dim W = 0. In this case, Y ∼ = q∈W Λ q Y by (1.16), and Λ q Y consists of flat cotorsion R-modules having cosupport contained in {q} by (1.12). For each q ∈ W , we apply Lemma 1.7 to obtain a decomposition Λ q Y = Y ′ (q) ⊕ Y ′′ (q), where κ(q) ⊗ R Y ′ (q) has zero differential and Y ′′ (q) is contractible. Taking a product over q ∈ W , we obtain a decomposition for every p ∈ W , and the latter has zero differential.
Next suppose dim W = n > 0. Set Z = q∈max W Λ q Y . By (1.15), there is a degreewise split exact sequence of complexes of flat cotorsion R-modules: The complexes X and Z are complexes of flat cotorsion R-modules with cosupport in W \ max W and max W , respectively. As dim(W \ max W ) < n and dim(max W ) = 0 < n, we may apply the inductive hypothesis to obtain decompositions X = X ′ ⊕ X ′′ and Z = Z ′ ⊕ Z ′′ , where κ(p) ⊗ R Hom R (R p , X ′ ) and κ(p) ⊗ R Hom R (R p , Z ′ ) have zero differential for every p ∈ W and X ′′ and Z ′′ are contractible. Letting π : X → X ′ be the canonical projection, there exists a complex P of flat cotorsion R-modules making the following push-out diagram commute: The snake lemma yields an exact sequence of complexes of flat cotorsion R-modules 0 → X ′′ → Y f − → P → 0; evidently, this sequence is degreewise split, and it follows from the proof of [3, Lemma 1.6] (see also [10,Proposition 2.6]) that the sequence splits in C(R) and f is a homotopy equivalence.
On the other hand, letting ι : Z ′ → Z be the canonical inclusion, we obtain a complex Q of flat cotorsion R-modules making the pull-back diagram commute: The snake lemma yields a degreewise split exact sequence 0 → Q g − → P → Z ′′ → 0 of flat cotorsion R-modules. As Z ′′ is contractible, this sequence splits in C(R) and g is a homotopy equivalence by the dual argument of the proof of [3,Lemma 1.6] (see also [10,Proposition 2.6]); let g ′ : P → Q be a splitting of g in C(R), and note that g ′ is also a homotopy equivalence. Thus we have a split exact sequence It remains to show that for every p ∈ W , the complex κ(p) ⊗ R Hom R (R p , Q) has zero differential. To do so, we use minimality of X ′ and Z ′ and the degreewise split exact sequence in the bottom row of (1.10). The modules in X ′ have cosupport contained in W \max W as X ′ is a direct summand of X; similarly the modules in Z ′ have cosupport contained in max W . If p ∈ max W , then κ(p)⊗ R Hom R (R p , X ′ ) = 0 by (1.14). This implies that and that both complexes have zero differential. If p ∈ W \ max W , then we have Hom R (R p , Z ′ ) = 0 by (1.13) and hence where both complexes have zero differential. Remark 1.11. It is known that a complex of objects in an abelian category admitting injective envelopes can be decomposed as a direct sum of a minimal complex and a contractible complex; see [26,Proposition B.2]. In our situation, however, it is not clear how the arguments of loc. cit. can be employed; indeed, flat envelopes may not exist, and although flat covers do exist over any ring [6], there exists a minimal complex of flat cotorsion modules that is not built from flat covers, see Example 2.7. This is one motivation for modelling the arguments here on that of finitely generated free modules over a local ring.
On the other hand, minimality of a complex of flat cotorsion modules with cosupport in {p} can be characterized by flat covers; see Theorem 2.3.
In the remainder of this section, we summarize several basic facts concerning flat cotorsion R-modules which are often used in this paper. Let F = q∈Spec R T q be a flat cotorsion R-module, where T q is the q-adic completion of a free R q -module. The functors LΛ p and RHom R (R p , −) are useful for working with such a module; in particular, the following hold: [43,Lemma 2.2]. Not surprisingly, the above formulas extend to bounded complexes of flat cotorsion R-modules, see also (5.2). If for each We next explain a useful reduction technique for complexes of flat cotorsion Rmodules that will be used a number of times. Let W be a subset of Spec R and Y be a complex of flat cotorsion R-modules with cosupp Y i ⊆ W . We may then write where Y i = p∈W T i p and each T i p is the p-adic completion of a free R p -module. We denote by max W the subset of W consisting of prime ideals which are maximal with respect to the inclusion relation in W . If p ∈ max W , then by (1.12). Thus, the chain map Y → p∈max W Λ p Y induced by the canonical chain maps Y → Λ p Y yields a degreewise split exact sequence:

Minimality criteria for complexes of flat cotorsion modules
We now aim to refine [43,Theorem 3.5], which gives minimality criteria for complexes of flat cotorsion modules; our approach uses tools from the previous section.
Let Y be an R-complex. There are important bi-implications about cosupport: These characterizations were essentially shown in [5]; see also [37,Proposition 4.4].
The next lemma is a version of [43, Lemma 3.1]; the proof given here instead uses the notion of cosupport. (1) f is an isomorphism.
