ON THE RELATION BETWEEN K-FLATNESS AND K-PROJECTIVITY

IOANNIS EMMANOUIL

Abstract. Krause [19] has proved that the homotopy category K(R-PureProj) of pure projective modules over an associative ring R is compactly generated and equivalent to the Verdier quotient K(R)/T(R), where T(R) is the triangulated subcategory of the homotopy category K(R) consisting of the pure acyclic complexes. We provide an alternative proof of this result. When restricted to the homotopy category of projective modules, this equivalence reduces to that obtained by Neeman [26]. As an application of this description of K(R-PureProj), we show that the K-flat complexes defined by Spaltenstein [28] are, up to homotopy equivalence, the filtered colimits of K-projective complexes of pure projective modules.

0. Introduction

The concepts of projective, injective and flat modules are of fundamental importance in the study of the homological properties of modules. The passage from the category of modules to that of complexes of modules demands for suitable replacements of these fundamental notions. The categorical definitions of projective and injective complexes are too rigid to provide the proper substitutes. It turns out that the projective complexes are precisely the contractible complexes of projective modules and hence, given a non-acyclic complex X, there is no quasiisomorphism P ⟶ X from a projective complex P. On the other hand, if the complex X is right-bounded, there is a standard technique of obtaining a quasi-isomorphism P ⟶ X, where P is a right-bounded complex of projective modules. Moreover, such resolutions are unique up to homotopy equivalence. Dual results apply to left-bounded complexes and resolutions of these by left-bounded complexes of injective modules.

In the applications of the theory, one is often confronted with problems which can be modeled using unbounded (on both sides) complexes. For example, the study of group actions on homotopy spheres turns out to be closely related to periodicity phenomena in the cohomology of the group; cf. [1], [7], [21], [30]. These phenomena are better understood in terms of the generalized Tate cohomology groups, which are themselves often computed by means of certain unbounded complexes of projective modules, the so-called complete projective resolutions [29]. Such complexes are also of fundamental importance in Gorenstein homological algebra, which is the relative homological theory based on the Gorenstein projective and Gorenstein injective modules. This theory originates from the work of Auslander and Bridger [2] on the Gdimension of finitely generated modules over commutative Noetherian rings and was extended by Enochs and Jenda [9], who removed all finiteness conditions and defined the Gorenstein projective and Gorenstein injective modules; see also [3] and [12]. These notions have found important applications in representation theory and algebraic geometry. Other examples that show the importance of unbounded complexes in algebraic geometry, include the extension by Krause [18] of Orlov’s result on the dependence of the bounded derived category of a scheme only upon its singular locus [27] to the full subcategory of all acyclic complexes of injective sheaves, as well as the approach to Grothendieck duality for schemes that admit a dualizing complex by Iyengar and Krause [15], Neeman [26] and Murfet [22], which is expressed as a certain equivalence of triangulated categories of unbounded complexes of sheaves.

Going back to the general discussion in the beginning, it appears that the proper generalization of projectivity and injectivity for unbounded complexes of modules over a ring R consists of the notions introduced by Spaltenstein in [28]. A complex P is called K-projective if the complex of abelian groups Hom_{R}(P, C) is acyclic for any acyclic complex C, i.e. if P is in the orthogonal ^{⊥}K_{ac}(R) of the acyclic complexes K_{ac}(R) in the homotopy category K(R) of all unbounded complexes. Any complex X admits a quasi-isomorphism P ⟶ X from a K-projective complex of projective modules P, which is unique up to homotopy equivalence. Moreover, any right-bounded complex of projective modules is K-projective. Dually, one defines the notion of a K-injective complex and obtains analogous results. A complex F is called K-flat if the complex of abelian groups L ⊗_{R} F is acyclic for any acyclic complex of right modules L. All right-bounded complexes of flat modules and all K-projective complexes are K-flat. Hence, for any complex X there exists a quasi-isomorphism F ⟶ X from a K-flat complex of flat modules F. Nevertheless, these resolutions are not unique up to homotopy equivalence. To remedy this defect, one has to pass to the quotient K(R-Flat)/S(R), where S(R) is the triangulated subcategory of the homotopy category K(R-Flat) of flat modules, which consists of the acyclic and K-flat complexes of flat modules; this is explained in [22]. The complexes in S(R) have been studied in [10], [23] and [26]; as shown by Neeman, these are precisely the complexes in the relative right-orthogonal of the homootpy category K(R-Proj) of projective modules inside K(R-Flat). Neeman proved that the embedding of K(R-Proj) in K(R-Flat) has a right adjoint and concluded that the composition

(1)  K(R-Proj) ↪ K(R-Flat) ⟶ K(R-Flat)/S(R)

is an equivalence of categories.

