A Note on Cellular Approximations and Torsion Theory

Shoham Shamir

June 16, 2008

Abstract

Let R be a ring and A a dg-R-module. The inclusion functor of the localizing subcategory generated by A into the derived category of R has a right adjoint, denoted Cell_{A}. In [1], Benson shows how to compute Cell_{A}R when A is a simple module and R is an Artinian ring. We generalize this construction to the case where A is the direct sum of all cyclic torsion modules for some hereditary torsion theory T on R. In this case we assert that for every R-module M there exists an injective R-module E such that:

H^{n}(Cell_{A}M) ≅ Ext_{End_{R}(E)}^{n−1}(hom_{R}(M, E), E) for n ≥ 2

1 introduction

Let R be a ring and let D_{R} be the (unbounded) derived category of chain complexes of left R-modules. Fix an object A of D_{R}. We recall the following definitions from [2]. An object N of D_{R} is called A-null if Ext_{R}^{∗}(A, N) = 0. An object C of D_{R} is called A-cellular if Ext_{R}^{∗}(C, N) = 0 for every A-null N. An A-cellular object C is an A-cellular approximation of X ∈ D_{R} if there is a map μ : C → X such that Ext_{R}^{∗}(A, μ) is an isomorphism. Finally, we say an A-null object N is an A-nullification of X if there is a map ν : X → N which is universal among maps in D_{R} from X to A-null objects. We denote an A-cellular approximation of X by Cell_{A}X and an A-nullification of X by Null_{A}X. Note there is an exact triangle Cell_{A}X → X → Null_{A}X for every X ∈ D_{R}.

It turns out that the full subcategory DCell_{A} of A-cellular objects is the localizing subcategory of D_{R} generated by A. Moreover, one can construct a functor ℭ𝔢𝔩𝔩_{A} that for every object X of D_{R} produces an Acellular approximation of X. One readily sees that ℭ𝔢𝔩𝔩_{A} is right adjoint to the inclusion of the localizing subcategory generated by A into D_{R}. Similarly, it is possible to construct a left adjoint to the inclusion of the full subcategory of A-null objects (commonly denoted A^{⊥}) into D_{R}. This left adjoint functor gives for every object of D_{R} its A-nullification.

One method for calculating A-cellular approximations is the formula given by Dwyer and Greenlees in [2], which holds whenever A is a perfect chain-complex. This condition was generalized by Dwyer, Greenlees and Iyengar in [3]. A new method for calculating the A-cellular approximation for R-modules has been recently constructed by Benson in [1], dubbed k-squeezed resolutions. This method can be applied whenever A is a set of simple modules and R is an Artinian ring. In fact, it is more natural to use Benson’s method to construct the A-nullification of a module, rather than its A-cellular approximation.

We generalize Benson’s construction so that it applies whenever A is the direct sum of the set of cyclic torsion modules for a hereditary torsion theory. This gives an interesting relation between hereditary torsion theories and their cellular approximations. To be specific, let T be a hereditary torsion theory on left R-modules and let A be the direct sum of all the R-modules of the form R/a where a is a left ideal of R and R/a is T -torsion. We construct the A-nullification of an R-module using the module of quotients functor (a functor which assigns to each module a specific torsion free module, see 2.1 below). This construction shows, for example, that H^{0}(Null_{A}M) ≅ M_{F} for every R-module M. We also get the following result.

Theorem 1.1. Let T be a hereditary torsion theory and let A_{T} be the direct sum of the set of cyclic T -torsion modules. For every R-module M there exists an injective module E such that RHom_{End(E)}(hom_{R}(M, E), E) is an A_{T} -nullification of M. In particular, the differential graded algebra REnd_{End(E)}(E) is an A_{T} -nullification of R.

The formula given in the abstract follows immediately from the exact triangle: Cell_{A}M → M → Null_{A}M mentioned above.

As an example we study the case where R is an Artinian ring in Section 4. In this case every hereditary torsion theory is determined by the set of simple modules it contains and hence the analysis is quite simple. In that section we also give a different view and different proof to a result of Benson ([1, Theorem 5.1]).

