COMPLETE MODULES AND TORSION MODULES

W. G. DWYER AND J. P. C. GREENLEES

Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded R-modules there are naturally defined subcategories of A-torsion objects and of A-complete objects. Under a finiteness condition on A, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case R = ℤ and A = ℤ/p.

1. Introduction

Let R be a ring and R-mod the derived category of chain complexes of left R-modules (see 1.2). We choose a fixed complex A and declare another complex N to be A-trivial if Hom_{R}(A, N) ≅ 0, where Hom_{R}(·, ·) denotes the chain complex of morphisms in R-mod. Going further, we say that X is A-torsion if Hom_{R}(X, N) ≅ 0 for all A-trivial N, and that X is A-complete if Hom_{R}(N, X) ≅ 0 for all A-trivial N. The goal of this paper is to study the category A_{tors} of A-torsion complexes and the category A_{comp} of A-complete complexes (both full subcategories of R-mod). We relate these categories to one another, construct approximation functors R-mod → A_{tors} and R-mod → A_{comp}, provide algebraic formulas for the functors, and find that the functors are related in interesting ways. If R is a commutative ring, I ⊂ R is a finitely generated ideal, and A = R/I, we identify the functors in terms of local homology and local cohomology. In this case it turns out that a chain complex is A-torsion or A-complete if and only if its homology groups have appropriate torsion or completeness properties. One remarkable aspect of the theory we describe is how much can be said in general. In fact, the general case seems to shed some light on local homology and cohomology, and on the meaning of torsion and completeness: the local cohomology of M is a universal torsion object mapping to M, and the local homology of M is a universal complete object accepting a map from M (see §6). The first is a cellular approximation, the second a Bousfield localization. We also have some long-term applications in mind (1.3).

Date: April 28, 2000.

The authors are grateful to MIT for its hospitality in May 1999 when this work was begun, and the second author thanks J.D. Christensen for useful conversations.

The first author was partially supported by the National Science Foundation.

In order to make much progress, though, we must assume that A is perfect, i.e., that A is isomorphic in R-mod to a complex of finite length in which the entries are finitely generated projective R-modules. Complexes like this could just as well be called finite R-complexes, since they represent the elements of R-mod which can be built in a finite number of steps from R itself by taking suspensions, cofibration sequences, and retracts. They are the analogs in R-mod of finite complexes in the stable homotopy category.

1.1. Organization of the paper. In §2 we develop a Morita theory which shows that A_{tors} and A_{comp} are both equivalent to the derived category of modules over the endomorphism algebra of A; in particular, A_{tors} and A_{comp} are equivalent to one another. Section 3 describes some special cases of this Morita theory, and in particular a striking one with R = ℤ and A = ℤ/p. Section 4 establishes new notation for some of the functors from §2 and interprets these functors in terms of standard constructions: cellular approximations, homology localizations, and periodicizations. The functors fit into a homotopy fibre square (4.13) which generalizes the arithmetic square of abelian group theory (4.2). The next section establishes conditions under which the question of whether or not a chain complex belongs to A_{tors} or A_{comp} can be answered by examining its homology groups. Finally, §6 specializes to the case in which R is commutative, I ⊂ R is a finitely generated ideal, and A = R/I. A device involving the Koszul complex allows the previous theory to be applied, even though R/I is not necessarily perfect (i.e. R/I does not necessarily have a finite projective resolution over R). The functors from §4 in this case turn out to be classical local cohomology, or its dual local homology, at the ideal I; a chain complex M belongs to A_{tors} or A_{comp} if and only if its homology groups are I-torsion or I-complete in an appropriate sense.

1.2. Remark. The derived category R-mod is obtained from the category of unbounded chain complexes of R-modules by formally inverting the maps which induce isomorphisms on homology. See [15] or [1] for algebraic ways to look at this, and [11] for a topological approach. The statements in this paper are expressed almost exclusively in terms of such derived categories. In particular, Hom is the derived homomorphism complex (sometimes written RHom) considered as an object of the appropriate derived category, and ⊗ refers to the left derived tensor product (sometimes written ⊗^{L}). The convention of working in the derived category has some startling effects and should not be forgotten.

There is one exception to our convention. If A is an object of R-mod, then End_{R}(A) denotes the actual differential graded algebra obtained by taking a projective resolution of A and forming the usual DGA of endomorphisms of this resolution. Up to multiplicative homology equivalence, End_{R}(A) does not depend upon the choice of projective resolution.

1.3. Remark. This paper is intended to establish a framework and introduce some terminology in a simple but relatively general setting. With little change, our key arguments can be extended to cover a number of important categorically similar cases. For instance, R could be replaced by a graded ring (this comes up in some examples from §3), a differential graded algebra, or a ring spectrum; with some adjustments in terminology R could even be replaced by a ring with many objects, or a ring spectrum with many objects [14]. These extensions are important in constructing an algebraic model for rational equivariant cohomology theories. Some of the results in this paper, especially in §6, are related to results in [5], although here we take a somewhat different point of view.

2. Morita Theory

In this section we show that the categories A_{tors} and A_{comp} are equivalent, by relating them to a third category which is at least as interesting as the other two. Let E = End_{R}(A). This is a differential graded ring (1.2), and there is a derived category mod-E of right E-modules formed as usual by taking differential graded E-modules and inverting homology isomorphisms. We may define a functor

E : R-mod ⟶ mod-E E(M) = Hom_{R}(A, M) .

Note here that A is naturally a left E-module; this left module structure commutes with the action of R on A and passes to a right E-module structure on Hom_{R}(A, M). Let

(·)^{♯} : R-mod ⟶ mod-R

be the duality functor defined by M^{♯} = Hom_{R}(M, R), and note that A^{♯} is an object of mod-E, as well as an object of mod-R. Define functors

T : mod-E ⟶ R-mod T (X) = X ⊗_{E} A ,
C : mod-E ⟶ R-mod C(X) = Hom_{E}(A^{♯}, X) .

The main result of this section is the following theorem.

