CHERN CLASSES FROM MORAVA K-THEORIES TO P N $P^N$ upper P Superscript upper N-TYPICAL ORIENTED THEORIES

Source abstract

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.