---
abstract: 'Let $R$ be a ring and let $\ab$ be a hereditary torsion class of $R$-modules.  The inclusion of the localizing subcategory generated by $\ab$ into the derived category of $R$ has a right adjoint, denoted $\cell_\ab$. In [@Benson], Benson shows how to compute $\cell_\ab R$ when $R$ is a group ring of a finite group over a prime field and $\ab$ is the hereditary torsion class generated by a simple module.  We generalize Benson’s construction to the case where $\ab$ is any hereditary torsion class on $R$. It is shown that for every $R$-module $M$ there exists an injective $R$-module $E$ such that: $$H^n(\cell_\ab M) \cong \ext_{\End_R(E)}^{n-1}(\Hom_R(M,E),E) \text{ for } n\geq 2$$'
address: 'Department of Mathematics, University of Bergen, 5008 Bergen, Norway'
author:
- Shoham Shamir
bibliography:
- 'bib2007.bib'
title: Colocalization Functors in Derived Categories and Torsion Theories
---

Introduction {#sec: Introduction}
============

Let $R$ be a ring and let $\Derived_R$ be the (unbounded) derived category of chain complexes of left $R$-modules.  Fix a class $\ab$ of objects of $\Derived_R$. We recall some definitions of Dwyer and Greenlees from [@DwyerGreenlees].  An object $N$ of $\Derived_R$ is *$\ab$-null* if $\ext^*_R(A,N)=0$ for every $A\in \ab$. An object $C$ of $\Derived_R$ is *$\ab$-cellular* if $\ext^*_R(C,N)=0$ for every $\ab$-null $N$. An $\ab$-cellular object $C$ is an *$\ab$-cellular approximation* of $X\in \Derived_R$ if there is a map $\mu:C \to X$ such that $\ext^*_R(A,\mu)$ is an isomorphism for all $A\in\ab$. Finally, an $\ab$-null object $N$ is an *$\ab$-nullification* of $X$ if there is a map $\nu:X \to N$ which is universal among maps in $\Derived_R$ from $X$ to $\ab$-null objects.  Denote an $\ab$-cellular approximation of $X$ by $\cell_\ab X$ and an $A$-nullification of $X$ by $\Null_\ab X$.

The following properties are easy to check.  A map $\mu:C \to X$ is an $\ab$-cellular approximation of $X$ if and only if it is universal among all maps from $\ab$-cellular objects to $X$. There is an exact triangle $\cell_\ab X \to X \to \Null_\ab X$ whenever $\cell_\ab X$ or $\Null_\ab X$ exists.  An $\ab$-cellular approximation of some object $X$ is unique up to isomorphism and the same goes for an $\ab$-nullification of $X$.

Now suppose $\ab$ is a set, then it turns out that the full subcategory of $\ab$-cellular objects is the localizing subcategory generated by $\ab$ (see [@DwyerGreenleesIyengar] and [@Hirschhorn 5.1.5]).  Moreover, when $\ab$ is a set the inclusion functor of the full subcategory of $A$-cellular objects into $\Derived_R$ has a right adjoint, which is $\cell_\ab X$ for every $X\in \Derived_R$, see [@Hirschhorn] or [@FarjounBook].  Hence $\cell_\ab$ can be constructed as a *colocalization* functor (the right adjoint of an inclusion functor), and it follows that $\ab$-cellular approximation and $\ab$-nullification exist for any object of $\Derived_R$.

Similarly, when $\ab$ is a set there exists a left adjoint to the inclusion of the full subcategory of $\ab$-null objects, see for example Neeman’s book [@NeemanBook Section 9].  This functor is in fact $\ab$-nullification and it is a *localization* functor (the left adjoint to an inclusion functor).  One method for calculating $\ab$-cellular approximations is the formula given by Dwyer and Greenlees in [@DwyerGreenlees], which holds whenever $\ab=\{A\}$ and $A$ is a perfect complex.  This was later generalized by Dwyer, Greenlees and Iyengar in [@DwyerGreenleesIyengar].  A new method for calculating the $\ab$-cellular approximation for $R$-modules has been constructed by Benson in [@Benson], dubbed *$k$-squeezed resolutions*.  This method can be applied whenever $\ab$ is a set of simple modules and $R$ is an Artinian ring.  One major benefit of Benson’s construction is that it allows for explicit calculations. 

As we will see, it is more natural to use Benson’s method to construct the $\ab$-nullification of a module, rather than its $\ab$-cellular approximation.  We generalize Benson’s construction so that it applies whenever $\ab$ is a hereditary torsion class of modules.  A *hereditary torsion class* of modules is a class of modules that is closed under submodules, quotient modules, coproducts and extensions.  The main result of this paper is the following. 

\[the: Nullification as double endomorphism\] Let $\tc$ be a hereditary torsion class on left $R$-modules.  For every left $R$-module $M$ there exists an injective left $R$-module $E$ such that the complex $$\mathbf{R}\Hom_{\End_R(E)}(\Hom_R(M,E),E)$$ is a $\tc$-nullification of $M$. In particular, the differential graded algebra $\mathbf{R}\End_{\End_R(E)}(E)$ is a $\tc$-nullification of $R$.

The formula given in the abstract follows immediately from the distinguished triangle $\cell_\tc M \to M \to \Null_\tc M$ mentioned above. 

The layout of this paper is as follows.  The necessary background on hereditary torsion classes and the background on cellular approximations and nullifications is given in Section \[sec: Background\].  In Section \[sec: The Construction\] we describe the construction of nullification with respect to a hereditary torsion class and prove Theorem \[the: Nullification as double endomorphism\].  We study the case where $R$ is an Artinian ring in Section \[sec: Torsion Theories and Cellular Approximation in Artinian Rings\].  This section offers a different proof to a result of Benson ([@Benson Theorem 5.1]).  Finally, Section \[sec: Examples\] provides several examples. 