2.1. Theorem. Let A be a perfect complex of R-modules, and let E = End_{R}(A). Then the above functors E, T, and C give two pairs of adjoint equivalences of categories

\xymatrix{A_{tors} \ar[r]_{E} & mod-E \ar[l]_{T} \ar[r]_{C} & A_{comp} \ar[l]_{E}}

(where the left adjoints are displayed above the right ones).

2.2. Remark. If R is a commutative ring, then in the situation of 2.1 it has already been shown by Hovey, Palmieri, and Strickland [9, 3.3.5] that A_{tors} and A_{comp} are equivalent categories.

2.3. Remark. Theorem 2.1 is a variant of Morita theory. It is not hard to see that a complex A of R-modules is perfect if and only if A is small, in the sense that Hom_{R}(A, ·) commutes with arbitrary coproducts. Ordinary Morita theory implies that if A is a small generator of R-mod then the category of modules over E = End_{R}(A) is equivalent to R-mod itself. Theorem 2.1 states that if A is small but not necessarily a generator, then the category of modules over E is equivalent both to the subcategory A_{tors} and the subcategory A_{comp} of R-mod. This is particularly plausible in the case of A_{tors}, since this is just the category of chain complexes which can be built from A (4.5).

Before beginning with the proof of 2.1, it is useful to point out a few simple facts.

2.4. A-equivalences. It is convenient to say that a map X → Y in R-mod is an A-equivalence if its cofibre is A-trivial, or equivalently if E(X) ≅ E(Y). We leave it to the reader to verify by a formal argument that an A-equivalence between A-torsion objects of R-mod is an isomorphism, and that an A-equivalence between A-complete objects of R-mod is an isomorphism.

2.5. Adjunctions. Note that for each left R-module M and right Emodule X there is an adjunction isomorphism

Hom_{E}(X, Hom_{R}(A, M)) ≅ Hom_{R}(X ⊗_{E} A, M).

The unit map

X ⟶ Hom_{R}(A, X ⊗_{E} A) (2.6)

is obtained by applying the functor ·⊗_{E}A to pass from X ≅ Hom_{E}(E, X) to Hom_{R}(A, X ⊗_{E} A). The counit map

Hom_{R}(A, M) ⊗_{E} A ⟶ M (2.7) is obtained from evaluation. Similarly, for any left R-module M and right E-module X there is an adjunction isomorphism

(2.8) Hom_{R}(M, Hom_{E}(A^{♯}, X)) ≅ Hom_{E}(A^{♯} ⊗_{R} M, X).

2.9. Maps and tensors. Finally, note that since A is perfect there are isomorphisms

Hom_{R}(A, N) ≅ A^{♯} ⊗_{R} N and in particular E ≅ A^{♯} ⊗_{R} A.

This last is an isomorphism inside the derived category of either left or right E-modules, with E acting on the left on A^{♯} ⊗_{R} A via its action on A, and on the right via its action on A^{♯}.

2.10. The left side of 2.1. We begin by observing that for any right E-module X, the module T (X) = X ⊗_{E} A is in fact an object of A_{tors}. In fact, one can use 2.5 to calculate that for any A-trivial N,

Hom_{R}(T (X), N) = Hom_{R}(X ⊗_{E} A, N) = Hom_{E}(X, Hom_{R}(A, N)) ≅ 0.

This adjunction also shows that T is left adjoint to E.

We now need to show that the unit map (2.6) is always an equivalence, and that the counit (2.7) is an equivalence if M is an object of A_{tors}. The unit is a natural map and preserves both cofibre sequences and coproducts (since A is small (2.3)) and therefore it suffices to check the result for X = E, where it is clear. The counit is a natural map and its domain is A-torsion. In order to prove that it is an isomorphism if its range is A-torsion, it is enough (2.4) to show that the counit map is always an A-equivalence. To prove this, we use 2.9 to calculate

Hom_{R}(A, T E(M)) ≅ A^{♯} ⊗_{R} A ⊗_{E} Hom_{R}(A, M) ≅ Hom_{R}(A, M).

2.11. The right side of 2.1. First we show that C(X) = Hom_{E}(A^{♯}, X) belongs in fact to A_{comp}. Suppose that N is A-trivial. We may use 2.8 and 2.9 to calculate

Hom_{R}(N, C(X)) = Hom_{R}(N, Hom_{E}(A^{♯}, X)) ≅ Hom_{E}(A^{♯} ⊗_{R} N, X) ≅ Hom_{E}(Hom_{R}(A, N), X) ≅ 0 ..

This reasoning also shows that C is right adjoint to E.

We now have to show that the counit map EC(X) → X is an isomorphism for all right E-modules X. This follows from 2.5 and 2.9:

EC(X) = Hom_{R}(A, Hom_{E}(A^{♯}, X)) ≅ Hom_{E}(A^{♯} ⊗_{R} A, X) ≅ X.

Finally, we have to check that the unit map CE(M) → M is an isomorphism for each A-complete left R-module M. Since the target of this map is A-complete, it is enough by 2.4 to verify that the map itself is always an A-equivalence, i.e., becomes an isomorphism when the functor E is applied. But as above, EC(X) ≅ X, so ECE(M) ≅ E(M).

3. Sample applications of the Morita theory

We describe three situations in which Theorem 2.1 holds.

3.1. The paradoxical case of ℤ/p. This simplest nontrivial application of Theorem 2.1 may well leave the reader nodding in disbelief. Let R = ℤ and A = ℤ/p, this last considered as a chain complex concentrated in degree 0. The complex A is perfect because it is isomorphic in ℤ-mod to the resolution ℤ ⟶^{p}ℤ. An object N of ℤ-mod is A-trivial if and only if all of its homology groups are uniquely p-divisible, or equivalently if and only if the natural map N → ℤ[1/p] ⊗_{ℤ} N is an isomorphism. From this and the isomorphism

Hom_{ℤ}(X, ℤ[1/p] ⊗_{ℤ} Y) ≅ Hom_{ℤ}(ℤ[1/p] ⊗_{ℤ} X, ℤ[1/p] ⊗_{ℤ} Y)

it is easy to see that A_{tors} is the subcategory of ℤ-mod consisting of objects X with ℤ[1/p] ⊗_{ℤ} X ≅ 0, i.e., objects X which have p-primary torsion homology groups. By inspection E is a DG algebra whose homology algebra is isomorphic to Ext_{ℤ}^{∗}(ℤ/p, ℤ/p); from a multiplicative point of view this is an exterior algebra over ℤ/p on one generator of dimension (−1). We are asserting that the functor E gives an equivalence between the category A_{tors} and the category of E-modules.