1.1 Notation and Terminology

By a ring we always mean and associative ring with a unit, not necessarily commutative. Unless otherwise noted all modules considered are left modules. A triangle always means a distinguished (exact) triangle in the unbounded derived category of left R-modules, denoted D_{R}. A differential graded R-module, or dg-R-module, is just a chain-complex of R-modules. For graded and differential-graded objects we use the standard convention that subscript grading is the negative of the superscript grading, i.e. □_{−i} = □^{i}. For dg-R-modules X and Y the notation Hom_{R}(X, Y) stands for the usual chain-complex of morphism groups. Every R-module is also considered as a dg-R-module concentrated in degree 0 and with zero differential.

2 Background on Hereditary Torsion Theories and Cellular-Approximation, Nullification and Completion

We give the necessary background on hereditary torsion theories and on cellular-approximation, nullification and completion. At the end of this section we also begin discussing the example of cellular-approximation with respect to a hereditary torsion theory.

2.1 Hereditary Torsion Theories

Below is a recollection of the definition and main properties of hereditary torsion theories. A thorough review of this material can be found in [6].

Definition 2.1. A torsion theory T is a class of R-modules that is closed under quotient modules, coproducts and extensions. Closure under extensions means that if 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence with M_{1} and M_{3} in T , then so is M_{2}. A torsion theory is called hereditary if it is also closed under submodules. The modules in T will be called T -torsion modules (or just torsion modules when the torsion theory is clear from the context). The class of torsion-free modules F is the class of all modules F satisfying hom_{R}(C, F) = 0 for every C ∈ T . To every torsion theory there is an associated radical t, where t(M) is the largest torsion submodule of M.

Every class of modules C generates a torsion theory in the following manner. Define first the class of torsion-free module as

F = {F | hom_{R}(C, F) = 0 for all C ∈ C}

and then define the torsion modules as

T = {M| hom_{R}(M, F) = 0 for all F ∈ F}

Every hereditary torsion T theory has an injective cogenerator (see [6, VI.3.7]). This means there exists an injective module E such that a module M is torsion if and only if hom_{R}(M, E) = 0.

Let T be a hereditary torsion theory, let t be the associated radical and let F be the class of torsion-free modules. For every R-module M there is a natural map M → M_{F} where M_{F} is the module of quotients of M (see [6, IX.1]). The map M → M_{F} has the following properties: the kernel of the map M → M_{F} is t(M), M_{F} is torsion-free and the cokernel of this map is a torsion module. In fact, the module of quotients is a functor which is left adjoint to the inclusion of the full subcategory of F-closed modules. An R-module M is called F-closed if, for every left ideal a ⊂ R such that a ∈ F, the induced map M = Hom_{R}(R, M) → Hom_{R}(a, M) is an isomorphism.

2.2 Cellular-Approximation, Nullification and Completion

The following bit recalls the definitions and basic properties of cellular approximation and completion, as given by Dwyer and Greenlees in [2]. Note that in [2] A-cellular dgmodules are also called A-torsion dg-modules, we avoided that terminology for obvious reasons. The definition of nullification is also given here. Note that nullification is a specific type of localization with respect to a map. Such localization is described by Dwyer in [4] in the same algebraic context considered here.

Definition 2.2. Let R be a ring and let A be a dg-R-module. A map f : X → Y is called an A-equivalence if Ext_{R}^{∗}(A, f) is an isomorphism. A dg-R-module N is called A-null if Ext_{R}^{∗}(A, N) = 0. A dg-R-module C is called A-cellular if Ext_{R}^{∗}(C, N) = 0 for any A-null dg-module N. A dg-R-module C is called A-complete if Ext_{R}^{∗}(N, C) = 0 for any A-null dg-module N. We say that a dg-R-module C is an A-cellular approximation of a dg-R-module X if there is a map μ : C → X such that μ is an A-equivalence and C is A-cellular. This definition of nullification is equivalent to the definition given in the introduction and easier to check. We say that a dg-R-module N is an A-nullification of X if there is a triangle C → X → N such that C is A-cellular and N is A-null, note this immediately implies that C is an A-cellular approximation of X. Finally, a dg-R-module C is an A-completion of a dg-R-module X if there is a map κ : X → C such that κ is an A-equivalence and C is A-complete.