The (not necessarily acyclic) K-flat complexes of flat modules are studied in [5], whereas the acyclic and K-flat (but not necessarily flat) complexes are the pure acyclic complexes studied in [8]. Our goal in this paper is to examine the K-flat complexes (without imposing any additional assumption on them). The class of these complexes contains all K-projective complexes and is closed under filtered colimits and homotopy equivalence. It follows that any complex which is homotopy equivalent to a filtered colimit of K-projective complexes is K-flat. Conversely, we show that: Any K-flat complex is homotopy equivalent to a filtered colimit of K-projective complexes. This result is reminiscent of the description by Lazard [20] and Govorov [11] of flat modules as the filtered colimits of projective modules. It is of some interest apart from its macroscopic resemblance to the relation between flat and projective modules, since a K-projective (resp. K-flat) complex is not necessarily a complex of projective (resp. flat) modules.

Our main technical tool, in order to prove the result above, involves the notion of purity and the associated class of pure projective modules. Purity was introduced by Cohn [6]; it is closely related to flatness and has been successfully used in order to prove the flat cover conjecture [4]. It is the basis of a relative homological theory, which has been studied extensively in the literature. The reader may consult [17], [31] and the monograph [16] for more details. Krause [19] has proved that the homotopy category K(R-PureProj) of pure projective modules is always compactly generated. We provide an alternative proof of this result in the sequel. In the special case where R has finite pure global dimension, Krause’s result was obtained by Holm and Jørgensen in [13]. We also note that certain properties of the full subcategory of K(R-PureProj), which consists of the right-bounded complexes of pure projective modules, are examined in [32]. Krause’s result should be contrasted with the case of the homotopy category K(R-Proj) of projective modules, which is not always compactly generated; cf. [26]. Brown representability [24] implies that the embedding ı of K(R-PureProj) in K(R) admits a right adjoint ı^{∗}. When restricted to the homotopy category K(R-Flat) of flat modules, ı^{∗} coincides with the right adjoint of the embedding K(R-Proj) ↪ K(R-Flat), obtained by Neeman. The kernel T(R) = K(R-PureProj)^{⊥} of ı^{∗} is the triangulated subcategory consisting of the acyclic K-flat complexes (i.e. of the pure acyclic complexes [8]) and the composition

K(R-PureProj) ↪ K(R) ⟶ K(R)/T(R)

is an equivalence of categories. When restricted to the homotopy category K(R-Proj) of projective modules, this equivalence reduces to Neeman’s equivalence (1). The Verdier quotient K(R)/T(R) seems to provide the proper setting for studying K-flat complexes. Indeed, any K-flat complex is isomorphic to a K-projective complex therein and hence the K-flat complexes are the complexes which are left-orthogonal to the class of acyclic complexes in K(R)/T(R). It turns out that a complex of pure projective modules is K-projective if it is left-orthogonal to the class of all acyclic complexes of pure projective modules in K(R). We conclude that:

A complex of pure projective modules is K-flat if and only if it is K-projective. This noteworthy property that complexes of pure projective modules enjoy is reminiscent of the well-known fact that any flat and pure projective module is necessarily projective. Of course, we remind the reader that not all K-projective (resp. K-flat) complexes consist necessarily of projective (resp. flat) modules. Using that property of complexes of pure projective modules and the existence of the adjoint ı^{∗}, we obtain the description, up to homotopy equivalence, of any K-flat complex as a filtered colimit of K-projective complexes.

Here is a brief outline of the contents of the paper: In Section 1, we show that the homotopy category K(R-PureProj) of pure projective modules is compactly generated and describe an explicit set of compact generators. As a consequence of this, we conclude in Section 2 that the embedding ı of K(R-PureProj) in the homotopy category K(R) of R has a right adjoint ı^{∗}. We describe the kernel T(R) of ı^{∗}, study some basic properties of it and examine its restriction to the homotopy category of flat modules. Finally, in Section 3, we show that the Verdier quotient K(R)/T(R) provides an appropriate setting for the study of K-flat complexes and relate these complexes to K-projective complexes.

Notations and terminology. Unless otherwise specified, all modules considered in this paper are left modules over a fixed associative unital ring R and all complexes are chain complexes of modules. For any complex X, we denote by ΣX (or X[1]) its suspension. The orthogonality in a triangulated category is understood to be that provided by the Hom-pairing therein.

1. K(R-PureProj) is compactly generated

The homotopy category K(R-PureProj) has objects the complexes of pure projective modules and morphisms the homotopy classes of chain maps between them. Since the class of pure projective modules is closed under (set-indexed) direct sums and direct summands, K(R-PureProj) is a triangulated subcategory of the homotopy category K(R) of the ring R, which is closed under coproducts.