For instance, E(ℤ/p^{∞}) = ℤ/p. Accordingly, it follows that

H_{∗}(End_{E}(ℤ/p)) = H_{∗}(End_{ℤ}(ℤ/p^{∞})) = ℤ_{p̂}.

This seems very hard to believe, since the identity map of ℤ/p has additive order p.

The issue, though, revolves around what “additive order” means. Although p times the identity map of ℤ/p is null-homotopic as a ℤ-map, it is not null-homotopic as an E-map. To calculate E = End_{ℤ}(ℤ/p) as a strict DGA we have to replace the abelian group ℤ/p by the above resolution C = (ℤ ⟶^{p} ℤ). Viewing an element of C as a column vector with the top entry from the ℤ in homological degree 0, we may view E = End_{ℤ}(C) as the algebra of 2 × 2 matrices (appropriately graded). One then calculates

d ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}0 & p \\ 0 & 0 \end{matrix} ) , d ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) ,

d ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) = ( \begin{matrix}p & 0 \\ 0 & p \end{matrix} ) , d ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}0 & −p \\ 0 & 0 \end{matrix} ) .

The elements of C^{♯} are row vectors, with E acting on the right. The map p : C^{♯} ⟶ C^{♯} is indeed the boundary of right multiplication

by ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ). However this matrix is not a central element of E and

therefore right multiplication by it does not represent an E-map. In fact, the identity map of C has infinite order as an E-map.

We may also give a constructive interpretation. Observe that the usual construction of an Adams spectral sequence gives a conditionally convergent spectral sequence

Ext_{H_{∗}E}^{s,t}(H_{∗}(ℤ/p), H_{∗}(ℤ/p)) ⇒ H_{t−s}End_{E}(ℤ/p).

This may be identified as an unravelled Bockstein spectral sequence. The generator of H_{−1}E is a Bockstein operator, and the multiplicationby-p map described above is represented by an element of bidegree (1, 1) which a stable homotopy theorist would denote h_{0}. In effect we may regard the category mod-E as encoding Bockstein spectral sequences for both p-torsion and p-complete modules, and Theorem 2.1 as stating that the Bockstein spectral sequence determines a p-torsion or a pcomplete module.

3.2. Example. Let T be the circle group. This second example suggests an approach to studying T -equivariant rational cohomology theories. We content ourselves here with sketching an analogy which is refined elsewhere to a theorem and proved in greater generality.

We take R = k[c], where k is a field, c is of degree −2, I = (c), and A = R/I. If k = ℚ it is natural to think of R/I = k as analogous to the free T -cell T_{+}. Thus E is analogous to the DGA of self-maps of T_{+}, and mod-E to the category of free rational T -spectra. The left-hand equivalence of 2.1 is then analogous to the theorem of [6] stating

(c)-torsion-ℚ[c]-mod ≃ free rational T -spectra.

From this point of view, the great attraction of the equivalence is that the category of torsion ℚ[c]-modules is of injective dimension 1, whilst that of E-modules is of infinite homological dimension. The analogue of the Adams spectral sequence in Example 3.1 is the descent spectral sequence. Generally speaking this is much less useful than the Adams spectral sequence based on the torsion module k[c, c^{−1}]/k[c], which collapses to a short exact sequence.

3.3. Example. The third example is connected with chromatic stable homotopy theory [13]. We take R = ℤ[v_{1}, v_{2}, …, v_{n−1}, v_{n}, v_{n}^{−1}], I = (p, v_{1}, …, v_{n−1}), and A = R/I. The category of I-primary torsion modules is analogous to the n-th monochromatic category, whilst the category of I-complete modules is analogous to the category of Bousfield K(n)-complete modules. In this case Theorem 2.1 gives something analogous to the equivalence [10, 6.19]

nth monochromatic category ≃ K(n)-complete spectra,

with the intermediate category of E-modules being analogous to a category of modules over the ring of self-maps of K(n).

4. Cellular approximations and homology localizations

We work in the setting of §2, but in order to organize the results more clearly we will introduce some new notation. For an object M of R-mod, let CW_{A}(M) stand for T E(M) and M_{ Â} for CE(M). There is a natural A-equivalence (2.4) M → M_{ Â}. There is also a natural A-equivalence CW_{A}(M) → M, and we will denote the cofibre (which is A-trivial) by M_{1/A}.

It follows easily from the arguments in §2 that the functor CW_{A}(·) is idempotent and is right adjoint to the inclusion A_{tors} → R-mod. Similarly, the completion functor (·)_{Â} is idempotent and is left adjoint to the inclusion A_{comp} → R-mod.