It is easy to show that all A-cellular approximations of a dg-R-module X are isomorphic to each other in D_{R}. Similarly all A-nullifications of X are isomorphic to each other in D_{R} and also all A-completions of X are isomorphic to each other in D_{R}. Denote an A-cellular approximation of X by Cell_{A}X, an A-nullification of X by is denoted by Null_{A}X and an A-completion of X is denoted by X_{A}^{∧}.

As mentioned above, nullification is a type of localization with respect to a map (see [4]). Indeed, A-nullification is the same as localization with respect to the map A → 0. A general method for constructing A-nullification is described in [4], at least for the case when A is a perfect dg-R-module (i.e. a finite chain-complex composed of finitely generated projective modules). Farjoun describes a general method of constructing such a nullification functor in [5], though there it is given in a topological context. Also note that from a method of constructing A-nullification one can immediately infer a way of constructing A-cellular approximations.

Definition 2.3. Suppose T is a hereditary torsion-theory on R-modules. Let A_{T} be the direct sum of all cyclic T -torsion modules. Say a map f : X → Y is a T -equivalence if f is an A_{T} equivalence. Similarly we say that a dg-R-module is T -cellular (respectively T - null) if it is A_{T} -cellular (A_{T} -null). We shall use the notation Cell_{T} and Null_{T} for Cell_{A_{T}} and Null_{A_{T}} respectively. Similarly by a T -cellular approximation and T -nullification we mean A_{T} -cellular approximation and A_{T} -nullification respectively.

2.3 Cellular-Approximation with respect to a Hereditary Torsion Theory

We make a small detour to note that for a hereditary torsion theory T , an R-module is T -torsion if and only if it is T -cellular. This is the result of the next two lemmas.

Lemma 2.4. Let T be a hereditary torsion theory, then every T -torsion module is T - cellular.

Proof. First note that every cyclic T -torsion module is T -cellular, simply because such a module is a retract of A_{T}. Therefore every direct sum of cyclic T -torsion modules is T - cellular. Let M be a T -torsion module, then there is a surjection ⊕_{m∈M}R/ann(m) ↠ M. Since every hereditary torsion theory is closed under submodules, then R/ann(m) is T -torsion for every m ∈ M. Clearly C_{M} = ⊕_{m∈M}R/ann(m) is T -cellular and T - torsion. Next we build a resolution C of M using T -cellular modules. Let C_{0} be C_{M}, and let d_{0} : C_{0} ↠ M the map defined above. The kernel ker(d_{0}) is T -torsion, so there is an epimorphism C_{ker(d_{0})} ↠ ker(d_{0}). Let C_{1} = C_{ker(d_{0})} and let d_{1} be the composition C_{1} ↠ ker(d_{0}) ↪ C_{0}. Thus we inductively build C and it is clear that C is quasi-isomorphic to M. Since each C_{n} is A_{T} -cellular, it is easy to show (using, for example, spectral sequence arguments) that Ext_{R}^{∗}(C, N) = 0 for every A_{T} -null dg-R-module N.

Lemma 2.5. Let T be a hereditary torsion theory and let X be a T -cellular dg-R-module, then the homology groups of X are T -torsion R-modules.

Proof. Let C be the subcategory of D_{R} whose homology groups are T -torsion R-modules. Clearly C is localizing subcategory that contains A_{T}, hence C also contains all tc-cellular objects.

Corollary 2.6. Let T be a hereditary torsion theory, then an R-module M is T -torsion if and only if M is T -cellular. In particular, the localizing subcategory of D_{R} generated by all the T -torsion modules is equal to the localizing subcategory generated by A_{T}.