An R-complex X is minimal if every homotopy equivalence X → X is an isomorphism in C(R); see [3]. Compare the next result to [43, Theorems 3.5 and 4.1]; conditions (2) and (4) here are new. For a homomorphism f of R-modules, denote its cokernel by coker(f ). Moreover, R p simply denotes the p-adic completion of R p . Theorem 2.3. Let Y be a complex of flat cotorsion R-modules. The following conditions are equivalent: (1) The complex Y is minimal.
(2) ⇒ (3): Fix p ∈ Spec R and set X = Hom R (R p , Y ). As X is a complex of flat cotorsion R p -modules, see (1.13), we may apply Theorem 1.8 to X to obtain a decomposition X = X ′ ⊕ X ′′ such that κ(p) ⊗ R X ′ has zero differential and X ′′ is contractible. From the canonical projection π : X → X ′ , form a push-out diagram: As in the proof of Theorem 1.8, the snake lemma yields a split exact sequence The assumption (2) now implies X ′′ = 0, thus X = X ′ . Hence it holds that is also a homotopy equivalence for every p ∈ Spec R. However, since the complex κ(p)⊗ R Hom R (R p , Y ) has zero differential, it follows that in fact κ(p) ⊗ R Hom R (R p , f ) is an isomorphism for every p ∈ Spec R. Lemma 2.2 now yields that f is an isomorphism.
The next lemma is needed for the equivalence (3) ⇔ (4) in Theorem 2.3 above; notice that its proof shows an R p -module M having a presentation by flat cotorsion modules with cosupport in {p} in fact has a resolution by such modules.
Lemma 2.5. Let p ∈ Spec R and let T 0 and T 1 be p-adic completions of free R p - Conversely, suppose that κ(p)⊗ R f = 0; this is equivalent to saying that κ(p)⊗ R g is an isomorphism. Suppose that there is a commutative diagram: By assumption, all maps in this diagram become isomorphisms upon application of κ(p) ⊗ R −, and so Lemma 1.1 (2) implies that the map T 1 → T 1 is an isomorphism. Hence it remains to show that g is a flat precover, or equivalently, ker(g) is cotorsion. To show this, we will prove that there is an exact sequence where all T i are p-adic completions of free R p -modules. Then the truncated complex · · · → T −2 → T −1 → T 0 → 0 is a resolution of ker(g), and we can easily verify that ker(g) is cotorsion, by using Remark 3.2 and (A.1).
Set K = ker(f ). The p-adic completion functor induces an isomorphism on both T 0 and T 1 , hence we obtain the following commutative diagram: By a simple diagram chase, the image of the canonical map K ∧ p → T 0 is precisely K, hence the second row is exact. Choose a surjection from a free R p -module F → K; this induces by Simon [38, Lemma p. 232] a surjection F ∧ p = T −1 → K ∧ p , hence we obtain a surjection T −1 → K. Repeating this process, we can construct an exact sequence as in (2.6).
The existence of the exact sequence (2.6) is also a consequence of a result of Dwyer and Greenlees [11,Proposition 5.2], which implies that M ∼ = LΛ p (M ) in this setting.
We end the section with an example showing that statement (4) in Theorem 2.3 is almost the best possible: Example 2.7. Let k be an uncountable field and R = k[x, y]. The minimal pureinjective resolution of R is a minimal complex of flat cotorsion R-modules of the

Functorial constructions of semi-flat-cotorsion replacements
In this section, we give two functorial ways to construct a chain map from a complex of flat modules to a complex of flat cotorsion modules such that its mapping cone is pure acyclic; recall that a complex P is pure acyclic if M ⊗ R P is acyclic for any R-module M . In particular, this approach yields a replacement of a semi-flat complex that is both semi-flat and semi-cotorsion (defined below).
Although the setting of this first construction is a bit restricted, the construction itself is not complicated; moreover, it plays a key role in Example 5.12 below.
every term of ( m∈W Λ m P )/P is a flat cotorsion module, see [16, §8.5]. See also Remark 3.6. Regarding the above sequence as a double complex, denote its totalization by X P . The rows of this double complex are pure exact, that is, they are exact upon application of M ⊗ R − for any R-module M . A basic argument [25,Theorem 12.5.4] of double complexes shows that X P is pure acyclic. On the other hand, there is a commutative diagram where all arrows express the canonical chain maps. Regarding both rows as double complexes, this morphism between double complexes naturally induces a chain map P → Y P , where Y P denotes the total complex of the second row. The mapping cone of the canonical map P → Y P can be identified with X P , hence P → Y P is a quasi-isomorphism with pure acyclic mapping cone. Moreover, Y P is a complex of flat cotorsion R-modules.
A complex P is semi-projective if Hom R (P, −) preserves acyclicity and P i is projective for every i ∈ Z; a complex F is semi-flat if − ⊗ R F preserves acyclicity and F i is flat for every i ∈ Z. Semi-projective complexes and pure acyclic complexes of flat modules are both semi-flat. It follows that if P is semi-projective in the above construction, then Y P is semi-flat.
A complex C is semi-cotorsion if Hom R (−, C) preserves acyclicity of pure acyclic complexes of flat modules and C i is cotorsion for every i ∈ Z (see Appendix A). By the construction, Y P consists of flat cotorsion R-modules. The next remark shows that Y P is also semi-cotorsion.