4.1. Another look at ℤ/p. One way to get some insight into these constructions is to consider the special case R = ℤ, A = ℤ/p; this can be analyzed either by direct calculation (3.1) or by applying the results of §6. We describe the situation in this case, using terminology which will be explained below. For simplicity, write CW_{p}(·), (·)_{1/p}, and (·)_{p̂}, for CW_{A}(·), (·)_{1/A}, and (·)_{Â} when A = ℤ/p. For any object M of ℤ-mod, there are isomorphisms

\begin{aligned}CW_{p}(M) & ≅ Σ^{−1}ℤ/p^{∞} ⊗_{ℤ} M \\ M_{1/p} & ≅ ℤ[1/p] ⊗_{ℤ} M \\ M_{ p̂} & ≅ Hom_{ℤ}(Σ^{−1}ℤ/p^{∞}, M). \end{aligned}

The map CW_{p}(M) → M is cellular approximation (4.5) with respect to ℤ/p (i.e. gives the universal p-torsion approximation to M). The map M → M_{ p̂} is Bousfield localization (4.7) with respect to the homology theory on ℤ-mod given by M ↦ H_{∗}(ℤ/p ⊗_{ℤ} M) (the complex M_{ p̂} is sometimes called the Ext-p-completion of M.) . The map M → M_{1/p} is nullification (4.10) with respect to ℤ/p. Since ℤ/p is perfect, this is a smashing localization [12], and so M → M_{1/p} is also Bousfield localization with respect to the homology theory M ↦ H_{∗}(ℤ[1/p] ⊗_{ℤ} M). Finally, there is a homotopy fibre square

\begin{matrix}M & ⟶ & M_{ p̂} \\ ↓ & & ↓ \\ M_{1/p} & ⟶ & (M_{ p̂})_{1/p} \end{matrix} (4.2)

The square is obtained by applying the natural map X → X_{1/p} to the upper row.

The goal of this section is to obtain the above results in the general case. We assume as usual that A is a perfect object of R-mod.

4.3. Proposition. For any object M of R-mod, there are natural isomorphisms

CW_{A}(M) ≅ CW_{A}(R) ⊗_{R} M
M_{1/A} ≅ R_{1/A} ⊗_{R} M
M_{ Â} ≅ Hom_{R}(CW_{A}(R), M)

4.4. Remark. Implicit in the above formulas is the fact that CW_{A}(R) and R_{1/A} can be constructed as objects in the derived category of Rbimodules. This is a consequence of the fact that R itself is an Rbimodule. It is easier to explain this in a slightly more general setting. Suppose that M is a left module over R and a right module over S, in such a way that the two actions commute. Recall the formula

CW_{A}(M) = Hom_{R}(A, M) ⊗_{E} A .

The extra right action of S on M persists to a right action of S on Hom_{R}(A, M). This S-action commutes with the action of E on Hom_{R}(A, M) (because the action of E works on the A variable) and so passes through the tensor product to give a right action of S on CW_{A}(M) which commutes with the normal left action of R. Applying this in the special case M = R and S = R gives the R-bimodule structure on CW_{A}(R). The bimodule structure on R_{1/A} is obtained similarly.

If R is commutative, these bimodules are obtained from ordinary modules by taking the action on one side to be the same as the action on the other.

Proof. For the first isomorphism we use the chain

CW_{A}(M) = Hom_{R}(A, M) ⊗_{E} A
≅ (A^{♯} ⊗_{R} M) ⊗_{E} A
≅ (A^{♯} ⊗_{E} A) ⊗_{R} M

where the last isomorphism comes from the fact that the action of E on A^{♯} ⊗_{R} M is induced by an action of E on A^{♯} which commutes with the right action of R. Now note that A^{♯} ⊗_{E} A is CW_{A}(R). For the second, use a naturality argument and tensor the exact triangle

CW_{A}(R) → R → R_{1/A}

over R with M. For the third, note that M → M_{ Â} is an A-equivalence and so induces an isomorphism

Hom_{R}(CW_{A}(R), M) ≅ Hom_{R}(CW_{A}(R), M_{ Â}).

Since M_{ Â} is A-complete and CW_{A}(R) → R is an A-equivalence, we conclude that Hom_{R}(CW_{A}(R), M_{ Â}) ≅ Hom_{R}(R, M_{ Â}) ≅ M_{ Â}. Composing the two isomorphisms gives M_{ Â} ≅ Hom_{R}(CW_{A}(R), M).

4.5. Cellular approximation. An object M of R-mod is said to be A-cellular if M is built from A in the sense that it belongs to the smallest subcategory of R-mod which contains A and is closed under isomorphisms, suspensions, coproducts, and cofibre sequences. If M is A-cellular then any A-equivalence (2.4) X → Y induces an isomorphism Hom_{R}(M, X) → Hom_{R}(M, Y) (in particular, M is A-torsion). A map M′ → M is said to be an A-cellular approximation map if M′ is A-cellular and the map M′ → M is an A-equivalence. A formal argument shows that A-cellular approximations are unique, if they exist; the arguments of Dror-Farjoun show that they do exist [4].

4.6. Proposition. The objects of A_{tors} are exactly the A-cellular objects of R-mod. For any object M of R-mod, the natural map CW_{A}(M) → M is an A-cellular approximation map.

Proof. We noted above that any A-cellular object of R-mod belongs to A_{tors}. If M is A-torsion, we can choose an A-cellular approximation map M′ → M. This is an A-equivalence between A-torsion complexes, and so it is an isomorphism. Therefore, M is A-cellular.

The second statement is proved by observing that CW_{A}(M) is Atorsion, hence A-cellular, and that the map CW_{A}(M) → M is an A-equivalence (2.10).

4.7. Homology localization. For our purposes a homology theory on R-mod is a functor S_{∗} from R-mod to graded abelian groups determined by the recipe

S_{∗}(M) = H_{∗}(S ⊗_{R} M)

for some object S in the derived category of right R-modules. An object M is said to be S_{∗}-acyclic if S_{∗}(M) = 0. A map M → M′ is an S_{∗}-isomorphism if it induces an isomorphism S_{∗}(M) ≅ S_{∗}(M′), or equivalently if its cofibre is S_{∗}-acyclic. An object M is S_{∗}-local if Hom_{R}(N, M) ≅ 0 for each S_{∗}-acyclic N. An S_{∗}-localization of M is an S_{∗}-equivalence M → M′ with the property that M′ is S_{∗}-local. A formal argument shows that S_{∗}-localizations are unique, if they exist; Bousfield’s arguments show that in fact they do exist [2] [3] [4].