3 Nullification Construction

In [1], Benson gives a construction called k-squeezed resolution which yields k-cellular approximations over the ring kG, where k is a prime field and G is a finite group. We generalize Benson’s construction so as to produce T -cellular approximations over any ring R, where T is a hereditary torsion theory. This construction is described next.

Let R be a ring and let T be a hereditary torsion theory on R with an associated radical t. The class of torsion-free modules will be denoted by F. For an R-module M we construct the T -nullification of M as a cochain complex I^{0} ⟶^{d} I^{1} ⟶^{d} I^{2} ⟶^{d} ⋯ inductively.

Let N^{0} = M_{F} be the the module of quotients with respect to the given torsion theory. Note there is a natural map M → N^{0} whose kernel is t(M). Let I^{0} be the injective hull of N^{0}. Since N^{0} is torsion free, so is I^{0}. We continue by induction. Let N^{i+1} be the module of quotients of I^{i}/d(I^{i−1}). As before, the kernel of the map I^{i}/d(I^{i−1}) → N^{i+1} is t(I^{i}/d(I^{i−1})). Let I^{i+1} be the injective hull of N^{i+1}, again I^{i+1} is torsion free - because N^{i+1} is. The differential d : I^{i} → I^{i+1} is the composition I^{i} → I^{i}/d(I^{i−1}) → N^{i+1} → I^{i+1}. Denote the resulting cochain-complex by I.

One can describe this construction in a slightly simpler way. For any R-module L, the injective hull of L_{F} and the injective-hull of L/t(L) are the same. This is because (L/t(L))_{F} = L_{F} and L/t(L) is an essential submodule of (L/t(L))_{F} (see [6, IX.2.4]). Hence we can define M^{0} = M and let I^{0} be the injective envelope of M^{0}/t(M^{0}). Then set M^{i+1} = I^{i}/d(I^{i−1}) and let I^{i+1} be the injective envelope of M^{i+1}/t(M^{i+1}). However, to define M^{1} we must use the module of quotients M_{F} and set M^{1} = I^{0}/M_{F}.

Lemma 3.1. The dg-R-module I constructed above is the T -nullification of of the module M. In particular, H_{0}(Null_{T} M) ≅ M_{F} for any R-module M.

Proof. It is clear that I is T -null, in fact I is A-null with respect to any T -torsion module A, because I is composed of torsion free injective modules. Let C be an object of D_{R} such that there is a triangle C → R → I. The homology of I is: H^{i}(I) = t(M^{i}) = t(I_{i}/d(I^{i−1})) for i > 0 and H^{0}(I) = M^{0} = M_{F}. Using the long exact sequence in homology we get: H^{0}(C) = t(M), H^{1}(C) = M_{F}/M and H^{i}(C) = H^{i−1}(I) = t(I^{i−1}/d(I^{i−2})) for i > 1. Note that M_{F}/M is a T -torsion module ([6, IX.1.5]). A simple argument (see [2, proof of Proposition 5.2]) shows that C belongs to the localizing category generated by the homology groups of C. Since the homology groups of C belong to the localizing subcategory generated by all T -cellular module, we conclude that C is T - cellular and therefore a T -cellular approximation of M. Hence I is the T -nullification of M.

Using the construction above we can give a different description of T -nullification, the one shown in Theorem 1.1. Before proving this theorem we note that for an Rmodule E, the functor Hom_{R}(−, E) is a contravariant functor from left dg-R-modules to left dg-End_{R}(E)-modules and the functor Hom_{End_{R}(E)}(−, E) is a contravariant functor from left dg-End_{R}(E)-modules to left dg-R-modules.

Proof of Theorem 1.1. Given an R-module M, construct a T -nullification of M in the way prescribed above. This results in a cochain-complex I, with I^{n} being an injective torsion-free module. The first step is to show there is an injective module E such that I^{n} is a direct summand of E for all n.