Remark 3.2. Let p ∈ Spec R. As R is noetherian, Λ p is left adjoint to the inclusion of p-adically complete modules into Mod R (this follows from [38, Theorem 1.1] which states that Λ p is idempotent, see also [40, §2.2.5]); in addition, the functor R p ⊗ R − is left adjoint to the inclusion of p-local modules into Mod R. Hence, if M is any R-module and T p is the p-adic completion of a free R p -module, then Let T be a complex of p-adic completions of free R p -modules. For any pure acyclic complex X of flat R-modules, we have Hom R (X, T ) ∼ = Hom R (Λ p (X p ), T ), and so Hom R (X, T ) is acyclic because Λ p (X p ) is contractible. To see this, we only need to notice that all cycle modules of X are flat, and Λ p (R p ⊗ R −) assigns a short exact sequence of flat R-modules to a split short exact sequence of flat cotorsion R-modules, see Remark 1.2. Therefore T is semi-cotorsion.
Let W be a subset of Spec R with dim W < ∞ and let Y be a complex of flat cotorsion R-modules with cosupp Y i ⊆ W for every i ∈ Z. Then we can easily show that Y is semi-cotorsion by an inductive argument on dim W , using the above fact, (1.15), and (1.16). In particular, it follows that all complexes of flat cotorsion R-modules are semi-cotorsion when dim R < ∞.
We define a complex Y to be semi-flat-cotorsion if it is both semi-flat and semicotorsion. The above remark shows that any semi-flat complex of flat cotorsion R-modules is semi-flat-cotorsion as long as R has finite Krull dimension. 2 In particular, if P is assumed to be semi-projective (or semi-flat) in Construction 3.1, then the complex Y P constructed therein is semi-flat-cotorsion.
Assume dim R < ∞. We now aim to give the construction of a functor from [33], which in particular sends semi-flat R-complexes to semi-flat-cotorsion ones. If W is a subset of Spec R with dim W = 0, then writeλ W = p∈W Λ p (R p ⊗ R −). There is a canonical morphism id C(R) →λ W ; see [33,Notation 7.1]. For a non-empty subset W of Spec R, a family of subsets W = {W i } 0≤i≤n is a system of slices of W if W = 0≤i≤n W i , the intersections W i ∩ W j are empty for i = j, dim W i = 0 for 0 ≤ i ≤ n, and W i is specialization-closed in W ; see [33,Definition 7.6].
For an R-complex X, we naturally get a double complex L W X, and the canonical chain maps X →λ Wi X induce a morphism X → L W X of double complexes. Totalization yields a chain map ℓ W X : X → tot L W X. Set λ W = tot L W , as in [30]; we see that λ W is a functor on C(R) and there is a natural transformation ℓ W : id C(R) → λ W .
If M is an R-module, then λ W M = L W M . If F is a flat R-module, then the R-moduleλ Wi F = p∈Wi Λ p F p is flat cotorsion, see Remark 1.2; thus if X is a complex of flat R-modules, then λ W X is a complex of flat cotorsion R-modules. 2 In fact, it follows from [41] or [4] that every complex of flat cotorsion modules is semi-cotorsion without any additional assumptions on the ring, and so every semi-flat complex of flat cotorsion modules is semi-flat-cotorsion; see Lemma A.7. We provide the more elementary observation above for the reader's convenience.
Assume now that W = Spec R, so d = n. For each flat R-module X i , it follows from [33,Corollary 7.12] that ℓ W X i : X i → λ W X i is a quasi-isomorphism; we give a more elementary proof of this in Fact 3.5 below. Moreover, cone(ℓ W X) is the totalization of the double complex whose rows are pure exact, and so the totalization cone(ℓ W X) is pure acyclic; see for example [25,Theorem 12.5.4]. It then follows that λ W assigns any semi-flat complex to a semi-flat complex of flat cotorsion R-modules (cf. [33, Remark 7.13]), that is, a semi-flat-cotorsion complex per Remark 3.2.
A semi-flat-cotorsion replacement of an R-complex X is an isomorphism in D(R) between X and a semi-flat-cotorsion R-complex; see Definition A.4. Proof. Let X be an R-complex with semi-flat resolution F → X. Construction 3.3 yields a semi-flat-cotorsion replacement F − → λ W F . By Corollary 2.4, the complex In the precedent work [33], theČech complex L W naturally appeared as a consequence of the (generalized) Mayer-Vietoris triangles [33,Theorem 3.15]. For the reader's convenience, we provide an alternative proof of the following fact from [33], which we used in Construction 3.3.
Fact 3.5. Assume dim R < ∞ and let W be a system of slices of Spec R. If F is a flat R-module, then the map ℓ W F : F → λ W F is a quasi-isomorphism. In particular, the mapping cone of ℓ W F is a pure acyclic complex of flat R-modules.