Notation and Terminology
------------------------

By a ring we always mean an associative ring with a unit, not necessarily commutative.  Unless otherwise noted all modules considered are left modules.  A triangle always means an exact (distinguished) triangle in the unbounded derived category of left $R$-modules, denoted $\Derived_R$. A complex is always a chain-complex of $R$-modules.  For complexes we use the standard convention that subscript grading is the negative of the superscript grading, i.e. $\square_{-i}=\square^i$. It is taken for granted that every $R$-module is a complex concentrated in degree 0 and with zero differential.  A complex $X$ is *bounded-above* if for some $n$ and for all $i>n$, $H_i(X)=0$. For complexes $X$ and $Y$ the notation $\Hom_R(X,Y)$ stands for the usual chain complex of homomorphisms.  The notation $\mathbf{R}\Hom_R(-,-)$ stands for the derived functor of the $\Hom_R(-,-)$ functor.  By $\End_R(M)$ we mean the endomorphisms ring of an $R$-module $M$. The symbol $\simeq$ stands for quasi-isomorphism of complexes. 

Background on Hereditary Torsion Theories and Cellular-Approximation, Nullification and Completion {#sec: Background}
==================================================================================================

Hereditary Torsion Theories {#sub: Recollection about 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 [@Stenstrom]. 

A *hereditary torsion class* $\tc$ is a class of $R$-modules that is closed under submodules, quotient modules, coproducts and extensions.  Closure under extensions means that if $0 \to M_1 \to M_2 \to M_3 \to 0$ is a short exact sequence with $M_1$ and $M_3$ in $\tc$, then so is $M_2$. The modules in $\tc$ will be called *$\tc$-torsion modules* (or just torsion modules when the torsion theory is clear from the context).  The class of *torsion-free* modules $\ff$ is the class of all modules $F$ satisfying $\Hom_R(C,F)=0$ for every $C \in \tc$. The pair $(\tc,\ff)$ is referred to as a *hereditary torsion theory*.  To every hereditary torsion theory there is an associated radical $t$, where $t(M)$ is the maximal torsion submodule of $M$. Note that $M/t(M)$ is therefore torsion-free. 

Every hereditary torsion class $\tc$ has an *injective cogenerator* (see [@Stenstrom VI.3.7]).  This means there exists an injective module $E$ such that a module $M$ is torsion if and only if $\Hom_R(M,E)=0$. It is also important to note that in any hereditary torsion theory, the class of torsion-free modules is closed under injective hulls (see [@Stenstrom VI.3.2]).  Thus, if $F$ is a torsion-free module then the injective hull of $F$ is also torsion-free. 

\[def: Module of quotients\] Let $(\tc,\ff)$ be a hereditary torsion theory and let $t$ be the associated radical.  An $R$-module $M$ is called *$\ff$-closed* if, for every left ideal $\aideal \subset R$ such that $R/\aideal \in \tc$, the induced map $M=\Hom_R(R,M) \to \Hom_R(\aideal,M)$ is an isomorphism.  The inclusion of the full subcategory of $\ff$-closed modules has a left adjoint $M \mapsto M_\ff$. The module $M_\ff$ is called the *module of quotients* of $M$ (see [@Stenstrom IX.1]).  The unit of this adjunction has the following properties: the kernel of the map $M \to M_\ff$ is $t(M)$, $M_\ff$ is torsion-free and the cokernel of this map is a torsion module. 

Cellular-Approximation, Nullification and Completion
----------------------------------------------------

The following recalls the basic properties of cellular approximation, as well as the definition of completion given by Dwyer and Greenlees in [@DwyerGreenlees]. 

\[def: cellularization, nullification and completion\] Let $R$ be a ring and let $\ab$ be a class of $R$-complexes.  We say an $R$-complex $X$ is *$\ab$-complete* if $\ext_R^*(N,X)=0$ for any $\ab$-null object $N$. An $R$-complex $C$ is an *$\ab$-completion* of $X$ if $C$ is $\ab$-complete and there is an $\ab$-equivalence $X \to C$.  It is easy to see that an $\ab$-completion of a complex $X$ is unique up to an isomorphism in $\Derived_R$. As in [@DwyerGreenlees], we denote an $\ab$-completion of $X$ by $X^\wedge_\ab$.

The following criterion for nullification is usually easier to check than the original definition.  Its proof is easy and therefore omitted. 

\[lem: Triangle criterion for nullification\] Let $R$ be a ring and let $\ab$ be a class of $R$-complexes.  A complex $N$ is an *$\ab$-nullification* of $X$ if there is a triangle $C \to X \to N$ such that $C$ is $\ab$-cellular and $N$ is $\ab$-null.  In this case it also follows that $C$ an $A$-cellular approximation of $X$.

Recall that when $\ab$ is a set, the full subcategory of $\ab$-cellular objects of $\Derived_R$ is the localizing subcategory of generated by $A$.  The *localizing category generated by $\ab$*, denoted $\langle \ab \rangle$, is the smallest full triangulated subcategory of $\Derived_R$ that is closed under triangles, direct sums and retracts.  Closure under triangles means that for every distinguished triangle in $\Derived_R$, if two of the objects are in the localizing subcategory, so is the third.  The proof of the following lemma is clear. 

Let $\ab$ be a class of $R$-complexes, then every object of $\langle \ab \rangle$ is $\ab$-cellular.  If $\bb$ is another class of $R$-complexes such that $\langle \ab \rangle= \langle \bb \rangle$, then $\ab$-cellular approximation is the same as $\bb$-cellular approximation. 

In [@DwyerGreenlees] $\ab$-cellular complexes were called *$\ab$-torsion* while the term *$\ab$-cellular* was reserved for complexes in $\langle\ab\rangle$. When $\tc$ is a hereditary torsion theory, the two terms agree (by Lemma \[lem: Every torsion is cellular\] below). 

Cellular-Approximation with respect to a Hereditary Torsion Theory {#sec: Cellular-Approximation with respect to a Hereditary Torsion Theory}
------------------------------------------------------------------

Let $\tc$ be a hereditary torsion class.  It is not immediately apparent that $\tc$-cellular approximation exists.  Below, in Lemma \[lem: Every torsion is cellular\], we show that $\langle\tc\rangle$ is the same as the localizing subcategory generated by a set $\ab_\tc$. This immediately implies that $\tc$-cellular approximation and $\tc$-nullification exist for any $R$-complex, see Corollary \[cor: Consequence of being generated by AT\]. 

Given a hereditary torsion class of $R$-modules $\tc$, we denote by $\ab_\tc$ the set of all cyclic $\tc$-torsion modules. 

\[lem: Every torsion is cellular\] Let $\tc$ be a hereditary torsion class, then every $\tc$-torsion module is $\ab_\tc$-cellular and hence $\langle\tc\rangle=\langle\ab_\tc\rangle$.

Clearly, every cyclic $\tc$-torsion module is $\ab_\tc$-cellular.  Therefore every direct sum of cyclic $\tc$-torsion modules is $\ab_\tc$-cellular.  Let $M$ be a $\tc$-torsion module, then there is a surjection $C(M)=\oplus_{m\in M} R/\mathrm{ann}(m) \twoheadrightarrow M$. Since every hereditary torsion theory is closed under submodules, $R/\mathrm{ann}(m)$ is $\tc$-torsion for every $m\in M$. Clearly $C(M)$ is $\ab_\tc$-cellular and $\tc$-torsion.  Next we build a resolution $X$ of $M$ using $\ab_\tc$-cellular modules.  Let $X_0=C(M)$, and let $d_0:X_0 \twoheadrightarrow M$ the map defined above.  The kernel of $d_0$ is $\tc$-torsion, so there is an epimorphism $C(\ker(d_0)) \twoheadrightarrow \ker(d_0)$. Let $X_1=C(\ker(d_0))$ and let $d_1$ be the composition $X_1 \twoheadrightarrow \ker(d_0) \hookrightarrow X_0$. In this way $X$ is built inductively and it is clear that $X$ is quasi-isomorphic to $M$. By construction, $X$ is in the localizing subcategory generated by $\ab_\tc$.

\[cor: Consequence of being generated by AT\] Let $\tc$ be a hereditary torsion class, then $\tc$-cellular approximation, $\tc$-nullification and $\tc$-completion exist for every complex.  Moreover, a complex $X$ is $\tc$-cellular if and only if $X\in\langle\tc\rangle$.

Lemma \[lem: Every torsion is cellular\] implies that $\tc$-cellular approximation is the same as $\ab_\tc$-cellular approximation.  As mentioned in Section \[sec: Introduction\], $\ab_\tc$-cellular approximation exists for every complex.  The proof of the other claims is similar. 

\[lem: Bounded-above is cellular\] Let $\tc$ be a hereditary torsion class. 

1.   If $X$ is a $\tc$-cellular complex then the homology groups of $X$ are $\tc$-torsion $R$-modules. 

2.   If $X$ is a bounded-above complex such that the homology groups of $X$ are $\tc$-torsion then $X$ is $\tc$-cellular. 

Let $\cc$ be the full subcategory of $\Derived_R$ containing all objects whose homology groups are $\tc$-torsion $R$-modules.  The properties of a hereditary torsion theory show that $\cc$ is localizing subcategory.  Since $\cc$ contains $\tc$, then $\cc$ also contains $\langle \tc \rangle$. This proves the first statement. 

Now suppose $X$ is a bounded-above complex and that $H_i X \in \tc$ for all $i$. Because $X$ is bounded-above, $X$ belongs to the localizing subcategory generated by the homology groups of $X$ (see for example [@DwyerGreenlees 5.2]).  Since the homology groups of $X$ all belong to $\langle \tc \rangle$, so does $X$.

If $R$ is a commutative Noetherian ring, then a complex $X$ is $\tc$-cellular if and only if all the homology groups of $X$ are $\tc$-torsion.  This easily follows from a result of Neeman [@NeemanChromatic Theorem 2.8].  However, Example \[exa: Noncommutative\] shows a noncommutative ring $R$ and a complex $X$ such that $H_i(X)$ is $\tc$-torsion for all $i$ but $X$ is not $\tc$-cellular. 

Nullification Construction {#sec: The Construction}
==========================

In [@Benson], 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 $\tc$-cellular approximations over any ring $R$, where $\tc$ is a hereditary torsion class.  In fact, we give two isomorphic constructions. 

\[Construction\] Let $(\tc,\ff)$ be a hereditary torsion theory with radical $t$. For an $R$-module $M$ we construct the $\tc$-nullification of $M$ as a cochain complex $I^0 \xrightarrow{d} I^1 \xrightarrow{d} I^2 \xrightarrow{d} \cdots$ inductively. 

Let $M^0=M$, let $F^0=M^0/t(M^0)$ and let $I^0$ be the injective hull of $F^0$. Note that since $F^0$ is torsion-free, so is $I^0$. We proceed by induction, set $$M^{n+1}=I^n/F^n, \qquad F^{n+1}=M^{n+1}/t(M^{n+1})$$ and let $I^{n+1}$ be the injective hull of $F^{n+1}$. Again $I^{n+1}$ is torsion-free because $F^{n+1}$ is.  The differential $d:I^n \to I^{n+1}$ is the composition $I^n \to M^{n+1} \to F^{n+1} \to I^{n+1}$. The image of $d:I^n \to I^{n+1}$ is $F^{n+1}$ and therefore $d\circ d=0$. Denote the resulting complex by $I$. The natural map $M \to I^0$ extends to a map of complexes $M\to I$.

\[Construction2\] For an $R$-module $M$ we construct a cochain complex $J^0 \xrightarrow{d^0} J^1 \xrightarrow{d^1} J^2 \xrightarrow{d^2} \cdots$ inductively. 

Let $Q^0=M$, let $N^0=(Q^0)_\ff$ and let $J^0$ be the injective hull of $N^0$. Denote by $d^{-1}$ the map $M \to J^0$. Now proceed by induction, let $$Q^{n+1}=J^n/\im(d^{n-1}), \qquad N^{n+1}=(Q^{n+1})_\ff$$ and let $J^{n+1}$ be the injective hull of $J^n$. The differential $d^n:J^n \to J^{n+1}$ is the composition $J^n \to Q^{n+1} \to N^{n+1} \to J^{n+1}$. Clearly, $d^{n+1}\circ d^n=0$. Denote the resulting complex by $J$. The natural map $M \to J^0$ extends to a map of complexes $M\to J$. Note that for every $n$, $J^n$ is torsion-free because $N^n$ is. 

Let $J$ be the complex constructed from $M$ in \[Construction2\], then $H_0(J) \cong M_\ff$.