Let K be any triangulated category with small coproducts; the standard reference in [25]. An object c of K is called compact if the following condition is satisfied: For any (set-indexed) family (x_{i})_{i∈I} of objects of K with coproduct ⨆_{ı∈I} x_{i} and any morphism f : c ⟶ ⨆_{i∈I} x_{i} in K there exists a finite subset J ⊆ I, such that f admits a factorization through the coproduct ⨆_{ı∈J} x_{i} as a composition of the form c ⟶ ⨆_{i∈J} x_{i} ⟶^{ȷ} ⨆_{i∈I} x_{i}, where ȷ is the natural map. We say that K is compactly generated if there exists a set ℭ of compact objects in K which generates K, in the sense that the following condition is satisfied: Whenever an object x of K is such that Hom_{K}(c, x) = 0 for all c ∈ ℭ, then the object x is isomorphic to 0. In that case, we also say that ℭ is a set of compact generators of K.

Even though the homotopy category K(R-Proj) of projective modules is compactly generated if R is right coherent (cf. [14], [26, Theorem 7.14]), there are commutative rings R over which K(R-Proj) is not compactly generated; for a counterexample, see [26, Example 7.16]. In this section, we provide an alternative and simple proof of Krause’s result [19], which asserts that the homotopy category K(R-PureProj) is compactly generated and exhibits a set ℭ of compact generators therein. The set ℭ consists of the bounded complexes of finitely presented modules. In order to ensure that ℭ is really a set, we also demand that the modules of chains of the complexes in ℭ are all quotients of the modules R^{n}, n ≥ 0, by suitable finitely generated submodules of them.

Theorem 1.1. (cf. [19]) For any ring R the set ℭ of all bounded complexes of finitely presented modules is a set of compact generators of the homotopy category K(R-PureProj).

Proof. First of all, we show that any bounded complex N of finitely presented modules is a compact object of K(R-PureProj). Let (X_{i})_{i∈I} be a family of complexes of pure projective modules and consider a morphism N ⟶ ⨆_{ı∈I} X_{i} in K(R-PureProj), which is represented by a suitable chain map g : N ⟶ ⨁_{i∈I} X_{i}. Since N is a bounded complex of finitely presented (and hence finitely generated) modules, there exists a finite subset J ⊆ I, such that g factors through the direct sum ⨁_{i∈J} X_{i}, as a composition of the form N ⟶ ⨁_{i∈J} X_{i} ⟶^{ȷ} ⨁_{i∈I} X_{i}, where ȷ is the natural map. It follows readily that N is compact in K(R-PureProj).

We now prove that the set ℭ generates K(R-PureProj). To that end, we consider a complex of pure projective modules X and assume that Hom_{K}(N, X) = 0 for any bounded complex N of finitely presented modules. In other words, we assume that any chain map N ⟶ X is null-homotopic for any bounded complex N of finitely presented modules. Then, the complex X is pure acyclic, in view of [8, Proposition 2.8]. Invoking [8, Corollary 3.7], we conclude that the pure acyclic complex of pure projective modules X is, in fact, contractible; in other words, X ≃ 0 in the homotopy category K(R-PureProj). □

Remark 1.2. The argument used in the first part of the proof of Theorem 1.1 shows that the bounded complexes of finitely presented modules are also compact when viewed as objects in the homotopy category K(R). In other words, the embedding ı : K(R-PureProj) ↪ K(R) maps the compact generating set ℭ of K(R-PureProj) onto compact objects of K(R).

2. The right adjoint ı^{∗}

In this section, we examine certain properties of the right adjoint ı^{∗} of the embedding ı of K(R-PureProj) into the full homotopy category K(R) of R and relate its restriction to the homotopy category K(R-Flat) of flat modules to Neeman’s results.

First of all, we explain that the existence of ı^{∗} is an immediate consequence of Theorem 1.1.

Theorem 2.1. (cf. [19]) For any ring R the embedding ı : K(R-PureProj) ↪ K(R) has a right adjoint ı^{∗} : K(R) ⟶ K(R-PureProj), which preserves coproducts.

Proof. Theorem 1.1 states that the homotopy category K(R-PureProj) is compactly generated; therefore, it satisfies Brown representability (cf. [24, Theorem 3.1]). Since the embedding ı preserved coproducts, the existence of the right adjoint ı^{∗} follows from [24, Theorem 4.1]. On the other hand, as noted in Remark 1.2, the embedding ı maps a compact set of generators of K(R-PureProj) onto compact objects of K(R). Invoking [24, Theorem 5.1], we conclude that ı^{∗} preserves coproducts. □

For any complex X the counit of the adjunction established in Theorem 2.1 provides us with a morphism ı^{∗}X ⟶ X, which induces a bijection Hom_{K}(P, ı^{∗}X) ⟶ Hom_{K}(P, X) for any complex P of pure projective modules. It follows that the kernel of ı^{∗}, i.e. the class of all complexes X for which ı^{∗}X = 0, coincides with the right-orthogonal K(R-PureProj)^{⊥} of K(R-PureProj) in K(R). The complexes in K(R-PureProj)^{⊥} are precisely those complexes X for which any chain map P ⟶ X is null-homotopic for any complex P of pure projective modules. Invoking [8, Theorem 3.6], we conclude that K(R-PureProj)^{⊥} is the class of pure acyclic complexes. For brevity, we denote K(R-PureProj)^{⊥} by T(R) in the sequel.