4.8. Proposition. For any object M of R-mod, the natural map M → M_{ Â} is an S_{∗}-localization map for S = A^{♯}.

Proof. It is necessary to show that M_{ Â} is S_{∗}-local and that M → M_{ Â} is an S_{∗}-equivalence. Given that A^{♯} ⊗_{R} M ≅ Hom_{R}(A, M), the first statement follows from the fact that M_{ Â} is A-complete, and the second from the arguments of 2.11.

4.9. Proposition. For any object M of R-mod, the natural map M → M_{1/A} is an S_{∗}-localization map for S = R_{1/A}

Proof. In this case S ⊗_{R} M ≅ M_{1/A} (4.3), so it is enough to show that (M_{1/A})_{1/A} ≅ M_{1/A} (i.e., that M → M_{1/A} is an S_{∗}-equivalence), and that if N_{1/A} ≅ 0, then Hom_{R}(N, M_{1/A}) ≅ 0 (i.e., that M_{1/A} is S_{∗}- local). The first statement follows from the fact that CW_{A}(CW_{A} M) ≅ CW_{A}(M), so that (CW_{A} M)_{1/A} ≅ 0. The second follows from the fact that if N_{1/A} ≅ 0 then N ≅ CW_{A} N, i.e., N is A-torsion, so that Hom_{R}(N, K) ≅ 0 for any A-trivial object K, in particular, for K = M_{1/A}.

4.10. Nullification. Suppose that W is an object of R-mod. An object N of R-mod is said to be W -null if Hom_{R}(W, N) ≅ 0. A map M → M′ is a W -null equivalence if it induces an isomorphism Hom_{R}(M′, N) → Hom_{R}(M, N) for each W -null object N. A map f : M → M′ is a W -nullification of M if M′ is W -null and f is a W -null equivalence. A formal argument shows that W -nullifications are unique up to isomorphism if they exist; the arguments of Bousfield and Dror-Farjoun show that in fact they do exist [4].

4.11. Proposition. For any object M of R-mod, the map M → M_{1/A} is a W -nullification of M for W = A.

Proof. Note that for W = A, “W -null” is the same as “A-trivial”. The object M_{1/A} is A-trivial because CW_{A}(M) → M is an A-equivalence. The map M → M_{1/A} is a W -null equivalence because its fibre CW_{A}(M) is A-torsion, and so Hom_{R}(CW_{A}(M), N) ≅ 0 for any A-trivial N.

4.12. An arithmetic square. For our purposes, the term “homotopy fibre square” means a commutative square which can be completed in such a way as to induce an isomorphism between the fibre of the right vertical map and the fibre of the left vertical map.

4.13. Proposition. For any object M of R-mod, there is a homotopy fibre square

\begin{matrix}M & ⟶ & M_{ Â} \\ ↓ & & ↓ \\ M_{1/A} & ⟶ & (M_{ Â})_{1/A} \end{matrix}.

Proof. Since M → M_{ Â} is an A-equivalence, it induces an isomorphism CW_{A}(M) → CW_{A}(M_{ Â}). This is the required isomorphism between the fibres.

5. Homology groups

In this section we describe conditions under which it is possible to determine whether an object of R-mod is A-torsion or A-complete by examining its homology groups one by one.

We emphasize that in this section an R-module, as opposed to an object of R-mod, is an ordinary classical left R-module. Of course, an R-module M can be viewed as a chain complex concentrated in degree 0, and thus treated as an object of R-mod.

5.1. Definition. An R-module M is homotopically A-complete if the natural map M → M_{ Â} is an isomorphism in R-mod. An R-module M is homotopically A-torsion if the natural map CW_{A}(M) → M is an isomorphism in R-mod.

The main results of this section are the following ones. Recall that CW_{A}(R) can be considered to be an object of the derived category of Rbimodules (4.4). For the statements, we will say that a chain complex is essentially concentrated between dimensions i and j (i ≤ j) if it is isomorphic in the appropriate derived category to a chain complex of projective modules which vanishes except between dimension i and dimension j.

5.2. Proposition. Suppose there exists an n ≥ 0 such that as an object of R-mod the complex CW_{A}(R) is essentially concentrated between dimensions (−n) and 0. Then an object X of R-mod is A-complete if and only if each homology group H_{i}(X) is homotopically A-complete.

5.3. Proposition. Suppose there exists an n ≥ 0 such that as an object of mod-R the complex CW_{A}(R) is essentially concentrated between dimensions (−n) and 0. Then an object X of R-mod is A-torsion if and only if each homology group H_{i}(X) is homotopically A-torsion.

5.4. Remark. Somewhat surprisingly, an R-module M is homotopically A-complete if and only if H_{i}(M_{ Â}) = 0 for i < 0 and the natural map M → H_{0}(M_{ Â}) is an isomorphism. The surprise is that under these conditions the groups H_{i}(M_{ Â}) vanish for i > 0. For suppose the conditions are satisfied. Let X be the quotient of M_{ Â} obtained by dividing out by the cycles in dimension 1 and by all elements in dimensions > 1. Then X ≅ M, and the composite

M → M_{ Â} → X

exhibits M as a retract of the A-complete complex M_{ Â}. It follows that M is A-complete, so that M_{ Â} ≅ M and M is homotopically Acomplete. It is useful to note that (by 4.3) the condition H_{i}(M_{ Â}) = 0 for i < 0 is automatically satisfied under the assumptions of 5.2.

A similar argument shows that M is homotopically A-torsion if and only if H_{i} CW_{A}(M) vanishes for i > 0 and the map H_{0} CW_{A}(M) → M is an isomorphism. Again, it is useful to note that (by 4.3) the condition H_{i} CW_{A}(M) = 0 for i > 0 is automatically satisfied under the assumptions of 5.3.

