. __|_|
4.3.The algebraicization of the category of T-spectra over H.
In this section we quickly complete the proof of Theorem 3.4.3, by observing *
*that
T-Spec=H is a split linear triangulated category arising as the homotopy categ*
*ory of a
model category. We also prove the analogous theorem for F-spectra, once we have*
* intro-
duced the appropriate abelian category.
Of course we take D = T-Spec=H , A = torsQ[cH ]-mod, and H*(X) = ssT*(X). It *
*is easily
verified that torsQ[cH ]-modis abelian and has injective dimension 1. The split*
* condition
arises since Q[cH ]is concentrated in even degrees, so that for any Q[cH-]modul*
*e M we may
take M+ to be the even-graded part of M and M- to be the odd graded part. The A*
*dams
short exact sequence 3.1.1 shows that A is the linearization of D.
Finally, it follows from the construction of Lewis-May that T-Spec=H is the *
*homotopy
category of a model category. The model category we have in mind is the catego*
*ry of
F-spectra, with weak equivalences created by the functor X 7-! eH ssT*(X). Sinc*
*e ssT*(X) =
eH ssT*(X) for objects of T-Spec=H ,we see that homology is well defined. We co*
*nclude that
Theorem 4.2.6 applies to show there is an equivalence
p : T-Spec=H -'! D(torsQ[cH ])
of triangulated categories; this completes the proof of Theorem 3.4.3. __|_|
It may be instructive to give an example illustrating how maps are represente*
*d in
D(torsQ[cH ]).
52 4. CATEGORICAL REPROCESSING.
Example 4.3.1.Consider a non-trivial map f : oe1H-! oe0H. First we note that *
*ssT*(oe0H) =
Q, and there is an exact sequence
0 -! Q -! I(H) cH-!3I(H) -! 0:
Thus p(oe0H) = F (cH ), which we may illustrate:
.. .
. ..
| #
| Q degree 11
# . |
Q | degree 10
| #
| Q degree 9
# . |
Q | degree 8
| #
| Q degree 7
# . |
Q | degree 6
| #
| Q degree 5
# . |
Q | degree 4
| #
| Q degree 3
# . |
Q | degree 2
#
Q degree 1
Here degree is displayed vertically, the vertical arrows are multiplication by *
*cH , and the
differential is displayed diagonally. Now the map f : oe1H-! oe0Hhas zero d-inv*
*ariant, and
its e-invariant is a non-zero element of Q ~=Ext(3Q; Q). Thus p(f) is non-trivi*
*al in all
even degrees and zero in all odd degrees. The cofibre of f is equivalent to E