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K
K
K
-theoretic counterexamples to Ravenel's telescope conjecture
Authors
Robert Burklund
Jeremy Hahn
Ishan Levy
Tomer M. Schlank
Publication date
26 October 2023
Publisher
View
on
arXiv
Abstract
At each prime
p
p
p
and height
n
+
1
β₯
2
n+1 \ge 2
n
+
1
β₯
2
, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for
Z
\mathbb{Z}
Z
acting by Adams operations on
B
P
β¨
n
β©
\mathrm{BP}\langle n \rangle
BP
β¨
n
β©
, we prove that the
T
(
n
+
1
)
T(n+1)
T
(
n
+
1
)
-localized algebraic
K
K
K
-theory of
B
P
β¨
n
β©
h
Z
\mathrm{BP}\langle n \rangle^{h\mathbb{Z}}
BP
β¨
n
β©
h
Z
is not
K
(
n
+
1
)
K(n+1)
K
(
n
+
1
)
-local. We also show that Galois hyperdescent,
A
1
\mathbb{A}^1
A
1
-invariance, and nil-invariance fail for the
K
(
n
+
1
)
K(n+1)
K
(
n
+
1
)
-localized algebraic
K
K
K
-theory of
K
(
n
)
K(n)
K
(
n
)
-local
E
β
\mathbb{E}_{\infty}
E
β
β
-rings. In the case
n
=
1
n=1
n
=
1
and
p
β₯
7
p \ge 7
p
β₯
7
we make complete computations of
T
(
2
)
β
K
(
R
)
T(2)_*\mathrm{K}(R)
T
(
2
)
β
β
K
(
R
)
, for
R
R
R
certain finite Galois extensions of the
K
(
1
)
K(1)
K
(
1
)
-local sphere. We show for
p
β₯
5
p\geq 5
p
β₯
5
that the algebraic
K
K
K
-theory of the
K
(
1
)
K(1)
K
(
1
)
-local sphere is asymptotically
L
2
f
L_2^{f}
L
2
f
β
-local.Comment: 100 pages. Comments very welcom
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oai:arXiv.org:2310.17459
Last time updated on 16/01/2024