The hereditary torsion theory T has an injective cogenerator Ẽ ([6, VI.3.7]). By [6, VI.3.9], an R-module torsion-free if and only if it is a submodule of a product of copies of Ẽ. Hence for every n ≥ 0 there is an ordinal α_{n} such that I^{n} is a submodule of a product of α_{n} copies of Ẽ. Let α be an ordinal such that α ≥ α_{n} for every n (e.g. α = ∪_{n}α_{n}) and let E be the product of α copies of Ẽ. Then I^{n} is a submodule of E and since I^{n} is injective it is a direct summand of E. Denote by E the endomorphism ring End_{R}(E).

Now consider the triangle C → M → I (recall C is a T -cellular approximation of M). Applying the functor Hom_{R}(−, E) to this triangle yields a triangle in D_{E}:

Hom_{R}(C, E) ← Hom_{R}(M, E) ← Hom_{R}(I, E)

Since E is injective, H_{i}(Hom_{R}(C, E)) ≅ hom_{R}(H_{i}(C), E). Since the homology groups of C are torsion, hom_{R}(H_{i}(C), E) = 0. Therefore the map Hom_{R}(I, E) → Hom_{R}(M, E) is a quasi-isomorphism of dg-E-modules.

Since I^{n} is a direct summand of E, hom_{R}(I^{n}, E) is a direct summand of E and therefore a projective E-module. Thus the map Hom_{R}(I, E) → hom_{R}(M, E) is a projective resolution of hom_{R}(M, E) in the category of E-modules. We conclude that Hom_{E}(Hom_{R}(I, E), E) is the derived functor RHom_{E}(hom_{R}(M, E), E).

Because I^{n} is a direct summand of E, one readily sees that

hom_{E}(hom_{R}(I^{n}, E), E)

is naturally isomorphic to I^{n}. Hence Hom_{E}(Hom_{R}(I, E), E) ≅ I.

Remark 3.2. It is known that for any dg-R-module A, the A-nullification of R is quasiisomorphic to a differential graded algebra, see for example [4].

Remark 3.3. Theorem 1.1 can be seen as a generalization of [6, IX.3.3], which states that there is an injective R-module E such that R_{F} ≅ End_{End_{R}(E)}(E).

4 Torsion Theories and Cellular Approximation in Artinian Rings

Throughout this section R is an Artinian ring, S is a set of non-isomorphic simple modules of R and S′ denotes its complement. The torsion theory generated by S is denoted T , it is easy to see this is a hereditary torsion theory. Note that every hereditary torsion theory of R-modules is generated by a set of simple modules (see [6, VIII]), so this context covers all hereditary torsion theories over R. We give here several results regarding T -nullification as well as give a different proof of a result of Benson.

Let E be the product of the injective hulls of the simple modules in S′ and let P be the direct sum of the projective covers of those modules. We show that E is an injective cogenerator of T and that being T -cellular is the same as being (⊕_{S∈S}S)-cellular.

Lemma 4.1. Let X be a dg-R-module, then Ext_{R}^{∗}(P, X) = 0 if and only if Ext_{R}^{∗}(X, E) = 0.

Proof. This is well known when X is a finitely generated R-module. Now suppose X is any R-module. Since P is a finitely generated projective module we see that hom_{R}(P, X) = 0 if and only if hom_{R}(P, X′) = 0 for every finitely generated submodule X′ of X. Similarly, because E is injective, hom_{R}(X, E) = 0 if and only if hom_{R}(X′, E) = 0 for every finitely generated submodule X′ of X. Hence the lemma holds for any R-module. Finally, Let X be a dg-R-module, then Ext_{R}^{∗}(P, X) = hom_{R}(P, H_{∗}(X)). Similarly Ext_{R}^{∗}(X, E) = hom_{R}(H^{∗}(X), E).

Lemma 4.2. The module E is an injective cogenerator for the hereditary torsion theory T .

Proof. The class of modules M such that hom_{R}(M, E) = 0 is a hereditary torsion theory which contains S (since hom_{R}(S, S′) = 0 for every S ∈ S and S′ ∈ S′ and since E(S′) is the injective envelope of S′). Hence this class contains T - which is the torsion theory generated by S.