Proof. Set C = cone(ℓ W F ), which by definition is of the following form: The map ℓ W F is a quasi-isomorphism if and only if C is acyclic, and so it will be enough to show supp R C = ∅; see [17, Lemma 2.6]. The statement will then follow from the next more general claim, by setting W = Spec R: Claim: Let W be a non-empty subset of Spec R with dim W = n < ∞ and let W = {W i } 0≤i≤n be a system of slices of W . If F is a flat R-module, then we have Proof of Claim: We proceed by induction on n. If n = 0, then C is the complex 0 → F →λ W F → 0. As R/p ⊗ R − commutes with the direct product, it follows that κ(p) ⊗ R C is acyclic for any p ∈ W , hence W ∩ supp R C = ∅. Next, suppose that n > 0. Set U = 1≤i≤n W i and U i = W i+1 . We obtain a system of slices U = {U i } 0≤i≤n−1 of U , which yields aČech complex L U as in Construction 3.3. Set C ′ = cone(ℓ U F ) and C ′′ = cone(ℓ UλW0 F ). The canonical map F →λ W0 F between flat modules induces a chain map C ′ → C ′′ between mapping cones; here the first row is C ′ and the second row is C ′′ : If we regard the above diagram as a double complex, then its total complex is C. Thus to show that W ∩ supp R C = ∅, it is enough to justify: (i) If p ∈ U , then κ(p) ⊗ R C ′ and κ(p) ⊗ R C ′′ are acyclic.
As dim U = n − 1, the inductive hypothesis implies (i). For p ∈ W 0 , application of κ(p)⊗ R − to the above diagram leaves only the left column nonzero, which becomes an isomorphism by the argument for the n = 0 case above, thus (ii) also holds.
The construction of cone(ℓ W F ) as a totalization of a double complex above is just an analogue of the corresponding construction of classic (extended)Čech complexes: For a sequence x 1 , ..., x n ∈ R, theČech complexČ(x 1 , ..., x n ) (see [7, §5.1]) is naturally isomorphic toČ(x 1 , ..., x n−1 ) ⊗ RČ (x n ). Note however, that one must be a bit cautious: If X is an R-complex of finitely generated modules, then (L W R) ⊗ R X ∼ = λ W X, see [33, (8.4)], but this isomorphism need not hold for an arbitrary R-complex X. Moreover,λ WjλWi need not be isomorphic toλ WiλWj . Remark 3.6. If dim R < ∞, then the minimal pure-injective resolution of a flat module, constructed as in Constructed 3.3 and using Corollary 2.4, implies immediately that the pure-injective dimension of any flat R-module is at most dim R, see also [16,Corolary 8.5.12]. Recall that Construction 3.1 uses this fact under the assumption dim R ≤ 1; this case is enough for one of the main aims in Section 5, see Example 5.12.
On the other hand, Construction 3.1 also treats any countable ring R with dim R < ∞, as the pure-injective dimension of any flat R-module is at most 1. It is possible to deduce this fact by using Theorem 2.3 and [24, Lemma 2].
The next example shows the necessity for considering not semi-flat-cotorsion resolutions but semi-flat-cotorsion replacements. Indeed, semi-flat-cotorsion replacements always exist over any ring. This is almost directly deduced from Gillespie's work [19], which shows that pure acyclic complexes of flat modules and semi-cotorsion complexes form a complete cotorsion pair; see Theorem A.5. However, we do not know whether minimal ones can be always obtained as in Theorem 3.4, see Question A. 10. In addition, the constructions here yield additional information about the structure of semi-flat-cotorsion replacements, which we take advantage of in the next section.

Structure of semi-flat-cotorsion replacements and finitistic dimensions
The goal of this section is to describe the structure of semi-flat-cotorsion replacements using the construction of the functor λ W in Section 3, and give applications of this structure to finitistic flat and projective dimensions. If dim R = d < ∞, we set W i = {p ∈ Spec R | dim R/p = i}, and notice that W = {W i } 0≤i≤d is a system of slices for Spec R; see Section 3. In this setting, the functor λ W is now defined as in Construction 3.3; it assigns semi-flat complexes to semi-flat-cotorsion complexes.
X is a complex of flat R-modules with X i = 0 for i > 0, then λ W X has the form: where each T n p is the p-adic completion of a free R p -module. Moreover, if n ∈ Z and X i = 0 for i < n, then (λ W X) i = 0 for i < n.
Proof. This is a direct consequence of the construction of λ W .
If (R, m, k) is local, X is an R-complex that is isomorphic in D(R) to a bounded complex of flat R-modules (in particular, fd R X < ∞) and H(X) = 0, then the following is a version of the Auslander-Buchsbaum formula: This is a special case of the generalization given by Foxby and Iyengar [18,Theorem 2.4]. See loc. cit. for a number of characterizations of depth.
If p ∈ Spec R and Y is a bounded complex of flat cotorsion R-modules with H(Y ) = 0, then an immediate consequence of (1.13) and (4.2) is an equality: Recall that R p stands for the p-adic completion of R p . The first author noticed the formulation of the next lemma through a collaboration with Takahashi and Yassemi [32].
Lemma 4.4. Assume dim R < ∞ and let p ∈ Spec R. Let X be an R-complex that is isomorphic in D(R) to a bounded complex of flat R-modules and satisfies H(X) = 0. If Y is a minimal semi-flat-cotorsion replacement of X, then Proof. Let Y be a minimal semi-flat-cotorsion replacement of X; notice that Y is a bounded complex of flat cotorsion R-modules, by Corollary 2.4, Lemma 4.1 and Proposition A.8. By Theorem 2.3, the complex κ(p) ⊗ L R RHom R (R p , Y ) has zero differential for every p ∈ Spec R, and so The claim now follows from (4.3).