Lemma 2.2. Let X be a complex and f : ı^{∗}X ⟶ X a chain map representing the counit of adjunction morphism on X. Then, for any complex of right modules L, the chain map of complexes of abelian groups 1 ⊗ f : L ⊗_{R} ı^{∗}X ⟶ L ⊗_{R} X is a quasi-isomorphism. In particular, f is a quasi-isomorphism.

Proof. For any complex X the morphism ı^{∗}X ⟶ X is part of a distinguished triangle

Z ⟶ ı^{∗}X ⟶ X ⟶ ΣZ,

where Z is a complex in K(R-PureProj)^{⊥} = T(R). Then, the mapping cone C of the chain map f is homotopy equivalent with ΣZ; hence, it is contained in T(R). Let

0 ⟶ X ⟶ C ⟶ ı^{∗}X[1] ⟶ 0.

be the associated short exact sequence of complexes. Then, for any complex of right modules L there is an induced short exact sequence of complexes of abelian groups

0 ⟶ L ⊗_{R} X ⟶ L ⊗_{R} C ⟶ L ⊗_{R} ı^{∗}X[1] ⟶ 0.

It follows that the complex of abelian groups L⊗_{R} C may be identified with the mapping cone of the chain map 1 ⊗ f : L ⊗_{R} ı^{∗}X ⟶ L ⊗_{R} X. Since the complex C is contained in T(R), the complex L ⊗_{R} C is acyclic (see, for example, [8, Proposition 1.1]). We conclude that 1 ⊗ f is a quasi-isomorphism, as needed. □

Corollary 2.3. The functor ı^{∗} : K(R) ⟶ K(R-PureProj) preserves acyclic complexes, i.e. if X is an acyclic complex, then the complex ı^{∗}X is acyclic as well. □

If we let K(R)/T(R) be the corresponding Verdier quotient and p : K(R) ⟶ K(R)/T(R) the natural quotient map, then the composition

K(R-PureProj) ↪^{ı} K(R) ⟶^{p} K(R)/T(R)

is an equivalence of categories with quasi-inverse the functor obtained from the right adjoint ı^{∗} : K(R) ⟶ K(R-PureProj) by passage to the Verdier quotient. In this way, we may identify the categories K(R-PureProj) and K(R)/T(R); cf. [19]. Since both functors ı and ı^{∗} preserve the exactness of complexes (this is obvious for ı and Corollary 2.3 proves it for ı^{∗}), it follows that under this identification the acyclic complexes K_{ac}(R-PureProj) in K(R-PureProj) are identified with the acyclic complexes pK_{ac}(R) in K(R)/T(R).

We recall that Neeman has shown in [26] that the embedding ȷ : K(R-Proj) ↪ K(R-Flat) of the homotopy category of projective modules into that of flat modules has a right adjoint ȷ^{∗} : K(R-Flat) ⟶ K(R-Proj). The kernel of the functor ȷ^{∗} is the relative orthogonal S(R) = K(R-Flat)∩K(R-Proj)^{⊥}; Neeman proved that S(R) coincides with the class of all pure acyclic complexes of flat modules. As above, letting K(R-Flat)/S(R) be the corresponding Verdier quotient and q : K(R-Flat) ⟶ K(R-Flat)/S(R) the natural quotient map, the composition

K(R-Proj) ↪^{ȷ} K(R-Flat) ⟶^{q} K(R-Flat)/S(R)

is an equivalence of categories with quasi-inverse the functor obtained from the right adjoint ȷ^{∗} : K(R-Flat) ⟶ K(R-Proj) by passage to the Verdier quotient. In this way, the categories K(R-Proj) and K(R-Flat)/S(R) are identified; cf. [26].

As an immediate consequence of the explicit description of the complexes in S(R) and T(R), we conclude that S(R) = K(R-Flat) ∩ T(R). We may now explain the relation between the adjunction ı ⊢ ı^{∗} and Neeman’s result:

Proposition 2.4. Let R be a ring and consider the embedding ν : K(R-Flat) ↪ K(R) and its restriction ν|: K(R-Proj) ↪ K(R-PureProj). We keep the notation as above.

(i) There is an embedding ν′ : K(R-Flat)/S(R) ↪ K(R)/T(R), which fits into the following commutative diagram (whose horizontal compositions are both equivalences)

K(R-Proj) ⟶^{ȷ} K(R-Flat) ⟶^{q} K(R-Flat)/S(R)
ν| ↓ ν ↓ ↓ ν′
K(R-PureProj) ⟶^{ı} K(R) ⟶^{p} K(R)/T(R)

(ii) The following diagram is commutative up to natural equivalence

K(R-Flat) ⟶^{ȷ^{∗}} K(R-Proj)
ν ↓  ↓ ν|
K(R) ⟶^{ı^{∗}} K(R-PureProj)

Proof. (i) Since S(R) ⊆ T(R), any chain map between complexes (of flat modules) whose mapping cone is contained in S(R) defines an isomorphism in the quotient K(R)/T(R). The functor ν′ is then defined by the universal property of the Verdier quotient K(R-Flat)/S(R). It is full and faithful since the same is true for ν| and the compositions q ∘ ȷ and p ∘ ı are both equivalences of categories.