It easily follows from the definition of an $\ff$-closed module that any injective torsion-free module is $\ff$-closed, therefore $J^0$ is $\ff$-closed.  For any $\ff$-closed module $K$ there is an isomorphism $K\cong K_\ff$ (see [@Stenstrom page 198]), therefore $(J^0)_\ff\cong J^0$ and $(M_\ff)_\ff\cong M_\ff$.

The module of quotients functor is left exact (see [@Stenstrom page 199]).  Hence applying the module of quotients functor to the sequence $M_\ff \to J^0 \to J^0/M_\ff$ yields an exact sequence: $$0 \to M_\ff \to J^0 \to (J^0/M_\ff)_\ff$$ We see that $J^0/M_\ff$ is torsion-free, because it is isomorphic to a submodule of the torsion-free module $(J^0/M_\ff)_\ff$.

Now consider the short exact sequence $$M_\ff/\im(M) \to Q^1 \to J^0/M_\ff$$ The module $M_\ff/\im(M)$ is a torsion module (see Definition \[def: Module of quotients\]), while the module $J^0/M_\ff$ is torsion free.  From the definition of the radical $t$ it follows that $M_\ff/\im(M) \cong t(Q^1)$. Therefore $M_\ff$ is the kernel of $J^0 \to N^1$ and the proof is complete. 

\[lem: Construction of Nullification\] Let $M$ be an $R$-module, let $I$ be the complex constructed from $M$ in \[Construction\], let $C$ be a complex such that $C \to M \to I$ is a distinguished triangle and let $J$ be the complex constructed from $M$ in \[Construction2\].  Then $C$ is a $\tc$-cellular approximation of $M$ and both $I$ and $J$ are $\tc$-nullifications of $M$. In particular, $H_0(\Null_\tc M) \cong M_\ff$ for any $R$-module $M$.

We can choose $C$ to be the complex $M \to I^0 \to I^1 \to \cdots$ with $M$ in degree 0.  The homology of $C$ is easy to compute: $H^n(C)=t(M^n)$, with $M^n$ as defined in the Nullification Construction \[Construction\] above.  By Lemma \[lem: Bounded-above is cellular\], the complex $C$ is $\tc$-cellular.  The complex $I$ is $\tc$-null, simply because $I$ is composed of torsion-free injective modules.  Thus, by Lemma \[lem: Triangle criterion for nullification\], $I$ is a $\tc$-nullification of $M$ and $C$ is a $\tc$-cellular approximation of $M$.

Similar reasoning shows that $J$ is a $\tc$-nullification of $M$. The complex $J$ is $\tc$-null, simply because $J$ is composed of torsion-free injective modules.  The homology of $J$ is: $H^n(J)=t(Q^n)$ for $n>0$ and $H^0(J)=M_\ff$. Let $C'$ be a complex such that there is a distinguished triangle $C' \to M \to J$. The long exact sequence in homology yields: $H^0(C')=t(M)$, $H^1(C')= M_\ff/M$ and $H^n(C')=H^{n-1}(J)$ for $n>1$. Note that $M_\ff/M$ is a $\tc$-torsion module.  By Lemma \[lem: Bounded-above is cellular\], the complex $C'$ is $\tc$-cellular and hence $J$ is a $\tc$-nullification of $M$ and $C'$ is a $\tc$-cellular approximation of $M$. In particular, $H_0(\Null_\tc M) = H_0(J) \cong M_\ff$.

It follows that the complexes $I$ and $J$ in Lemma \[lem: Construction of Nullification\] are isomorphic in the derived category of $R$. In fact, they are isomorphic as complexes.  To construct this isomorphism one needs the following property: for any $R$-module $L$ the injective hull of $L_\ff$ and the injective-hull of $L/t(L)$ are the same; this is because $(L/t(L))_\ff = L_\ff$ and $L/t(L)$ is an essential submodule of $(L/t(L))_\ff$ (see [@Stenstrom IX.2.4]).  Using the aforementioned property it is a simple exercise to construct the isomorphism inductively. 

Using the construction above we can give a different description of $\tc$-nullification, the one shown in Theorem \[the: Nullification as double endomorphism\].  Before proving Theorem \[the: Nullification as double endomorphism\] it is necessary to note some properties of the functor $\Hom_R(-,E)$.

Let $E$ be an $R$-module and let $\ee$ be the endomorphism ring $\End_R(E)=\Hom_R(E,E)$. The functor $\Hom_R(-,E)$ is a contravariant functor from left $R$-complexes to left $\ee$-complexes.  This left $\ee$-action is simply composition on the left with the morphisms in $\ee$. In other words, the left $\ee$-action on $\Hom_R(-,E)$ is induced by the left $\ee$-action on $E$ itself.  Moreover, the functor $\Hom_{\ee}(-,E)$ is a contravariant functor, this time from left $\ee$-complexes to left $R$-complexes.  Here the left $R$-action on $\Hom_\ee(-,E)$ comes from the left $R$-action on $E$ (which commutes with the left $\ee$-action on $E$).  In particular, there is a derived version of this functor: $\mathbf{R}\Hom_\ee(-,E): \Derived_\ee \to \Derived_R$.

Given an $R$-module $M$, construct a $\tc$-nullification of $M$ in the way prescribed in \[Construction\].  This construction results in a cochain complex $I$, with $I^n$ being an injective torsion-free module.  Let $E$ be a torsion-free injective $R$-module such that for every $n$, $I^n$ is a direct summand of a finite direct sum of copies of $E$. For example, one can take $E$ to be the product $\prod_n I^n$. Denote by $\ee$ the endomorphism ring $\End_R(E)$.

Now consider the triangle $C \to M \to I$. Since $I$ is a $\tc$-nullification of $M$, $C$ is a $\tc$-cellular approximation of $M$. Applying the functor $\Hom_R(-,E)$ to this triangle yields a triangle in $\Derived_\ee$: $$\Hom_R(I,E) \to \Hom_R(M,E) \to \Hom_R(C,E)$$ Since $E$ is injective, $H_i(\Hom_R(C,E)) \cong \Hom_R(H_i (C),E)$. Since the homology groups of $C$ are torsion, $\Hom_R(H_i (C),E)=0$. Therefore the map $\Hom_R(I,E) \to \Hom_R(M,E)$ is a quasi-isomorphism of $\ee$-complexes. 

Because $I^n$ is a direct summand of a finite direct sum of copies of $E$, the $\ee$-module $\Hom_R(I^n,E)$ is projective.  Thus the map $\Hom_R(I,E) \to \Hom_R(M,E)$ is a projective resolution of $\Hom_R(M,E)$ in the category of $\ee$-modules.  We conclude that the complex $\Hom_\ee(\Hom_R(I,E),E)$ is the derived functor $\mathbf{R} Hom_\ee(\Hom_R(M,E),E)$.

Because $I^n$ is a direct summand of a finite direct sum of copies of $E$, one readily sees that the $R$-module $$\Hom_\ee(\Hom_R(I^n,E),E)$$ is naturally isomorphic to $I^n$ and therefore $\Hom_\ee(\Hom_R(I,E),E)\cong I$.

As noted in Theorem \[the: Nullification as double endomorphism\], $\Null_\tc R \simeq \mathbf{R}\End_{\ee}(E)$ and therefore $$R_\ff \cong H^0 (\Null_\tc R) \cong H^0 (\mathbf{R}\End_{\ee}(E)) = \End_{\ee}(E)$$ This isomorphism recovers [@Stenstrom IX.3.3], where it is stated that there is an injective $R$-module $E$ such that $R_\ff \cong \End_{\End_R(E)}(E)$. Also note that $\Null_\tc R$ is quasi-isomorphic to a differential graded algebra.  This also follows from a result of Dwyer [@DwyerNoncommutative Proposition 2.5], where it is shown that for any set of complexes $\ab$, the complex $\Null_\ab R$ is quasi-isomorphic to a differential graded algebra. 

Let $X= \cdots \to X_n \to X_{n-1} \to \cdots$ be a complex such that there exists some $m$ for which $X_n=0$ for all $n>m$. Then it is possible to generalize the Nullification Construction \[Construction\] to give the $\tc$-nullification of $X$. Moreover, this generalized construction of $\Null_\tc X$ can be done in such a way that for $n>m$ $(\Null_\tc X)_n=0$, while for $n\leq m$ $(\Null_\tc X)_n$ is a finite direct sum of torsion-free injective modules.  Therefore $(\Null_\tc X)_n$ is itself a torsion-free injective for $n\leq m$. Now it is easy to see that the proof of Theorem \[the: Nullification as double endomorphism\] works for $\Null_\tc X$ as well and yields the same result.  Namely, there exists an injective $R$-module $E$ such that $$\Null_\tc X \simeq \mathbf{R}\Hom_{\End_R(E)}(\Hom_R(X,E),E)$$ Clearly, this result carries over to any bounded-above complex $X$.

Say an injective module $E$ is *sufficient to compute the $\tc$-nullification of $M$* if $$\Null_\tc M \simeq \mathbf{R} Hom_\ee(\Hom_R(M,E),E)$$ where $\ee=\End_R(M)$. Given an $R$-module $M$ one can use the proof of Theorem \[the: Nullification as double endomorphism\] to construct an injective module $E$ which is sufficient to compute the $\tc$-nullification of $M$. However there are other injective modules sufficient to compute the $\tc$-nullification of $M$, as shown by the following proposition. 

Let $M$ be an $R$-module and let $E$ be an injective cogenerator of $\tc$. Denote by $\ee$ the ring $\End_R(E)$.

1.   If the $\ee$-module $\Hom_R(M,E)$ has a resolution composed of finitely generated projective modules in each degree, then $E$ is sufficient to compute the $\tc$-nullification of $M$.

2.   There exists an ordinal $\alpha$ such that the module $E'=\prod_{i<\alpha} E$ is sufficient to compute the $\tc$-nullification of $M$.

Let $P$ be a finitely generated projective $\ee$-module, then it is easy to see that $\Hom_R(\Hom_\ee(P,E),E)$ is naturally isomorphic to $P$. Now let $F$ be a projective resolution of $\Hom_R(M,E)$ over $\ee$ and assume $F$ is composed of finitely generated projective modules in each degree.  Then $\Hom_R(\Hom_\ee(F,E),E)$ is naturally isomorphic to $F$.

The quasi-isomorphism $\eta:F \to \Hom_R(M,E)$ induces a map $\Hom_\ee(\Hom_R(M,E),E) \to \Hom_\ee(F,E)$. Composing with the natural map $M \to \Hom_\ee(\Hom_R(M,E),E)$ yields a map $\mu:M \to \Hom_\ee(F,E)$. It is easy to see that $\Hom_R(\mu,E)$ is the quasi-isomorphism $\eta$.

Consider the triangle $C \to M \xrightarrow{\mu} \Hom_\ee(F,E)$. Clearly, $\Hom_\ee(F,E)$ is $\tc$-null.  Since $\Hom_R(\mu,E)$ is a quasi-isomorphism, $\Hom_R(C,E)$ is quasi-isomorphic to zero.  This implies $\Hom_R(H_i(C),E)=0$ for all $i$. Since $E$ is an injective cogenerator for $\tc$, $H_i(C)$ is torsion for all $i$. Clearly $C$ is bounded-above and so, by Lemma \[lem: Bounded-above is cellular\], $C$ is $\tc$-cellular.  We conclude that $\Hom_\ee(F,E)$ is a $\tc$-nullfication of $M$ and $E$ is sufficient to compute the $\tc$-nullification of $M$.

We now turn our attention to the second item in the proposition.  By [@Stenstrom VI.3.9], every torsion-free module has a monomorphism to some direct product of copies of $E$. In particular, every torsion-free injective is a isomorphic to a direct summand of some direct product of copies of $E$.

Let $I$ be the complex described in the Nullification Construction \[Construction\].  Let $E'$ be a direct product of copies of $E$ such that for every $n$, $I^n$ is isomorphic to a direct summand of $E'$. Clearly the $\End_R(E')$-complex $\Hom_R(I,E')$ is a projective resolution of $\Hom_R(M,E')$ which is composed of finitely generated projective modules in every degree.  Hence $E'$ is sufficient to compute the $\tc$-nullification of $M$.