Let X be an R-complex. The flat dimension of X is defined as In the next theorem, we simply write depth R p for depth Rp R p . Theorem 4.5. Assume dim R = d < ∞. If M is an R-module with fd R M < ∞, then the minimal semi-flat-cotorsion replacement of M has the following form: This result specializes to give the structure of a minimal pure-injective resolution of a flat module shown in [14, Theorem 2.1] provided dim R < ∞. Also (perhaps unsurprisingly) this implies that the finitistic flat dimension of R, defined to be sup{fd R M | M is an R-module with fd R M < ∞}, is at most dim R; this was shown by Auslander and Buchsbaum [1, Theorem 2.4]: Corollary 4.6 (Auslander and Buchsbaum). The finitistic flat dimension of R is at most dim R.
Proof of Theorem 4.5. Let M be an R-module with fd R M < ∞. The module M has a minimal semi-flat-cotorsion replacement Y ; moreover, it is isomorphic to a direct summand of the one in Lemma 4.1 by Corollary 2.4 and Proposition A.8, hence Y is bounded. Fix p ∈ Spec R and set T = Λ p Hom R (R p , Y ). It now follows from Lemma 4.4, because depth Rp RHom R (R p , M ) ≥ 0, that T −i = 0 for i > depth Rp R p as desired. It remains to justify the equation for B n p . Since Y is a bounded complex of flat cotorsion modules, we have κ(p) (1.14), and minimality of Y implies that this complex has zero differential by Theorem 2.3. Thus the κ(p)-dimension of the nth cohomology module of this complex coincides with B n p . Compare the next result with [9, Corollary 5.9].
Theorem 4.7. Assume dim R < ∞. If X is an R-complex that is isomorphic in D(R) to a bounded complex of flat R-modules, then Proof. If H(X) = 0, then fd R X = −∞ and depth Rp RHom R (R p , X) = ∞ for each p ∈ Spec R. Hence the above equality holds. Suppose that H(X) = 0. Set n = fd R X and let F be a semi-flat complex isomorphic to X in D(R) and satisfying both F i = 0 for i < −n and F i = 0 for i ≫ 0. We obtain from Lemma 4.1 that the semi-flat-cotorsion replacement λ W F of F satisfies (λ W F ) i = 0 for i < −n. In other words, we have −n ≤ inf{i | (λ W F ) i = 0}. Further, we can by Corollary 2.4 find a minimal semi-flat-cotorsion replacement Y of X as a direct summand of λ W F . Then we have −n ≤ inf{i | Y i = 0}, and this must be an equality since fd R X = n. It then holds that where the last equality follows from Lemma 4.4.
We end the section by recovering two classic facts: First, that the finitistic projective dimension of R is at most dim R-this was originally proven by Gruson   Proof. We may assume d = dim R is finite. Let M be an R-module with fd R M < ∞ and let N be any R-module. It is sufficient to show Ext i R (M, N ) = 0 for all i > d. Take a projective resolution P of N and set Y = λ W P ; this is a semi-flat-cotorsion replacement of N . Our goal is to show that H i (RHom R (M, Y )) = 0 for i > d.
As above, set W i = {p ∈ Spec R | dim R/p = i}. By iteration of (1.15), we can make a sequence of subcomplexes where each quotient complex Y i /Y i+1 is a complex of flat cotorsion modules with cosupport in W i ; that is, Y i /Y i+1 is isomorphic to p∈Wi T (p), where for each prime p ∈ Spec R, we have T (p) = Λ p Hom R (R p , Y ) is a complex of flat cotorsion modules with cosupport in {p}, see (1.16). Thus it is enough to show that H i (RHom R (M, T (p))) = 0 for i > d and p ∈ Spec R. Note that T (p) i = 0 for i > dim R/p by Lemma 4.1.
Now, since T (p) is a complex of R p -modules, we have By Corollary 4.6, the module M p has a flat resolution F over R p such that F i = 0 for i < − dim R p . Since F is semi-flat (over R) and T (p) is semi-cotorsion by Remark 3.2, we have RHom R (F, T (p)) ∼ = Hom R (F, T (p)), see (A.1). Combining these two isomorphisms, we obtain

Cosupport: a refinement, correction, and counterexample
The goal of this section is to examine the relationship between the cosupport of an R-complex and the prime ideals "appearing" in a minimal semi-flat-cotorsion replacement; in particular, we seek to correct and improve [42, Theorem 2.7]. As minimal semi-flat-cotorsion replacements exist at least for rings of finite Krull dimension, we will compare the cosupport of a minimal complex Y of flat cotorsion R-modules to the set i∈Z cosupp R Y i , which can be thought of as the prime ideals appearing in Y . We begin with a lemma showing one containment always holds: Lemma 5.1. Let Y be a complex of flat cotorsion R-modules. The inclusion holds: This is proved in [33, Proposition 6.3] under several conditions; for example, if dim R < ∞, or if Y i = 0 for i ≪ 0, then loc. cit. implies the above inclusion. The result above follows in general from a result ofŠtovíček [41,Theorem 5.4] that says any complex of flat cotorsion modules over any ring is semi-cotorsion; see also Bazzoni, Cortés Izurdiaga, and Estrada [4, Theorem 0.3]. For p ∈ Spec R and a complex Y of flat cotorsion R-modules, we thus have the second isomorphism follows by [41,Theorem 5.4] (or Remark 3.2 if dim R < ∞) and (A.1), see [33,Proposition 2.5] for the first isomorphism.
Proof of Lemma 5.1. By (5.2), we obtain the next isomorphisms: Hence, by (1.12), (1.13), and (2.1), if p ∈ cosupp R Y then p ∈ i∈Z cosupp R Y i . Let X be an R-complex. If H i (X) = 0 for i ≪ 0 and I is a minimal semiinjective resolution of X, then supp R X = i∈Z supp R I i . However, Chen and Iyengar provide in [8] an example of an unbounded R-complex X whose minimal semi-injective resolution I satisfies supp R X i∈Z supp R I i . Similarly, there exists a complex X with minimal semi-flat-cotorsion replacement Y such that cosupp R X i∈Z cosupp R Y i , see Example 5.12. In particular, this yields a counterexample to the statement of [42,Theorem 2.7]; indeed, the argument in [42, p. 257, l. 5] is incorrect. The author of that paper sincerely apologizes for his mistake. To salvage this result, we prove in Theorem 5.4 a correction (and improvement) to [42,Theorem 2.7]. In particular, our correction is sufficient to verify [42,Theorem 4.6].
Theorem 5.4. Let Y be a minimal complex of flat cotorsion R-modules. Suppose that one of the following conditions holds: (1) Hom R (R p , Y ) is semi-flat for all p ∈ Spec R; (2) Y i = 0 for i ≪ 0. Then one has an equality: The condition (2) is the same as the original statement. To salvage this case, we need the next lemma; it essentially follows from a result of Auslander and Buchsbaum [1] on finitistic flat dimension; this was reproved in Corollary 4.6 above.
Lemma 5.6. Let Y be a minimal complex of flat cotorsion R-modules. Assume that Y is acyclic and Y i = 0 for i ≪ 0. Then Y i = 0 for all i ∈ Z.
Proof. Suppose that Y = 0 in C(R) and deduce a contradiction. We can take a prime ideal p ∈ i∈Z cosupp R Y i . Since Y is acyclic, so is Λ p Hom R (R p , Y ) by (5.3); this complex satisfies the same condition as Y , and so we may replace Y by Λ p Hom R (R p , Y ). Thus we may assume Y is a complex of flat cotorsion modules with cosupport in {p}, hence consists of R p -modules. Moreover, without loss of generality, we may assume that Y i = 0 for i < 0 and Y 0 = 0. Now, fix an integer n > dim R p and consider the truncation Minimality of Y implies that κ(p) ⊗ Rp Y has zero differential, and hence κ(p) ⊗ Rp Y ′ has zero differential as well. Thus Tor Rp n (κ(p), C) ∼ = κ(p) ⊗ Rp Y 0 = 0. This implies that dim R p < n ≤ fd Rp C < ∞, contradicting Corollary 4.6.
Proof of Theorem 5.4. Let p ∈ Spec R, and assume that p / ∈ cosupp R Y . By Lemma 5.1, we only have to show that p / ∈ i∈Z cosupp R Y i . Suppose that (1) holds. We then have by (5.2) that . Minimality of Y implies that κ(p)⊗ R Hom R (R p , Y ) has zero differential by Theorem 2.3. In addition, κ(p) ⊗ R Hom R (R p , Y ) is acyclic by (2.1) since p / ∈ cosupp R Y . It follows that κ(p) ⊗ R Hom R (R p , Y ) = 0 as a complex. Hence p / ∈ i∈Z cosupp R Y i by (1.14).
Next suppose that (2) holds. By (2.1) and (5.3), the complex Λ p Hom R (R p , Y ) is an acyclic complex of flat cotorsion modules. Further, as Y i = 0 for i ≪ 0, we also have ( Remark 5.7. Using finitistic injective dimension it can also be shown that if I is a minimal R-complex of injective modules with I i = 0 for i ≫ 0, then there is an equality supp R I = i∈Z supp R I i . Compare this with [8, Proposition 2.1].
Our next task is to give an example of an R-complex X whose minimal semiflat-cotorsion replacement Y satisfies cosupp R X i∈Z cosupp R Y i . Although our example is analogous to [8,Proposition 2.7], a key role is played by Construction 3.1 and the following result which gives a condition for this construction to yield a minimal complex.
Proposition 5.8. Assume dim R ≤ 1, or R is countable and dim R < ∞. If P is a complex of projective R-modules such that R/p ⊗ R P has zero differential for every minimal prime p, then the complex Y P in Construction 3.1 is minimal.