Next, let M be a module such that hom_{R}(M, E) = 0. To show that M is a torsion module it is enough to show that every cyclic submodule of M is a torsion module. This is because M is a quotient of ⊕_{C⊆M cyclic}C.

So let C be a cyclic submodule of M, then hom_{R}(C, ⊕_{S′∈S′}S′) = 0 (since E is the injective envelope of ⊕_{S′∈S′}S′, see [6, VI.3.8]). In particular, none of the simple modules S′ can be a quotient of C. Therefore there is an epimorphism C ↠ S for some S ∈ S (because R is an Artinian ring, and C a finitely generated module, hence of finite length). Now C ≅ R/a for some left ideal a ⊂ R, so C is both Artinian and noetherian and hence C has a composition series:

0 = C_{0} ⊂ C_{1} ⊂ ⋯ ⊂ C_{m} = C ,

where all the quotients C_{i}/C_{i−1} are simple modules. Suppose that C_{i}/C_{i−1} ≅ S′ for some S′ ∈ S′. Let x ∈ C_{i} \ C_{i−1}, then the cyclic module generated by x has S′ as a quotient. This implies the cyclic submodule Rx of M has a non-zero map to ⊕_{S′∈S′}S′, in contradiction. Therefore C_{i}/C_{i−1} ≅ S for some S ∈ S.

We now proceed by induction on i to show that C_{i} ∈ T . Assume that C_{i−1} ∈ T , we know that C_{i}/C_{i−1} ∈ T , since T is closed under extensions (see [6, VI.2.1]) we conclude that C_{i} ∈ T . Hence C = C_{n} ∈ T .

Corollary 4.3. A dg-R-module X is T -cellular if and only if X is (⊕_{S∈S}S)-cellular.

Proof. It is enough to show that every cyclic R-module is (⊕_{S∈S}S)-cellular, but that is obvious from the proof of the previous lemma.

A result of Benson ([1, Theorem 5.1]) can, after an obvious generalization, be written as follows:

H^{n}(Cell_{T} R) ≅ Ext_{End_{R}(P)}^{n−1}(hom_{R}(P, R), hom_{R}(P, R))

In fact, Benson’s result can be reformulated as:

RHom_{End_{R}(P)}(hom_{R}(P, R), hom_{R}(P, R)) is a T -nullification of R.

Interestingly, by Dwyer and Greenlees [2, Theorem 2.1], this dg-R-module is also the P -completion of R. Below we give a different proof of this.

Corollary 4.4. For any R-module M, a T -nullification of M is also a P -completion of M and therefore is given by the formula of Dwyer and Greenlees for P -completion ([2, 2.1]):

M_{P}^{∧} ≃ RHom_{End_{R}(P)}(Hom_{R}(P, R), Hom_{R}(P, M))

Proof. Consider the triangle Cell_{T} M → M ⟶^{ν} Null_{T} M. In the course of the proof of theorem 1.1 we saw that Ext_{R}^{∗}(ν, E) is an isomorphism, this implies Ext_{R}^{∗}(Cell_{T} M, E) = 0. Hence we conclude that Cell_{T} M is P -null and ν is a P -equivalence.

It remains to show that Null_{T} M is P -complete. Let I be the T -nullification of M described in 3. The full subcategory of P -complete objects in D_{R} is closed under isomorphisms, triangles and products, we denote this subcategory by C. Clearly E ∈ C and also every product of E. Since I^{n} is a direct summand of some product of copies of E, I^{n} is also an object of C.

Let I(n) denote the cochain complex I^{0} → I^{1} → ⋯ → I^{n}. An inductive argument shows that I(n) ∈ C. There is a triangle I → ∏_{n} I(n) ⟶^{ϕ−1} ∏_{n} I(n) where the map ϕ is defined by ϕ(a_{1}, a_{2}, a_{3}, ...) = (a_{2}, a_{3}, ...) (the n’th coordinate is in I(n)). Hence I is P -complete.

References

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