Torsion Theories and Cellular Approximation in Artinian Rings {#sec: Torsion Theories and Cellular Approximation in Artinian Rings}
=============================================================

Throughout this section $R$ is an Artinian ring and $\simples$ is a set of non-isomorphic simple modules of $R$. Define a class $\ff$ of $R$-modules by $\ff=\{F \ |\ \Hom_R(S,F)=0 \text{ for all } S \in \simples\}$ and define a class $\tc$ by setting $ \tc=\{M \ |\ \Hom_R(M,F)=0 \text{ for all } F \in \ff\}$. By [@Stenstrom VIII.3], the pair $(\tc,\ff)$ forms a hereditary torsion theory (alternatively, one can easily deduce this from Lemma \[lem: Cogenerator for torsion of simples\] below).  Because $R$ is Artinian, every hereditary torsion theory of $R$-modules is generated by a set of simple modules (see [@Stenstrom VIII]), so this context covers all hereditary torsion theories over $R$. In this section we give several results regarding $\tc$-nullification.  We also give a different proof for a result of Benson [@Benson Theorem 5.1] in Corollary \[cor: Bensons result\]. 

Let $\Omega$ be the set of isomorphism classes of simple modules of $R$ and let $\simples'$ be the complement of $\simples$ in $\Omega$. We denote by $E$ the product of the injective hulls of the simple modules in $\simples'$ and denote by $P$ the direct sum of the projective covers of those simple modules.  We show that $E$ is an injective cogenerator of $\tc$ and that being $\tc$-cellular is the same as the being $\simples$-cellular. 

\[lem: Cyclic E-complete is simples cellular\] Let $C$ be a cyclic $R$-module such that $\Hom_R(C,E)=0$, then $C$ is $\simples$-cellular. 

Since $R$ is Artinian, $C$ admits a composition series $$0 = C_0 \subset C_1 \subset \cdots \subset C_m=C \ ,$$ where all the quotients $C_i/C_{i-1}$ are simple modules.  We next show that $C_i/C_{i-1} \in \simples$ for all $i$. Suppose that for some $i$, $C_i/C_{i-1}\cong S'$ for some $S'\in \simples'$. Let $x \in C_i \smallsetminus C_{i-1}$, then the cyclic module generated by $x$ has $S'$ as a quotient.  This implies the submodule $Rx$ of $C$ has a non-zero map to $E(S')$ - the injective hull of $S'$. Clearly such a map can be lifted to a non-zero map $C\to E$, in contradiction.  Therefore $C_i/C_{i-1}\cong S$ for some $S\in \simples$. Now a simple inductive argument on $i$ shows that $C_i \in \langle\simples\rangle$ for every $i$, and hence $C$ is $\simples$-cellular. 

\[cor: tc-cell is simples-cell\] A complex $X$ is $\tc$-cellular if and only if $X$ is $\simples$-cellular. 

We need to show that $\langle \tc \rangle = \langle \simples \rangle$. Since $\simples \subset \tc$, we only need to show that $\tc \subset \langle \simples \rangle$. By Lemma \[lem: Every torsion is cellular\] it is enough to show that every cyclic $R$-module is $\simples$-cellular, but that is immediate from Lemma \[lem: Cyclic E-complete is simples cellular\]. 

\[lem: Cogenerator for torsion of simples\] The module $E$ is an injective cogenerator for $\tc$.

Let $\mathcal{U}$ be the class of modules $M$ such that $\Hom_R(M,E)=0$. Then $\mathcal{U}$ is a hereditary torsion theory.  Because $\Hom_R(S,S')=0$ for every $S\in \simples$ and $S' \in \simples'$, we see that $\Hom_R(S,E(S'))=0$, where $E(S')$ is the injective envelope of $S'$. Hence $E \in \ff$ and therefore $\mathcal{U}$ contains $\tc$.

Next, let $M$ be in $\mathcal{U}$. To show that $M$ is a $\tc$-torsion module it is enough to show that every cyclic submodule of $M$ is a torsion module, because $M$ is a quotient of the direct sum of its cyclic submodules.  So let $C$ be a cyclic submodule of $M$. Since $E$ is injective, it follows that $\Hom_R(C,E)=0$. By Lemma \[lem: Cyclic E-complete is simples cellular\] $C$ is $\simples$-cellular.  Therefore $C$ is $\tc$-cellular and by Lemma \[lem: Bounded-above is cellular\] $C$ is $\tc$-torsion. 

\[lem: P-null is E-conull\] For any complex $X$, $\ext_R^*(P,X)=0$ if and only if $\ext^*_R(X,E)=0$.

This is known when $X$ is a finitely generated $R$-module, see Benson’s book [@BensonBookI 1.7.6 &1.7.7].  Now suppose $X$ is any $R$-module.  Since $P$ is a finitely generated projective module, $\Hom_R(P,X)=0$ 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 any complex, then $\ext^*_R(P,X)=\Hom_R(P,H_*(X))$. Similarly $\ext^*_R(X,E)=\Hom_R(H^*(X),E)$.

\[cor: Bensons result\] For any $R$-module $M$, a $\tc$-nullification of $M$ is also a $P$-completion of $M$ and is therefore given by $$\Null_\tc M \simeq \mathbf{R} \Hom_{\End_R(P)} (\Hom_R(P,R),\Hom_R(P,M))$$

Consider the triangle $\cell_\tc M \to M \xrightarrow{\nu} \Null_\tc M$. Lemma \[lem: Cogenerator for torsion of simples\] implies that $E$ is $\tc$-null and therefore $\ext_R^*(\cell_\tc M,E)=0$. By Lemma \[lem: P-null is E-conull\], $\cell_\tc M$ is $P$-null and $\nu$ is a $P$-equivalence. 

It remains to show that $\Null_\tc M$ is $P$-complete.  Let $I$ be the $\tc$-nullification of $M$ described in \[Construction\].  The full subcategory of $P$-complete objects in $\Derived_R$ is closed under isomorphisms, completion of triangles, products and retracts.  Denote this subcategory by $\cc$. From Lemma \[lem: P-null is E-conull\], we see that $E \in \cc$ and therefore every product of $E$ is also in $\cc$. Lemma \[lem: Cogenerator for torsion of simples\] and [@Stenstrom VI.3.9] imply that every torsion-free module is a submodule of a product of copies of $E$. Since $I^n$ is injective, it is a direct summand of some product of copies of $E$, hence $I^n$ is also an object of $\cc$.