First, for a prime ideal p, a finitely generated R-module M , and a flat cotorsion R-module F , we remark that there is a canonical isomorphism To see this, write F = q∈Spec R T q , where T q is a flat cotorsion module with cosupport in {q}. As M is finitely presented, the tensor product M ⊗ R − commutes with arbitrary direct products. Hence it is enough to show, for each prime ideal q, that M ⊗ R Hom R (R p , T q ) ∼ = Hom R (R p , M ⊗ R T q ). One can deduce this by (1.6).
Proof of Proposition 5.8. It is enough to show that κ(q)⊗ R Hom R (R q , Y P ) has zero differential for every prime ideal q of R, by Theorem 2.3. Denote by W the set of maximal ideals of R; let n ∈ W . Application of Λ n to the exact sequence 0 / / P / / m∈W Λ m P / / ( m∈W Λ m P )/P / / 0 preserves exactness per [33, Proposition 2.5], hence sends the map P → m∈W Λ m P to an isomorphism, see (1.12). It follows that the complex C = ( m∈W Λ m P )/P consists of flat cotorsion modules with cosupport in (Spec R) \ W , so by (1.14), Now fix q ∈ Spec R. Application of R/q ⊗ R − to the canonical chain map m∈W Λ m P → C yields a surjective chain map R/q ⊗ R m∈W Λ m P / / R/q ⊗ R C . Moreover, it holds that see [33,Lemma 2.3] for the second isomorphism. Taking a minimal prime ideal p with p ⊆ q, we have R/q ⊗ R P ∼ = R/q ⊗ R R/p ⊗ R P , therefore the assumption on P implies that Λ m (R/q ⊗ R P ) has zero differential, and so does m∈W Λ m (R/q ⊗ R P ). Thus both complexes appearing in (5.11) have zero differential.
To complete the proof, apply Hom R (R q , −) to (5.11): where the both complexes have zero differential. Regarding this chain map as a double complex, we see from (5.9) that its total complex is nothing but the complex κ(q) ⊗ R Hom R (R q , Y P ). Going back to (5.10), we conclude that Y P is minimal.
We now give a counterexample to [42,Theorem 2.7].
Example 5.12. Let k be a field and R = k[[x, y]]/(x 2 ). Set m = (x, y) and p = (x). Let M = R/(x). As R is m-adically complete, R is flat cotorsion hence the complex is semi-flat, one obtains that P is also semi-flat.
The differential of P is given by multiplication by x, hence R/p ⊗ R P has zero differential. Construction 3.1 applied to the complex P yields a quasi-isomorphism P → Y P with pure acyclic mapping cone, where Y P is semi-flat-cotorsion by construction and minimal by Proposition 5.8. Furthermore, for each n ∈ Z, one has P n ∼ = N R. Since dim R > 0, the direct sum N R is not isomorphic to its m-adic completion Λ m ( N R). Thus the quotient module Λ m ( N R)/( N R) is non-trivial, and we see from the proof of Proposition 5.8 that this is a flat cotorsion module with cosupport in {p}. Therefore, the minimal semi-flat-cotorsion replacement Y P of X contains non-trivial flat cotorsion modules with cosupport in both {m} and {p}. In other words, i∈Z cosupp R Y i P = {p, m}. However, we claim cosupp R X = {m}. Indeed, there is an isomorphism of complexes X ∼ = i∈Z M [i], and hence cosupp R X = i∈Z cosupp R M [i] = {m}. Consequently we have On the other hand, note that by (2.1), since p ∈ cosupp R X, the complex LΛ p RHom R (R p , X) is acyclic; however, by (5.3), so the complex Hom R (R p , Y P ) is a nonzero acyclic complex of flat cotorsion modules. Furthermore, the complex Hom R (R p , Y P ) is not semi-flat: if it were, it would be contractible by Remark 3.2, hence zero by minimality of Y P , a contradiction.
Appendix A. Semi-flat-cotorsion replacements for associative rings In this appendix, let A be an associative ring with identity. Here, left A-modules and complexes of left A-modules are simply referred to as A-modules and A-complexes, respectively. Let K(A) denote the homotopy category of A-complexes and let D(A) denote the derived category over A. We first recall what it means for an A-complex X to be semi-projective, semi-injective, or semi-flat; 4 these have assumptions on the components of X in addition to being K-projective, K-injective, or K-flat in the sense of Spaltenstein [39].
• X is semi-projective if Hom A (X, −) preserves acyclicity and X i is projective for every i ∈ Z. • X is semi-injective if Hom A (−, X) preserves acyclicity and X i is injective for every i ∈ Z. • X is semi-flat if − ⊗ A X preserves acyclicity and X i is flat for every i ∈ Z. There is a natural corresponding notion-see Enochs and García Rozas [15, Definition 3.3 and Proposition 3.4]-for a complex of cotorsion A-modules as well: an A-complex X is semi-cotorsion if Hom A (−, X) preserves acyclicity of pure acyclic complexes of flat A-modules and X i is cotorsion for every i ∈ Z; recall that a complex of flat A-modules is pure acyclic if and only if it is acyclic and semi-flat.