(ii) Let X be a complex of flat modules. Then, the adjunction ȷ ⊢ ȷ^{∗} provides a distinguished triangle in K(R-Flat) and hence in K(R)

Y ⟶ ȷ^{∗}X ⟶ X ⟶ ΣY,

where ȷ^{∗}X is a complex of projective modules and Y is in S(R). Since S(R) ⊆ T(R), the complex Y is in the right-orthogonal of K(R-PureProj), i.e. Hom_{K}(W, Y) = 0 for any complex W of pure projective modules. It follows that the induced map

(2)  Hom_{K}(W, ȷ^{∗}X) ⟶ Hom_{K}(W, X)

is bijective for any complex W of pure projective modules. Since K(R-Proj) ⊆ K(R-PureProj), we may regard ȷ^{∗}X as a complex of pure projective modules. Then, the isomorphism (2) above implies that the morphism ȷ^{∗}X ⟶ X has the universal property of the morphism ı^{∗}X ⟶ X, which is obtained by applying the counit of the adjunction ı ⊢ ı^{∗} to X. It follows readily that ȷ^{∗}X and ı^{∗}X are naturally isomorphic in the category K(R-PureProj). □

Let K_{ac}(R) be the class of all acyclic complexes. Following Spaltenstein [28], the complexes in the left-orthogonal ^{⊥}K_{ac}(R) are called K-projective. The characterization of the K-projective complexes of pure projective modules given below will be useful in the sequel.

Proposition 2.5. A complex X of pure projective modules is K-projective if and only if Hom_{K}(X, Y) = 0 for any acyclic complex Y of pure projective modules.

Proof. By definition, a K-projective complex (of pure projective modules) is left-orthogonal to any acyclic complex; in particular, it is left-orthogonal to any acyclic complex of pure projective modules.

Conversely, assume that X is a complex of pure projective modules and Hom_{K}(X, Y) = 0 for any acyclic complex Y of pure projective modules. For any acyclic complex C there exists a distinguished triangle in K(R)

Z ⟶ ı^{∗}C ⟶ C ⟶ ΣZ,

where ı^{∗}C is a complex of pure projective modules and Z is pure acyclic. Since the complex Z is in T(R) = K(R-PureProj)^{⊥} and X is a complex of pure projective modules, it follows that Hom_{K}(X, Z) = 0. The complex ı^{∗}C is acyclic (cf. Corollary 2.3) and hence our hypothesis on X implies that Hom_{K}(X, ı^{∗}C) = 0. Therefore, we also have Hom_{K}(X, C) = 0. We conclude that X is a complex in the left-orthogonal ^{⊥}K_{ac}(R); in other words, X is K-projective. □

A complex of right modules L is called K-flat if the complex of abelian groups L⊗_{R}X is acyclic for any acyclic complex X; cf. [28]. We show below that, in this definition, it is sufficient to demand the acyclicity of L ⊗_{R} X for all acyclic complexes X of pure projective modules.

Proposition 2.6. A complex L of right modules is K-flat if and only if the complex of abelian groups L ⊗_{R} X is acyclic for any acyclic complex X of pure projective modules.

Proof. By definition, if L is K-flat, then the complex of abelian groups L ⊗_{R} X is acyclic for any acyclic complex X. In particular, L ⊗_{R} X is acyclic for any acyclic complex X of pure projective modules.

Conversely, assume that the complex of right modules L is such that L ⊗_{R} X is acyclic for any acyclic complex X of pure projective modules. For any acyclic complex C, the complex ı^{∗}C consists of pure projective modules and is acyclic (cf. Corollary 2.3); hence, L ⊗_{R} ı^{∗}C is acyclic. Since the complexes L ⊗_{R} C and L ⊗_{R} ı^{∗}C are quasi-isomorphic, in view of Lemma 2.2, it follows that L ⊗_{R} C is acyclic. Therefore, the complex of right modules L is K-flat. □

3. K-flat complexes

In this section, we consider the class of K-flat complexes and illustrate the importance of the triangulated category K(R)/T(R) in their study. As shown in [28], any K-projective complex is K-flat. We prove that any K-flat complex is homotopy equivalent with a filtered colimit (in the category of complexes) of K-projective complexes of pure projective modules. In the level of objects, the relation between K-flatness and T(R) is described in [8, Proposition 1.1]: The complexes in T(R) are precisely the acyclic K-flat complexes.

