### Sorry, you do not have access to this eBook

A subscription is required to access the full text content of this book.

512

Topological cyclic homology is a manifestation of Waldhausen’s vision that the cyclic theory of Connes and Tsygan should be developed with the initial ring
S
of higher algebra as base. In his philosophy, such a theory should be meaningful integrally as opposed to rationally. Bökstedt realized this vision for Hochschild homology [9], and he made the fundamental calculation that
THH
∗
(
F
p
)
=
HH
∗
(
F
p
/
S
)
=
F
p
[
x
]
is a polynomial algebra on a generator *x* in degree two [10]. By comparison,
HH
∗
(
F
p
/
Z
)
=
F
p
⟨
x
⟩
is the divided power algebra,
^{1}
so Bökstedt’s periodicity theorem indeed shows that by replacing the base
Z
by the base
S
, denominators disappear. In fact, the base-change map
HH
∗
(
F
p
/
S
)
→
HH
∗
(
F
p
/
Z
)
can be identified with the edge homomorphism of a spectral sequence
E
i
,
j
2
=
HH
i
(
F
p
/
π
∗
(
S
)
)
j
⇒
HH
i
+
j
(
F
p
/
S
)
,
so apparently the stable homotopy groups of spheres have exactly the right size to eliminate the denominators in the divided power algebra.

A subscription is required to access the full text content of this book.

Other ways to access this content: