Limits over categories of extensions

Let k be a commutative ring with identity, and let C be one of the following categories: the category Gr of groups, the category Ab of abelian groups, the category Ass k of associative algebras over k. Given an object G ∈ C, let Ext C (G) be the category whose objects are the extensions H F ։ G in C with G as the cokernel, and the morphisms are the commutative diagrams of short exact sequences of the form

It is clearly natural to consider also the full subcategory Fext C (G) of Ext C (G) which consists of the short exact sequences H F ։ G where F is a free object in C. The category Ext Gr (G) has been studied extensively from the point of view of the theory of cohomology of groups (see, for example, [6], [7]). The general question which we wish to address here can be formulated as follows: how can one study the properties of objects G ∈ Ob(C) from the properties of the category Fext C (G)?

Let C be a small category and F : C -→ C a covariant functor. The inverse limit lim

In that case the limit of the functors over categories of extensions defines a functor C → Gr by setting

To clarify our point of view further, let us recall some known examples which have motivated our present investigation. Let C = Gr, G a group, Z[G] its integral group ring and M a Z[G]module. For n ≥ 1, define the functor

M, where R ab denotes the abelianization of R, the action of G on R ⊗n ab is diagonal and is defined via conjugation in F . It is shown by I. Emmanouil and R. Mikhailov in [4] that lim

For any group G and field k of characteristic 0 viewed as a trivial G-module, the homology groups H n (G, k), n ≥ 2, appear as inverse limits for suitable natural functors defined on the category Ext Gr (G) (see [5] for details).

Consider the category C = Ass Q of associative algebras over Q, the field of rationals. For integers n ≥ 1, and an associative algebra A over Q, consider the functors

given by setting ( 5)

where [R, R] is the Q-submodule of R, generated by the elements rs -sr with r, s ∈ R. It has been shown by D. Quillen in [11] that the inverse limit lim

F n is isomorphic to the even cyclic homology group HC 2n (A, Q):

Furthermore, the Connes suspension map S : HC 2n (A, Q) → HC 2n-2 (A, Q) can be obtained as follows:

where the right hand side map is induced by the natural projection

The motivation for our investigation should now be clear from the above examples: one takes quite simple functors, like (3) and ( 5), on the appropriate categories of extensions and asks for the corresponding inverse limits. It is then also natural to consider the derived functors lim ←-i : Ab Fext(G) → Ab of the limit functor. Every functor F : Fext C (G) → Ab which is natural in the above-mentioned sense (see diagrams (1) and ( 2)) determines a series of functors C → Ab by setting G → lim

We now briefly describe the contents of the present paper. We begin by recalling in Section 2 two properties of the limits given in [4] and [5]. The first (Lemma 2.1) provides a set of vanishing conditions for lim ←-F while, the second states that lim ←-F embeds in F(c 0 ) for every quasi-initial object c 0 . In Section 3 (Theorems 3.1 and 3.4) we show how the derived functors in the sense of A. Dold and D. Puppe [3] of certain standard non-additive functors on Ab, like tensor power, symmetric power, exterior power, are realized as limits of suitable functors over extension categories. In Section 4 we discuss higher limits and prove (Theorem 4.4) that for the functor

not a perfect group. We conclude with some remarks and possibilities for further work in Section 5.

Recall that the coproduct of two objects a and b in a category C is an object a ⋆ b which is endowed with two morphisms ι a : a -→ a ⋆ b and ι b : b -→ a ⋆ b having the following universal property:

For any object c of C and any pair of morphisms f : a -→ c and g : b -→ c, there is a unique morphism h :

The morphism h is usually denoted by (f, g).

We recall from [4] the following Lemma which provides certain conditions which imply the triviality of the inverse limit. Lemma 2.1. [4] Let C be a small category and F : C -→ Ab a functor to the category of abelian groups, and suppose that the following conditions are satisfied:

(i) Any two objects a, b of C have a coproduct (a ⋆ b, ι a , ι b ) as above.

(ii) For any two objects a, b of C the morphisms ι a : a -→ a ⋆ b and

of abelian groups. Then, the inverse limit lim

F be a compatible family and fix an object a of C. We consider the coproduct a ⋆ a of two copies of a and the morphisms ι 1 : a -→ a ⋆ a and ι 2 : a -→ a ⋆ a. Then, we have

) and hence the element (x a , -x a ) is contained in the kernel of the additive map

In view of our assumption, this latter map is injective and hence x a = 0. Since this is the case for any object a of C, we conclude that the compatible family (x c ) c∈Ob(C) vanishes, as asserted.

We next mention another property of the limit functor. Recall that an object c 0 of a category C is called quasi-ini

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