Consider a short exact sequence of complexes

0 ⟶ X ⟶ Y ⟶ Z ⟶ 0,

which is split in each degree. It is easily seen that, whenever two of the complexes X, Y, Z are K-flat, then so is the third. It follows that the class of K-flat complexes forms a triangulated subcategory of the homotopy category K(R).

Lemma 3.1. Let X, Y be two complexes, which are isomorphic in the quotient K(R)/T(R). Then, X is K-flat if and only if Y is K-flat.

Proof. An isomorphism in K(R)/T(R) between X and Y is provided by a diagram of morphisms X ⟵^{a} W ⟶^{b} Y in K(R), which fit in distinguished triangles

W ⟶^{a} X ⟶ Z ⟶ ΣW and W ⟶^{b} Y ⟶ Z′ ⟶ ΣW,

where the complexes Z, Z′ are in T(R). Since all complexes in T(R) are K-flat and the class of K-flat complexes forms a triangulated subcategory of the homotopy category of R, it follows that X is K-flat if and only if W is K-flat if and only if Y is K-flat. □

We now explain in some detail the sense in which K-flat resolutions exist, are unique up to isomorphism and are functorial in the Verdier quotient K(R)/T(R).

Remarks 3.2. (i) As shown by Spaltenstein in [28], any complex X admits a surjective quasiisomorphism p : P ⟶ X, where P is a suitable K-projective complex of projective modules. In particular, for any complex X there is a surjective quasi-isomorphism q : F ⟶ X, where F is a K-flat complex.

(ii) Let F be a K-flat complex and consider a quasi-isomorphism p : P ⟶ F, where P is a K-projective complex of projective modules, as in (i) above. Then, the mapping cone C of p, being acyclic and K-flat, is contained in T(R). It follows that p induces an isomorphism in the Verdier quotient K(R)/T(R). Therefore, any K-flat complex is isomorphic in K(R)/T(R) with a K-projective complex of projective modules.

(iii) If P is a K-projective complex of projective modules and f : Y ⟶ X is a surjective quasi-isomorphism, then the induced chain map

f_{∗} : Hom_{R}(P, Y) ⟶ Hom_{R}(P, X)

is a surjective quasi-isomorphism as well. It follows that for any complex X there is a unique, up to homotopy equivalence, surjective quasi-isomorphism P ⟶ X from a K-projective complex of projective modules P. It also follows that these K-projective resolutions by complexes of projective modules are functorial in the homotopy category K(R) (cf. [28]).

(iv) Let X be a complex and fix a surjective quasi-isomorphism p : P ⟶ X, where P is a K-projective complex of projective modules. We also consider a surjective quasi-isomorphism q : F ⟶ X, where F is K-flat. Then, it follows from (iii) above that p may be factored as the composition P ⟶^{f} F ⟶^{q} X, for a suitable chain map f, which is unique up to homotopy. Since both q and p = q ∘ f are quasi-isomorphisms, it follows that f is a quasi-isomorphism as well. Then, the argument in (ii) shows that f induces an isomorphism in K(R)/T(R). Hence, K-flat resolutions are unique up to isomorphism in K(R)/T(R).

(v) Let f : X ⟶ Y be a chain map of complexes, whose mapping cone Z is contained in T(R). We choose K-projective complexes of projective modules P, Q and surjective quasiisomorphisms p : P ⟶ X and q : Q ⟶ Y. As explained in (iii) above, there exists a, unique up to homotopy, chain map ϕ : P ⟶ Q that makes the following diagram commutative (in the category of complexes)

\begin{matrix}P & ⟶^{ϕ} & Q \\ p ↓ & & ↓ q \\ X & ⟶^{f} & Y \end{matrix}

We claim that the mapping cone Π of ϕ is contained in T(R). First of all, we note that Π is a K-projective complex of projective modules and hence it is K-flat. On the other hand, the commutative diagram above induces a quasi-isomorphism π : Π ⟶ Z. The mapping cone C of π is an acyclic complex, which is K-flat (since both Z and Π are K-flat). Therefore, C is contained in T(R). Since Z is also contained in T(R), which is a triangulated subcategory of K(R), it follows that Π is contained in T(R) as well.

(vi) If X, Y are two complexes, then a morphism X ⟶ Y in the category K(R)/T(R) is provided by a diagram of morphisms X ⟵ W ⟶ Y in K(R), which are represented by chain maps X ⟵^{f} W ⟶^{g} Y, such that the mapping cone of f is a complex in T(R). As in (iii) above, we may choose K-projective complexes of projective modules P, S and Q, surjective quasi-isomorphisms p : P ⟶ X, s : S ⟶ W and q : Q ⟶ Y and chain maps ϕ : S ⟶ P and γ : S ⟶ Q that make the following diagram commutative (in the category of complexes)

P ⟵^{ϕ} S ⟶^{γ} Q
p ↓ s ↓ q ↓
X ⟵^{f} W ⟶^{g} Y

Since the mapping cone of ϕ is contained in T(R), as shown in (v), the top row of the diagram above represents a morphism P ⟶ Q in K(R)/T(R). The consequence of this is that K-flat resolutions are functorial in K(R)/T(R).