Let $I(n)$ denote the cochain complex $I^0 \to I^1 \to \cdots \to I^n$. An inductive argument shows that $I(n) \in \cc$. There is a triangle $ I \to \prod_n I(n) \xrightarrow{\phi-1} \prod_n I(n)$, where the map $\phi$ is induced by the maps $I(n+1)\to I(n)$. Hence $I$ is $P$-complete. 

By Dwyer and Greenlees [@DwyerGreenlees Theorem 2.1], the $P$-completion of an $R$-module $M$ is given by $$M^\wedge_P \simeq \mathbf{R} \Hom_{\End_R(P)} (\Hom_R(P,R),\Hom_R(P,M))$$

Corollary \[cor: Bensons result\] above implies Benson’s formula for $\tc$-cellular approximation given in [@Benson Theorem 5.1].  This corollary also explains the connection between Benson’s formula and Dwyer and Greenlees formula for $P$-completion from [@DwyerGreenlees Theorem 2.1]. 

Examples {#sec: Examples}
========

\[exa: Commutative example\] Let $I$ be a two-sided ideal of $R$ such that $I$ is finitely generated as a left $R$-module.  An $R$-module $M$ will be called *$I$-torsion* if for every $m\in M$ there exists some $n$ such that $I^n m=0$. It is not difficult to show that the class of $I$-torsion modules forms a hereditary torsion class $\tc$ (see [@Stenstrom VI.6.10]).  Using Lemma \[lem: Every torsion is cellular\] it is easy to conclude that $\langle\tc\rangle=\langle R/I \rangle$. Hence $\tc$-cellular approximation is the same as $R/I$-cellular approximation and the same goes for nullification.  Note that in this case the radical $t$ associated with $\tc$ has a simple description: for any $R$-module $M$ $$t(M)=\colim_{n\to \infty} \Hom_R(R/I^n,M)$$

Now suppose $R$ is a commutative Noetherian ring.  Dwyer and Greenlees have shown in [@DwyerGreenlees] that $R/I$-cellular approximation computes $I$-local cohomology, namely that there is a natural isomorphism $H_I^*(M)\cong H^*(\cell_{R/I} M)$. Recall there is an isomorphism: $$H_I^*(M) \cong \colim_{n\to \infty} \ext_R^*(R/I^n,M)$$ These facts show that $\tc$-cellular approximation is the derived functor of the radical $t$. Moreover, in this case an object $X\in \Derived_R$ is $\tc$-cellular if and only $H_n(X)$ is $\tc$-torsion for all $n$, see [@DwyerGreenlees 6.12]. 

\[exa: Noncommutative\] Here is an example of a case where $\tc$-cellular approximation is not the derived functor of the associated radical.  Let $G$ be the symmetric group on 3 elements, let $k$ be the field $\Int/3\Int$ and let $R$ be the group ring $k[G]$. There is an augmentation map $R \to k$, where $k$ has the trivial $G$-action.  Let $I$ be the augmentation ideal.  As before, denote the class of $I$-torsion modules by $\tc$ and the associated radical by $t$. Since $R$ is an Artinian ring, the sequence $I\supseteq I^2 \supseteq I^3 \supseteq \cdots$ stabilizes.  So there is a fixed index $m$ such that $t(M)=\Hom_R(R/I^m,M)$ for every $R$-module $M$. Therefore, the derived functors of the torsion radical $t$ are the functors $\ext_R^*(R/I^m,-)$. In particular, $\ext_R^i(R/I^m,R)=0$ for all $i>0$, because $R$ is injective.  On the other hand, a calculation using Benson’s methods from [@Benson] shows that $H^n(\cell_\tc R)$ is non-zero for infinitely many values of $n$; thereby showing that $\cell_\tc$ is not the derived functor of $t$. We describe this calculation next.  From the surjection $G \to \Int/2$ one sees that $R$ has two simple modules, the trivial module $k$ and a one dimensional simple module $\omega$. As a left module, $R \cong E_k\oplus E_\omega$ where $E_k$ and $E_\omega$ are the injective hulls of $k$ and $\omega$ respectively.  The module $E_k$ has a composition series $$k \subset B \subset E_k, \quad\text{where } B/k\cong \omega \ \text{ and }\ E_k/B\cong k$$ The composition series for $E_\omega$ is $$\omega \subset B' \subset E_\omega, \quad\text{where } B'/\omega\cong k \ \text{ and }\ E_\omega/B'\cong \omega$$ In addition $E_\omega/\omega \cong B$ and $E_k/k \cong B'$. Since $E_\omega$ is $k$-null (see Lemma \[lem: Cogenerator for torsion of simples\]), then $$\Null_\tc R \simeq \Null_\tc E_\omega \oplus \Null_\tc E_k \cong E_\omega \oplus \Null_\tc E_k$$ So we need only compute $\Null_\tc E_k$. Applying Construction \[Construction\] to the module $E_k$ we get the complex $I$ which is $E_\omega \xrightarrow{d} E_\omega \xrightarrow{d} E_\omega \xrightarrow{d} \cdots$ where $d$ is the composition $E_\omega \twoheadrightarrow \omega \hookrightarrow E_\omega$. Hence $H^n(\Null_\tc E_k)=k$ for $n>1$ and therefore $H^n(\cell_\tc R)$ is non-zero for infinitely many values of $n$. In fact $$H^n(\cell_\tc R)=\left\{
     \begin{array}{ll}
       k, & n=0; \\
       0, & n=1; \\
       k, & n>1. 
     \end{array}
   \right.$$ It is important to note that in this case, a complex $X$ such that $H_n(X)$ is $\tc$-torsion for all $n$ need not be $\tc$-cellular.  Consider, for example, the complex $R^\wedge_\tc$. As we explain below, the homology groups $H_n(R^\wedge_\tc)$ are $\tc$-torsion for all $n$. On the other hand, the $\tc$-equivalences $\cell_\tc R \to R$ and $R \to R^\wedge_\tc$ show that $\cell_\tc R$ is $\tc$-equivalent to $R^\wedge_\tc$. If $R^\wedge_\tc$ was $\tc$-cellular, then $R^\wedge_\tc$ would have been quasi-isomorphic to $\cell_\tc R$, because a $\tc$-equivalence between $\tc$-cellular complexes is a quasi-isomorphism.  As we show next, the complex $R^\wedge_\tc$ has no homology in negative degrees and so cannot be quasi-isomorphic to $\cell_\tc R$.

It remains to explain the properties of $R^\wedge_\tc$ used above.  From Corollary \[cor: tc-cell is simples-cell\] we learn that $R^\wedge_\tc \simeq R^\wedge_k$ and $\cell_\tc R\simeq \cell_k R$. Without going into details, combining [@DwyerGreenleesIyengar 5.9] with [@DwyerGreenlees 4.3] shows that $$R^\wedge_k \simeq \mathbf{R} \Hom_R(\cell_k R, R)$$ This immediately implies that $R^\wedge_\tc$ has no homology in negative degrees.  We next show that $H_n(R^\wedge_\tc)$ is $\tc$-torsion for all $n$. Since $E_\omega$ is a $\tc$-null module, $\ext^*_R(E_\omega,R^\wedge_\tc)=0$. Recall that $R$ is a group-algebra and therefore $E_\omega$ is also the projective cover of $\omega$. Because $E_\omega$ is projective we have $$\ext^{-n}_R(E_\omega, R^\wedge_\tc) \cong \Hom_R(E_\omega, H_n(R^\wedge_\tc))$$ Hence, by Lemma \[lem: P-null is E-conull\], $\ext^*_R(H_n(R^\wedge_\tc),E_\omega)=0$. Lemma \[lem: Cogenerator for torsion of simples\] shows $E_\omega$ is an injective cogenerator for $\tc$, therefore $H_n(R^\wedge_\tc)$ is $\tc$-torsion. 

This example relates $\tc$-nullification with Cohn localization.  We begin by recalling the definition of Cohn localization.  Let $S=\{f_\alpha:P_\alpha \to Q_\alpha\}$ be a set of maps between finitely generated projective $R$-modules.  Say a ring map $R\to R'$ is *$S$-inverting* if $\Hom_R(f,R')$ is an isomorphism for every $f \in S$. A *Cohn localization* of $R$ with respect to $S$ is a ring map $R\to S^{-1}R$ which is initial among all $S$-inverting ring maps.  Note that the definition given here is not the standard definition (see e.g. [@DwyerNoncommutative]), but it is equivalent to the standard one. 

Let $\cc_S$ be the set of cones of the maps $f_\alpha$. In [@DwyerNoncommutative], Dwyer considers $\cc_S$-nullification and shows that $H_0(\Null_{\cc_S}R)=S^{-1}R$ (see [@DwyerNoncommutative 3.2]).  Combining Dwyer’s results with Theorem \[the: Nullification as double endomorphism\] yields the following proposition. 

\[pro: Cohn localization\] Let $\tc$ be a hereditary torsion-class of $R$-modules.  If $\langle\tc\rangle=\langle\cc_S\rangle$ for some set of maps $S$ between finitely generated projective $R$-modules, then

1.   $\Null_\tc(-) \simeq (-)_\ff$,

2.   the module of quotients functor $(-)_\ff$ is exact and

3.   there is an isomorphism $S^{-1}R\otimes_R M \cong M_\ff$ for every module $M$.

By Theorem \[the: Nullification as double endomorphism\], for every $R$-module $M$ the complex $\Null_\tc M$ has no homology in positive degrees.  By [@DwyerNoncommutative Proposition 3.1], $\Null_\tc R$ has no homology in negative degrees.  Moreover, a result of Miller [@MillerFiniteLocalizations] (see also [@DwyerNoncommutative Proposition 2.10]) shows that for every $R$-module $M$, $\Null_\tc M \simeq \Null_\tc R \otimes_R^\mathbf{L} M$. This implies that $\Null_\tc M$ has no homology in negative degrees. 

We conclude that for every $R$-module $M$, $\Null_\tc M$ has homology only in degree zero and therefore, by Lemma \[lem: Construction of Nullification\], $\Null_\tc M$ is quasi-isomorphic to $M_\ff$. Since the functor $\Null_\tc$ is exact, so is $(-)_\ff$. Since $\Null_\tc R \simeq \Null_{\cc_S} R$, Dwyer’s result [@DwyerNoncommutative 3.2] shows that $R_\ff \cong S^{-1}R$. Finally, the quasi-isomorphism $\Null_\tc M \simeq \Null_\tc R \otimes_R^\mathbf{L} M$ implies $S^{-1}R\otimes_R M \cong M_\ff$.

This example is of a topological nature.  Let $M$ be a discrete monoid and let $k=\Int/p\Int$ for some prime $p$. The ring $R$ we consider is the monoid ring $R=k[M]$, it has a natural augmentation $R \to k$ with augmentation ideal $I$. We also make the following assumptions:

1.   The classifying space $\mathrm{B}M$ of $M$ has a finite fundamental group. 

2.   The augmentation ideal $I$ is finitely generated as a left $R$-module. 

3.   There is a projective resolution $P=\cdots P_2 \to P_1 \to P_0$ of $k$ over $R$ such that every $P_n$ is finitely generated as an $R$-module. 

Let $\tc$ be the hereditary torsion class of $I$-torsion $R$-modules, then $\langle\tc\rangle = \langle k \rangle$ and hence $\cell_\tc \simeq \cell_k$. Denote by $R^\vee$ the left $R$-module $\Hom_k(R,k)$. From the results of Dwyer, Greenlees and Iyengar [@DwyerGreenleesIyengar 6.15 and 7.5] it is easy to conclude that $\cell_k R^\vee$ is quasi-isomorphic to the cochain complex (with coefficients in $k$) of a certain space we describe next.  Let $(\mathrm{B}M)^\wedge_p$ be the Bousfield-Kan $p$-completion of the classifying space of $M$. The space $\Omega (\mathrm{B}M)^\wedge_p$ is the loop-space of $(\mathrm{B}M)^\wedge_p$. So, $\cell_k R^\vee$ is quasi-isomorphic to $\chains^*(\Omega (\mathrm{B}M)^\wedge_p;k)$ - the singular cochain complex of $\Omega (\mathrm{B}M)^\wedge_p$ with coefficients in $k$. By Theorem \[the: Nullification as double endomorphism\], there exists an injective $R$-module $E$ such that $$H^n(\Omega (\mathrm{B}M)^\wedge_p;k) \cong \ext_{\End_R(E)}^{n-1}(\Hom_R(R^\vee,E),E) \text{ for } n\geq 2.$$