Proof of 5.2. Let C = CW_{A}(R), which can be taken to be a projective chain complex supported between dimension (−n) and 0. For any object X of R-mod, X_{Â} is isomorphic to Hom_{R}(C, X) (see 4.3). Suppose that the homology groups of X are homotopically A-complete. Let X⟨i, j⟩ (i ≤ j) be the subquotient of X which agrees with X between dimensions i and j, has X_{j+1}/cycles in dimension j + 1, has ∂(X_{i}) in dimension i − 1, and has zero elsewhere. In particular, the homology groups of X⟨i, j⟩ agree with the homology groups of X between dimensions i and j and are otherwise zero. The cofibre sequences

X⟨j + 1, j + 1⟩ → X⟨i, j + 1⟩ → X⟨i, j⟩

allow X⟨i, j⟩ to be pieced together inductively out of complexes with only one nonzero homology group. By assumption, each of these complexes is A-complete. It follows that X⟨i, j⟩ is A-complete, and hence that the (evident) complexes X⟨i, ∞⟩ are A-complete, since

Hom_{R}(C, X⟨i, ∞⟩) ≅ Hom_{R}(C, holim_{j} X⟨i, j⟩) ≅ holim_{j} Hom_{R}(C, X⟨i, j⟩) ≅ holim_{j} X⟨i, j⟩ ≅ X⟨i, ∞⟩.

The identification X⟨i, ∞⟩ ≅ holim_{j} X⟨i, j⟩ is made by noting that before passing to the derived category the tower {X⟨i, j⟩}_{j≥i} of chain complexes is a tower of epimorphisms with inverse limit X⟨i, ∞⟩. Since C is concentrated in a finite number of dimensions, an easy connectivity argument shows that

Hom_{R}(C, X) ≅ Hom_{R}(C, colim_{i} X⟨i, ∞⟩)≅ colim_{i} Hom_{R}(C, X⟨i, ∞⟩).

The point is that the direct system {X⟨i, ∞⟩} is convergent in the sense that the homology groups above any given dimension stabilize after a certain point. It follows that X is A-complete.

Suppose on the other hand that X is A-complete. We need to show that for any k, the R-module H_{k}(X) is homotopically A-complete. By the dimensional assumption on C there are isomorphisms

H_{k}Hom_{R}(C, X⟨k − n − 1, ∞⟩) ≅ H_{k}Hom_{R}(C, X) = H_{k}(X)

and so after replacing X by the complex on the left, it is possible to assume that the homology groups of X vanish below a certain point. We now show by ascending induction on k that H_{i}(X) is homotopically A-complete for i ≤ k. This is true for k<<0, because in that case the groups all vanish. Suppose that the claim is true for k − 1; we must show that the module M = H_{k}(X) is homotopically A-complete. By the inductive assumption, X⟨−∞, k − 1⟩ is a complex with homology groups that are homotopically A-complete, and so, as above, this complex is A-complete. The cofibre sequence

X⟨k, ∞⟩ → X → X⟨−∞, k − 1⟩

shows that X⟨k, ∞⟩ is A-complete. Since C vanishes in positive dimensions and the map X⟨k, ∞⟩ → X⟨k, k⟩ ≅ Σ^{k}M is an isomorphism on homology up through dimension k, the induced map

M ≅ H_{k}Hom_{R}(C, X⟨k, ∞⟩) → H_{0}Hom_{R}(C, M)

is also an isomorphism. By 5.4, M is homotopically A-complete.

The proof of 5.3 is exactly parallel to the one above, with the opposite orientation (upward induction replaced by downward induction). It depends on the tensor product formula for CW_{A}(X) from 4.3.

6. Commutative Rings

In this section, we assume that R is commutative and that I ⊂ R is a finitely generated ideal. We wish to study the A-complete and Atorsion objects of R-mod in the special case A = R/I, but the theory from the rest of the paper does not immediately apply, because R/I need not be perfect, i.e., R/I need not have a finite length resolution by finitely generated projective R-modules. To get around this problem we construct an associated perfect complex K with the property that an object M of R-mod is K-torsion, K-complete, or K-trivial if and only if M is R/I-torsion, R/I-complete, or R/I-trivial. Elaborating a little on the construction of K leads to explicit formulas for the torsion functor CW_{K}(·) and the completion functor (·)_{K̂}; these turn out to be identical to the usual local cohomology and local homology functors with respect to the ideal I. This, for instance, gives an interpretation of local homology with respect to I as the Bousfield localization functor on the category R-mod associated to the homology theory M ↦ H_{∗}(R/I ⊗_{R} M).

The Koszul complex. For an element r in R, let K^{•}(r) denote the chain complex r : R → R, with the two copies of R in dimensions 0 and (−1), respectively. For a sequence r = (r_{1}, …, r_{n}), let K^{•}(r) be the tensor product K^{•}(r_{1})⊗_{R} ⋯⊗_{R} K^{•}(r_{n}). This is the Koszul complex associated to r.

Recall that if A and B are two objects of R-mod, then A is said to be built from B if A is in the smallest full subcategory of R-mod which contains B and is closed under isomorphisms, suspensions, coproducts, and cofibre sequences. If A is built from B then the class of A-trivial objects of R-mod contains the class of B-trivial objects. If A and B can each be built from one another, then the classes of A-trivial and B-trivial objects coincide, and it follows immediately that the classes of A-torsion and B-torsion objects also coincide, as do the classes of A-complete and B-complete objects.

The following is the property of K^{•}(r) which interests us.

6.1. Proposition. Suppose that R is a commutative ring and that I ⊂ R is an ideal generated by the sequence r = (r_{1}, …, r_{n}). Then the two objects R/I and K^{•}(r) of R-mod can each be built from one another.

6.2. Remark. Given 6.1 the results in the previous sections can be applied with A = K^{•}(r) to give conclusions involving R/I. It follows from 2.1, for instance, that the category of R/I-torsion objects of R-mod is equivalent to mod-E, where E = End_{R}(K^{•}(r)). The same is true of the category of R/I-complete objects of R-mod. Note that E is a matrix algebra of rank 2^{n} over R, graded so as to lie between dimensions −n and n, and provided with a suitable differential.