We describe below K-flatness, in terms of the orthogonality defined by the Hom-pairing in K(R)/T(R). Our arguments follow closely and, at the same time, generalize the arguments in [22, Corollary 5.14]. To avoid any possible confusion in the sequel, we denote by Hom\overline{K}( , ) the Hom-pairing in K(R)/T(R). Since the functor p ∘ ı : K(R-PureProj) ⟶ K(R)/T(R) is an equivalence of categories, for any two complexes X, Y of pure projective modules, there is an induced isomorphism of abelian groups Hom_{K}(X, Y) ≃ Hom\overline{K}(X, Y).

Proposition 3.3. A complex X is K-flat if and only if it is left-orthogonal with respect to the Hom-pairing in the category K(R)/T(R) to all acyclic complexes.

Proof. Assume that X is a K-flat complex and consider an acyclic complex C. We have to show that Hom\overline{K}(X, C) = 0. Since X is isomorphic in K(R)/T(R) with a K-projective complex of projective modules (cf. Remark 3.2(ii)), we may as well assume that X is itself K-projective and consists of projective modules. The complex C is isomorphic in K(R)/T(R) with the complex ı^{∗}C, which consists of pure projective modules and is acyclic (cf. Corollary 2.3). We thus reduce the problem to the case where C is an acyclic complex of pure projective modules. Then, the triviality of the group Hom\overline{K}(X, C) follows since that group is isomorphic with Hom_{K}(X, C); the latter group is trivial, as X is K-projective and C is acyclic.

Conversely, assume that Hom\overline{K}(X, C) = 0 for any acyclic complex C. Since X is isomorphic with ı^{∗}X in K(R)/T(R), it follows that Hom\overline{K}(ı^{∗}X, C) = 0 for any acyclic complex C. In particular, for any acyclic complex C of pure projective modules we have Hom_{K}(ı^{∗}X, C) ≃ Hom\overline{K}(ı^{∗}X, C) = 0. Then, Proposition 2.5 implies that the complex of pure projective modules ı^{∗}X is K-projective and hence K-flat. Since K-flatness is invariant under isomorphisms in the category K(R)/T(R) (cf. Lemma 3.1), it follows that X is K-flat. □

The following result implies that K-flatness is equivalent to K-projectivity, for complexes of pure projective modules; this will be an essential ingredient of the proof of Theorem 3.8 below.

Corollary 3.4. Any K-flat complex of pure projective modules is K-projective.

Proof. Let X be a K-flat complex of pure projective modules. Then, for any acyclic complex C of pure projective modules there is an isomorphism Hom_{K}(X, C) ≃ Hom\overline{K}(X, C). In view of Proposition 3.3, the latter group is trivial and hence Hom_{K}(X, C) = 0. Invoking Proposition 2.5, we conclude that X is K-projective. □

Corollary 3.5. The functor ı^{∗} maps K-flat complexes to K-projective complexes.

Proof. Let X be a K-flat complex. The counit of adjunction morphism ı^{∗}X ⟶ X induces an isomorphism in K(R)/T(R) and hence Lemma 3.1 implies that ı^{∗}X is K-flat as well. Since ı^{∗}X is a complex of pure projective modules, Corollary 3.4 finishes the proof. □

Corollary 3.6. The right adjoint functor ȷ^{∗} : K(R-Flat) ⟶ K(R-Proj) of the embedding ȷ of K(R-Proj) in K(R-Flat) maps K-flat complexes of flat modules to K-projective complexes of projective modules.

Proof. Let X be a K-flat complex of flat modules. Then, Proposition 2.4(ii) asserts that the complexes ı^{∗}X and ȷ^{∗}X are homotopy equivalent, whereas Corollary 3.5 asserts that ı^{∗}X is K-projective. It follows that ȷ^{∗}X is K-projective as well. □

Remarks 3.7. (i) We can provide an alternative proof of Corollary 3.4, by following the argument in the proof of [5, Theorem 7.8]: We consider a K-flat complex of pure projective modules X and fix a quasi-isomorphism f : P ⟶ X, where P is a K-projective complex of projective modules. Then, the mapping cone C of f is an acyclic complex, which fits into an exact sequence of complexes

0 ⟶ X ⟶ C ⟶ P [1] ⟶ 0.

Since X and P [1] are K-flat complexes of pure projective modules, the mapping cone C is also a K-flat complex of pure projective modules. Being also acyclic, the complex C is contractible (cf. [8, Corollary 3.7]); in particular, C is K-projective. Since P [1] is also K-projective, it follows that X is K-projective.