For a semi-flat A-complex F and semi-cotorsion A-complex C, we have an isomorphism this follows by noting that the mapping cone of a semi-projective resolution P → F is pure acyclic. It then follows from (A.1) that In particular, a morphism in D(A) between A-complexes that are both semi-flat and semi-cotorsion can be realized by a morphism in K(A). This naturally leads us to make the next definition: Definition A.3. An A-complex X is semi-flat-cotorsion if − ⊗ A X preserves acyclicity, Hom A (−, X) preserves acyclicity of pure acyclic complexes of flat modules, and X i is flat cotorsion for every i ∈ Z.
In other words, an A-complex is semi-flat-cotorsion if and only if it is semi-flat and semi-cotorsion.
Although every complex has a semi-flat resolution, not every complex has a semi-flat-cotorsion resolution (see Example 3.7); instead, we consider the following natural notion: Definition A.4. A semi-flat-cotorsion replacement of an A-complex X is an isomorphism in D(A) between X and a semi-flat-cotorsion A-complex.
The next result is due to Gillespie [19].
Theorem A.5. Every A-complex X has a semi-flat-cotorsion replacement.
Proof. Let F − → X be a semi-flat resolution. A result of Gillespie [19,Corollary 4.10] states 5 that the pair of pure acyclic complexes of flat modules and semi-cotorsion complexes forms a complete cotorsion pair on the category of A-complexes; in particular, this implies there is an exact sequence of A-complexes, where Y is semi-cotorsion and P is a pure acyclic complex of flat modules. The complex P is semi-flat, hence Y is semi-flat as well. It now follows that Y is a semi-flat-cotorsion replacement of X.
Analogous to the role of semi-injective complexes, (A.2) and Theorem A.5 show that semi-flat-cotorsion complexes also describe the derived category: Corollary A.6. The homotopy category of semi-flat-cotorsion A-complexes is equivalent to D(A).
Stovíček [41,Theorem 5.4] shows that every complex of flat cotorsion A-modules is semi-cotorsion; in fact, a recent result of Bazzoni, Cortés Izurdiaga, and Estrada shows that every complex of cotorsion A-modules is semi-cotorsion [4, Theorem 0.3]. We now have the following characterization: Lemma A.7. An A-complex X is semi-flat-cotorsion if and only if X is semi-flat and X i is cotorsion for every i ∈ Z.
A complex X of flat cotorsion A-modules such that X i = 0 for i ≫ 0 is semiflat-cotorsion. This follows from Lemma A.7.
The next proposition follows from (A.2) and the definition of minimality.
Proposition A.8. Let X and Y be minimal semi-flat-cotorsion complexes that are isomorphic in D(A). Then X ∼ = Y in C(A).
In particular, the zero complex is the only acyclic minimal semi-flat-cotorsion complex.
Remark A.9. Another important role is played by complexes of flat cotorsion modules; they describe the pure derived category of flat modules, as defined by Murfet and Salarian [29], whose work was motivated by Neeman [35]. The pure derived category is defined as the Verdier quotient of the homotopy category of complexes of flat A-modules by the subcategory of pure acyclic complexes. Gillespie's result [19,Corollary 4.10] implies that the pure derived category may be identified with a subcategory of the homotopy category of complexes of flat cotorsion A-modules, see also [20, Theorem 2.6 and Lemma 5.1]. If any flat A-module has finite projective dimension, as over a commutative noetherian ring of finite Krull dimension, then it is not hard to see (without using Lemma A.7) that any pure acyclic complex of flat cotorsion A-modules is contractible; this assumption implies that the pure derived category coincides with the homotopy category of complexes of flat cotorsion A-modules. Furthermore,Štovíček's [41,Corollary 5.8] implies that these two categories are equivalent over any ring.
By Corollary 2.4, if A is a commutative noetherian ring of finite Krull dimension, then each complex of flat cotorsion A-modules decomposes as a direct sum of a minimal one and a contractible one. However, we do not know whether this holds in general.
Question A. 10. Let Y be a complex of flat cotorsion A-modules. Does Y decompose as a direct sum Y ′ ⊕ Y ′′ , where Y ′ is minimal and Y ′′ is contractible?
For special complexes, this question has an affirmative answer.
Proposition A.11. Let F be a complex of flat A-modules and assume the canonical surjection F i → coker(d i+1 F ) is a flat precover for all i ∈ Z. Then F = F ′ ⊕ F ′′ , where F ′ is minimal and F ′′ is contractible.
In particular, if Y is an acyclic complex of flat cotorsion A-modules, then we have Y = Y ′ ⊕ Y ′′ , where Y ′ is minimal and Y ′′ is contractible.
Proof. One can use a method similar to [26, Appendix B] to find a decomposition F = F ′ ⊕ F ′′ where F ′′ is contractible and all differentials of F ′ are given by flat covers. It follows from [43,Theorem 4.1] that F ′ is minimal. The second assertion follows from the first and [41,Theorem 5.4] or [4,Theorem 0.3].
It follows that if A is a left perfect ring, in which case every left A-module is cotorsion and hence any surjection from a flat A-module is a flat (or projective) precover, one obtains an affirmative answer to Question A.10.
Complementary to the last statement of Proposition A.11, one may also consider a restriction of Question A.10 to the case of semi-flat-cotorsion complexes. Solving the restricted question is thus equivalent to showing the existence of minimal semi-flat-cotorsion replacements for all complexes, by (A.2), Theorem A.5, and the definition of minimality.