Convention. Suppose that R is a commutative ring and that I ⊂ R is an ideal generated by the finite sequence r. For the rest of this section we will write CW_{R/I}(·) and (·)_{R̂/I} instead of CW_{K}(·) and (·)_{K̂} with K = K^{•}(r). These objects are defined in terms of K (§2) but, in view of 6.1 and the results of §4, easy to interpret in terms of R/I. For instance, it follows from 4.6 that the natural map CW_{R/I}(M) → M is an R/Icellular approximation map. There is a corresponding interpretation of (M)_{R̂/I} in 6.13.

Proof of 6.1. Let K = K^{•}(r). Any object M of R-mod can be built out of R, and it follows immediately that K ⊗_{R} M can be built out of K. Since K ⊗_{R} (R/I) is a direct sum of shifted copies of R/I, we conclude that R/I can be built from K.

Conversely, note that for each i the map r_{i} : K → K is chain homotopic to zero, because r_{i} : K^{•}(r_{i}) → K^{•}(r_{i}) is chain homotopic to zero. It follows that the homology groups of K are modules over the ring R/I. Any module M over R/I can be built out of R/I as an object in the derived category of R/I, and the same recipe will build M out of R/I in the category R-mod. In the notation of the proof of 5.2, there are cofibre sequences

K⟨j, j⟩ → K⟨−∞, j⟩ → K⟨−∞, j − 1⟩.

As just noted, each one of the fibres is built from R/I. The fact that K itself is built from R/I follows from a finite induction, beginning with the fact that K⟨−∞, −(n + 1)⟩ ≅ 0 and ending with the fact that K⟨−∞, 0⟩ = K.

The dual Koszul complex. If r ∈ R, let K_{•}(r) denote the chain complex r : R → R, with the two copies of R in dimensions 1 and 0, respectively. Clearly K_{•}(r) = ΣK^{•}(r) ≅ K^{•}(r)^{♯}. For a sequence r = (r_{1}, …, r_{n}), let K_{•}(r) denote K_{•}(r_{1}) ⊗_{R} ⋯ ⊗_{R} K_{•}(r_{n}). Then K_{•}(r) = Σ^{n}K^{•}(r) ≅ K^{•}(r)^{♯}. Since K^{•}(r) and K_{•}(r) are suspensions of one another, the following proposition is a consequence of 6.1.

6.3. Proposition. Suppose that R is a commutative ring and I ⊂ R is an ideal generated by the sequence r = (r_{1}, …, r_{n}). Then the two objects R/I and K_{•}(r) of R-mod can each be built from one another.

As a consequence we obtain the following.

6.4. Proposition. Suppose that R is a commutative ring and I ⊂ R is an ideal generated by the sequence r = (r_{1}, …, r_{n}). Then for an object M of R-mod the following four conditions are equivalent:

1. R/I ⊗_{R} M ≅ 0,

2. K^{•}(r) ⊗_{R} M ≅ 0,

3. Hom_{R}(K_{•}(r), M) ≅ 0, and

4. Hom_{R}(R/I, M) ≅ 0.

Proof. The first and second are equivalent by 6.1, the third and fourth by 6.3, the second and third because K_{•}(r) = K^{•}(r)^{♯}.

6.5. Remark. It is a little surprising that the first and fourth conditions of 6.4 are equivalent in such generality.

Local cohomology. For r ∈ R, the commutative diagram

\begin{matrix}R & ⟶^{=} & R \\ _{r^{k}}↓ & & ↓_{r^{k+1}} \\ R & ⟶^{r} & R \end{matrix} (6.6)

gives a map K^{•}(r^{k}) → K^{•}(r^{k+1}). Let K^{•}(r^{∞}) denote colim_{k} K^{•}(r^{k}); this is just the flat complex r : R → R[1/r]. It is easy to see that K^{•}(r^{∞}) is isomorphic in R-mod to a free chain complex over R with nonzero groups only in dimensions 0 and −1; this can be taken to be the chain complex

\begin{matrix}d : ⊕_{i≥0}R → ⊕_{i≥0}R \\ d(x_{0}, x_{1}, …) = (x_{0} + x_{1}, x_{2} − rx_{1}, x_{3} − rx_{2}, …) \end{matrix}.

If r = (r_{1}, …, r_{n}) is a sequence of elements in R, we let r^{k} denote (r_{1}^{k}, …, r_{n}^{k}). Tensoring together the maps from 6.6 gives a map K^{•}(r^{k}) → K^{•}(r^{k+1}). Let K^{•}(r^{∞}) denote colim_{k} K^{•}(r^{k}), so that

K^{•}(r^{∞}) ≅ K^{•}(r_{1}^{∞}) ⊗_{R} ⋯ ⊗_{R} K^{•}(r_{n}^{∞}). (6.7)

The following proposition is clear.

6.8. Lemma. If r = (r_{1}, …, r_{n}), then K^{•}(r^{∞}) is isomorphic in R-mod to a free chain complex over R which is concentrated between dimensions (−n) and 0.

Suppose that I ⊂ R is the ideal generated by r. If M is an (ordinary) R-module, the local cohomology of M at I (see [8] or [7]) is denoted H_{I}^{∗}(M) and defined by the formula

H_{I}^{k}(M) = H_{−k}(K^{•}(r^{∞}) ⊗_{R} M).

In line with this, if M is an arbitrary object of R-mod we define H_{I}(M) = K^{•}(r^{∞}) ⊗_{R} M and call H_{I}(M) the derived local cohomology of M at I. There is a Kunneth spectral sequence

E_{p,q}^{2} = H_{I}^{−p}(H_{q}(M)) ⇒ H_{p+q}(H_{I}(M)).

This converges strongly because K^{•}(r^{∞}) is a flat chain complex concentrated in a finite range of dimensions.

6.9. Proposition. Suppose that R is a commutative ring and that I ⊂ R is the ideal generated by the sequence r = (r_{1}, …, r_{n}). Then CW_{R/I}(R) is isomorphic as an object of R-mod to K(r^{∞}).

Proof. Let K = K^{•}(r) and K^{∞} = K^{•}(r^{∞}). There is a natural map K^{∞} → R which amounts to taking a quotient by the elements in K^{∞} of strictly negative dimension. We have to show that K^{∞} is K-torsion, i.e., built from K (see 4.6) and that the map K^{∞} → R becomes an isomorphism when Hom_{R}(K, ·) is applied.

Fix k, and consider an ordinary R-module M on which the elements r_{1}^{k}, ⋯, r_{n}^{k} act trivially. Such a module is annihilated by the ideal I^{nk} and so has a finite filtration {I^{j}M} such that the associated graded modules are annihilated by I. In particular, each subquotient {I^{j}M/I^{j+1}M} is a module over R/I and so, as in the proof of 6.1, can be built from R/I. The cofibre sequences

I^{j}M/I^{j+1}M → M/I^{j+1}M → M/I^{j}M

allow for an inductive proof that the quotient modules M/I^{j}M, and hence also the module M itself, are built from R/I. Since each homology group of K^{•}(r^{k}) is annihilated by r_{1}^{k}, ⋯, r_{n}^{k}, it now follows from the argument in the proof of 6.1 that K^{•}(r^{k}) can be built from R/I. Passing to a directed (homotopy) colimit shows that K^{∞} can be built from R/I and hence (6.1) from K.

To finish the proof, it is enough, by 6.4, to show that tensoring the above map K^{∞} → R with R/I gives an equivalence

R/I ⊗_{R} K^{∞} → R/I ⊗_{R} R = R/I.

However, K^{∞} is given as a flat chain complex (6.7) and so we can compute the derived tensor product on the left as an ordinary tensor product. In this interpretation, the displayed map is an actual isomorphism of chain complexes, since R[1/r_{i}] ⊗_{R} R/I = 0 for i = 1, …, n.

The following proposition is a consequence of 6.9 and 4.3.

6.10. Proposition. Suppose that R is a commutative ring and that I ⊂ R is a finitely generated ideal. Then local cohomology at I computes R/I-cellular approximation, in the sense that for any object M of R-mod there is a natural isomorphism H_{I}(M) ≅ CW_{R/I}(M).

We can now apply the results of §5 to give a simple characterization of the R/I-cellular objects of mod-R. An ordinary R-module M is said to be an I-power torsion module if for each x ∈ M there is a k such that I^{k}x = 0.

6.11. Proposition. Suppose that R is a commutative ring and that I ⊂ R is a finitely generated ideal. Then an object M of R-mod is R/I-cellular if and only if each homology group of M is an I-power torsion module.

Proof of 6.11. Let r be a sequence of generators for I, and K = K^{•}(r). It follows from 4.6 and 6.1 that an object M of R-mod is R/I-cellular if and only if M is K-cellular, or, equivalently, K-torsion.

It is easy to argue from the definitions that if M is built from K then the homology groups of M are I-power torsion modules. Suppose that the homology groups of M are I-power torsion modules. Let K^{∞} = K^{•}(r^{∞}). The argument in the proof of 6.9 shows immediately that K^{∞} ⊗_{R} H_{i}(M) ≅ H_{i}(M) for all i, so by 6.9 CW_{K}(H_{i}M) ≅ H_{i}M. In other words, all of the homology groups of M are homotopically K-torsion. In view of 6.8 and 5.3, M is K-torsion.

Local homology. Suppose that I ⊂ R is an ideal generated by the sequence r. If M is an (ordinary) R-module, the local homology of M at I (see [7]) is denoted H_{∗}^{I}(M) and defined by the formula

H_{k}^{I}(M) = H_{k}Hom_{R}(K^{•}(r^{∞}), M).

In line with this, if M is an arbitrary object of R-mod we define H^{I}(M) = Hom_{R}(K^{•}(r^{∞}), M) and call H^{I}(M) the derived local homology of M at I. It is easy to construct a spectral sequence

E_{p,q}^{2} = H_{p}^{I}(H_{q}(M)) ⇒ H_{p+q}(H^{I}(M)) .

This converges strongly because K^{•}(r^{∞}) is equivalent to a projective chain complex of finite length.

The following proposition is a consequence of 6.9 and 4.3.

6.12. Proposition. Suppose that R is a commutative ring and that I ⊂ R is a finitely generated ideal. Then local homology at I computes R/I-completion, in the sense that for any object M of R-mod there is a natural isomorphism H^{I}(M) ≅ M_{ R̂/I}.

According to 4.8, the natural map M → M_{ K̂} is a Bousfield localization map for the homology theory on R-mod given by M ↦ H_{∗}(K^{♯} ⊗_{R} M). By 6.3, this homology theory has the same acyclic objects as the theory given by M ↦ H_{∗}(R/I ⊗_{R} M), and thus the same notion of localization. We have obtained the following interpretation of (·)_{R̂/I}, and hence of local homology

6.13. Proposition. Suppose that R is a commutative ring and that I ⊂ R is a finitely generated ideal. Then the natural map M → M_{ R̂/I} is a Bousfield localization map for the homology theory on R-mod given by M ↦ H_{∗}(R/I ⊗_{R} M). In particular, since H^{I}(M) ≅ M_{ R̂/I}, local homology at I computes Bousfield localization with respect to the homology theory determined by R/I.

Finally, along the lines of 6.11 we obtain the following characterization of the objects of R-mod which are local with respect to the above homology theory. It depends on combining 5.2 and 5.4 with the above observation that these local objects are exactly the objects which are K-complete.

6.14. Proposition. Suppose that R is a commutative ring and that I ⊂ R is a finitely generated ideal. Then an object M of R-mod is local with respect to the homology theory on R-mod given by M ↦ H_{∗}(R/I⊗_{R}M) if and only if for each integer k the natural map H_{k}(M) → H_{0}^{I}(H_{k}M) is an isomorphism.

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University of Notre Dame, Notre Dame, IN 46556, USA

E-mail address: dwyer.1@nd.edu

School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.

E-mail address: j.greenlees@sheffield.ac.uk