(ii) A ring R is called (left) pure semi-simple if any module is pure projective. These are precisely the rings over which any module is a direct sum of indecomposable modules (cf. [33]). It follows from Corollary 3.4 that the notions of K-flatness and K-projectivity are equivalent to each other over pure semi-simple rings.

We conclude with the description of K-flat complexes, in terms of filtered colimits of Kprojective complexes.

Theorem 3.8. Any K-flat complex is homotopy equivalent with a filtered colimit (in the category of complexes) of K-projective complexes of pure projective modules.

Proof. Let X be a K-flat complex. Then, Theorem 2.1 provides a distinguished triangle

(3)  Z ⟶ ı^{∗}X ⟶ X ⟶ ΣZ,

where ı^{∗}X is a complex of pure projective modules and Z is in T(R) (i.e. Z is pure acyclic). In view of Corollary 3.5, the complex ı^{∗}X is K-projective. The morphism Z ⟶ ı^{∗}X in (3) above is represented by a chain map f : Z ⟶ ı^{∗}X, whose mapping cone C is homotopy equivalent to X. We consider the associated short exact sequence of complexes

0 ⟶ ı^{∗}X ⟶ C ⟶ Z[1] ⟶ 0.

As shown in [8, Proposition 2.2], the pure acyclic complex Z[1] is isomorphic with a filtered colimit of contractible complexes of finitely presented modules. We fix such an isomorphism Z[1] ≃ lim_{⟶ λ} N_{λ}, for a suitable filtered system (N_{λ})_{λ} of contractible complexes of finitely

presented modules. For any index λ we may consider the canonical map N_{λ} ⟶ Z[1] and construct the commutative diagram of complexes with exact rows

(4)  \begin{aligned}0 & ⟶ & ı^{∗}X & ⟶ & C_{λ} & ⟶ & N_{λ} & ⟶ & 0 \\ & & ∥ & & ↓ & & ↓ \\ 0 & ⟶ & ı^{∗}X & ⟶ & C & ⟶ & Z[1] & ⟶ & 0 \end{aligned}

by letting C_{λ} be the pullback of the diagram of complexes

(5)  \begin{aligned}& & N_{λ} \\ & & ↓ \\ C & ⟶ & Z[1] \end{aligned}

Since the short exact sequence in the bottom of (4) is split in each degree, it follows that the same is true for the short exact sequence in the top of (4). The complex N_{λ} is contractible and hence K-projective. We conclude that C_{λ} is a K-projective complex of pure projective modules. In view of the construction of the top exact sequence of (4) by taking the pullback of (5), it is easily seen that there is a unique structure of a filtered system on the family of short exact sequences

(0 ⟶ ı^{∗}X ⟶ C_{λ} ⟶ N_{λ} ⟶ 0)_{λ},

which extends the filtered system (N_{λ})_{λ}, so that:

(i) all chain maps ı^{∗}X ⟶ ı^{∗}X in the filtered system are given by the identity of ı^{∗}X and (ii) the vertical morphisms in (4) are compatible with the filtered system. Since filtered colimits preserve exactness, there results a commutative diagram of complexes with exact rows

0 ⟶ ı^{∗}X ⟶ lim_{⟶ λ} C_{λ} ⟶ lim_{⟶ λ} N_{λ} ⟶ 0
∥ ↓ ↓
0 ⟶ ı^{∗}X ⟶ C  ⟶ Z[1] ⟶ 0

As the chain map lim_{⟶ λ} N_{λ} ⟶ Z[1] is an isomorphism of complexes, it follows that the chain map lim_{⟶ λ} C_{λ} ⟶ C is an isomorphism of complexes as well. □

Remark 3.9. Let X be a K-flat complex of flat modules and consider the triangle

Z ⟶ ȷ^{∗}X ⟶ X ⟶ ΣZ,

which is provided by the adjunction ȷ ⊢ ȷ^{∗}. Then, ȷ^{∗}X is a K-projective complex of projective modules (in view Corollary 3.6) and Z is in S(R). As shown in [26], the complex Z (and hence its suspension as well) is isomorphic with a filtered colimit of contractible complexes of finitely generated projective modules. Hence, following the arguments in the proof of Theorem 3.8, one can show that X is homotopy equivalent with a filtered colimit of K-projective complexes of projective modules. In fact, more is known in this case: The complex X is isomorphic (and not just homotopy equivalent) with a filtered colimit of bounded (and not just K-projective) complexes of projective modules; this is proved in [5].

Acknowledgments. Part of this work was completed while the author was visiting the Institut des Hautes Etudes Scientifiques in Bures-sur-Yvette. He wishes to express his gratitude to the Institute for its warm hospitality and for providing a stimulating research environment. It is a pleasure to thank A. Beligiannis for helpful comments and suggestions on an earlier draft of this work.

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Department of Mathematics, University of Athens, Athens 15784, Greece

  E-mail address: emmanoui@math.uoa.gr