Abstract
We study non-additive operations from algebraic Morava K-theories to oriented cohomology theories in algebraic geometry. For oriented cohomology theory
A
$A$
upper A
that has a
p
n
$p^n$
p Superscript n
-typical formal group law over a
Z
(
p
)
$\mathbb {Z}_{(p)}$
double struck upper Z Subscript left parenthesis p right parenthesis
-algebra we construct ‘Chern classes’ from the algebraic
n
$n$
n
-th Morava K-theory with
p
$p$
p
-local coefficients to
A
$A$
upper A
. If the coefficient ring of
A
$A$
upper A
is a free
Z
(
p
)
$\mathbb {Z}_{(p)}$
double struck upper Z Subscript left parenthesis p right parenthesis
-module we also prove that these Chern classes freely generate all operations from
K
(
n
)
i
n
t
∗
$\mathrm {K}(n)_{int}^*$
normal upper K left parenthesis n right parenthesis Subscript i n t Superscript asterisk
to
A
$A$
upper A
.
Examples of such theories are algebraic Morava K-theories
K
(
n
m
)
i
n
t
∗
$\mathrm {K}(nm)_{int}^*$
normal upper K left parenthesis n m right parenthesis Subscript i n t Superscript asterisk
for all
m
∈
N
$m\in \mathbb {N}$
m element of double struck upper N
and Chow groups with p-local coefficients. The universal
p
n
$p^n$
p Superscript n
-typical oriented theory is
B
P
{
n
}
∗
$BP\{n\}^*$
upper B upper P left brace n right brace Superscript asterisk
whose coefficient ring is also a free
Z
(
p
)
$\mathbb {Z}_{(p)}$
double struck upper Z Subscript left parenthesis p right parenthesis
-module.
Chern classes from the
n
$n$
n
-th algebraic Morava K-theory
K
(
n
)
i
n
t
∗
$\mathrm {K}(n)_{int}^*$
normal upper K left parenthesis n right parenthesis Subscript i n t Superscript asterisk
to itself allow us to introduce the gamma filtration on
K
(
n
)
i
n
t
∗
$\mathrm {K}(n)_{int}^*$
normal upper K left parenthesis n right parenthesis Subscript i n t Superscript asterisk
. This is the best approximation to the topological filtration obtained by values of operations and it satisfies properties similar to that of the classical gamma filtration on
K
0
$\mathrm {K}_0$
normal upper K 0
. The major difference from the classical case is that Chern classes from the graded factors
g
r
γ
i
K
(
n
)
i
n
t
∗
$gr^i_\gamma \mathrm {K}(n)_{int}^*$
g r Subscript gamma Superscript i Baseline normal upper K left parenthesis n right parenthesis Subscript i n t Superscript asterisk
to Chow groups with p-local coefficients are surjective for
i
≤
p
n
$i\le p^n$
i less than or equals p Superscript n
, which allows to estimate
p
$p$
p
-torsion in Chow groups of codimension up to
p
n
$p^n$
p Superscript n
of some varieties.