La “teoría de la estructuración” de Anthony Giddens

Definition 2.2.2. Let Tbe a triangulated category. A projective classinT consists of a class P of objects of T and a class I of morphisms of Tsuch that:

(i) P consists of exactly the objects P such that every compositeP →X→Y is zero for each X→Y in I,

(ii) I consists of exactly the maps X→ Y such that every composite P → X→ Y is zero for eachP inP.

(iii) for eachX inT, there is a cofibre sequenceP →X→Y withP inP and X→Y inI.

In this paper, we make the additional assumption that the projective class is stable, that is, that P (or equivalently I) is closed under suspension and desuspension. With slight alterations, most of our results remain true without this assumption, but the extra bookkeeping complicates the arguments. The one exception is that in Section 2.4.2 we make use of an unstable projective class.

Remark 2.2.3. It follows from the definition thatP is closed under arbitrary coproducts

and retracts, andI is an ideal.

We writeGfor the ideal of ghosts in the stable module category, andF for all retracts of direct sums of suspensions of k in StMod(kG). For a module M ∈ StMod(kG), since ˆHn(G, M) ∼= [Ωnk, M], we can form a map ⊕Ωik → M that is surjective on Tate cohomology by assembling sufficiently many homogeneous elements in ˆH∗(G, M). Completing this map into a triangle inStMod(kG)

ΩUM → ⊕Ωik→M −−→φM UM, (2.2.1)

we get a ghost φM :M → UM. The map φM is a (weakly) universal ghost in the sense

that every ghost out of M factors though it, but the factorization is not necessarily unique. It follows easily that (F,G) forms a projective class in StMod(kG). This is called the ghost projective class.

While the ghost projective class is the focus of this paper, some of our results apply to any projective class, so we mention two other examples at this point: The simple ghost projective class is the projective class whose projectives are generated by all simple objects, and it was proposed for study in [12] as a way to avoid focusing on

Thickhki. And the strong ghost projective class is the projective class whose ideal consists of the maps which are ghosts under restriction to every subgroup. (See [17] for more on this topic.)

For any projective class (P,I), there is a sequence of derived projective classes

(P_n,In_{) [21]. The ideal I}n _{consists of all n-fold composites of maps in I, and X is}

in Pn if and only if it is a retract of an object M that sits inside a cofibre sequence P →M →Q withP ∈ P₁ =P and Q∈ P_n−1. For n= 0, we letP0 consist of all zero

objects and I0 _{consist of all maps in}

T. Thelength lenP(X) of an object X of T with respect to (P,I) is the smallestnsuch thatXis inP_n, if this exists. The fact that each pair (Pn,In) is a projective class implies that the length of X is equal to the smallestn

such that every map inIn _{with domain X is trivial.}

The length of a module M with respect to the ghost projective class is called the

generating length of M, and this exists when M is in Thickhki. But since we are interested in the collection G_t of ghosts in Thickhki, we also get another invariant. We describe both invariants, and the associated invariants ofkG, in the following definition, generalizing the definition given in [19] forp-groups.

Definition 2.2.4.

• The generating length gel(M) of M ∈ Thickhki is the smallest n such that

M ∈ Fn. That is, gel(M) = lenF(M).

• Theghost lengthgl(M) ofM ∈Thickhkiis the smallest integernsuch that every map in (Gt)n with domain M is trivial.

• Thegenerating numberofkGis the least upper bound of the generating lengths of modules in Thickhki.

• Theghost numberofkGis the least upper bound of the ghost lengths of modules inThickhki.

With this terminology, the generating hypothesis is the statement that the ghost number ofkG is 1.

Let M be in Thickhki. Since each (Fn,Gn_{) is a projective class and (G}

t)n ⊆ Gn, it

follows that

gl(M)6gel(M)

and therefore that

When G has periodic Tate cohomology, the coproduct in (2.2.1) can be taken to be finite, and it follows that the ghost projective class restricts to a projective class in

Thickhki [19]. This implies that equality holds in this case. We don’t know whether equality holds in general, except for the trivial observation that M ∼= 0 if and only gel(M) = 0 if and only if gl(M) = 0 and the less trivial fact that gel(M) = 1 if and only if gl(M) = 1 (see Corollary 2.3.7or [16]). Thus the GH is equivalent to the generating number ofkGbeing 1. See Remark2.3.13for further discussion of whether ghost length equals generating length.