for stable co - categories
Freie Universiteit
of Gw sp '"
/ t u th
2) Prove the genuine Karroubi periodicity theorem
K Rob , I '"'
of additive functor)
e - Oi! ( Oi! - Cato )
functor IG * = Ice
where Eu has a Cz -action and
f- is equivariant with respect to the
trivial action on Ess .
Cz - eco) products if E, and Ec , admit finite
Ceo )products , f : E ,
o : Eez 't Eez
is C , - semi additive if Ese
, Eo , are
=
Examples : op
")
Ex :
"'
, hcz
(at - ) Fun CBG , Cato )
Cz Hy pl b)
Prop.I.7.4.ly
Suppose
-category .
Fun "
theorem font (spun (Fed , E!
" Mackey objects in Ep
base change Nar 16 along
op { bby Q2
s't ' t
, Hypllb )(gHyplY , ( Cz
ter,cats = Cat! -7 Sp .
Fat#yikes 't'tget"u Fs't'II ,et"
I s tCz Is
,g) is
fhyPff , E)
" E Lege)
I . Genuine Karroubi periodicity
Let F : Cat} → Sp be an additive factor,
then there is a natural equivalence
of genuine Cz -spectra
Cz FAYE, e) u Fsttlcqo)u8oFY"Gif) II
des - is a$4 F' "
1) Recall that Hype ( Hyply)) Elz Hypno)
2) F' "
is a- equivalence of genuine
(z -spectra
'
F'Hml Hyplyy) = Fh"THyplb) )
= Cz Fh"78,1)
(Cz FSH"fqqy)4k invariance .
D -
cited
, Etd)
KRIGE)
Cookin ' "
invertible module with involution over R ,
c. Eko (R) a subgroup closed under the
involution induced by M .
KRC Modi , Erm)=$"Farland,i , Emr) if R is connective ,
Krimodp , EE ")=siKRlmdiz.IE/Proof
7.7
(Ijm ) "'s Imr rose . S3
r connective
, S3
re Eg , s)
If R real orientable , M= -M
KRIM odd , Iai) is (2-20) -periodic
If R is connective ,
")
and 2 - periodic if R
is real orientable .
.