From 𝐾(𝑛+1)_{*}(𝑋) to 𝐾(𝑛)_{*}(𝑋)

Let X X be a space of finite type. Set q = 2 ( p − 1 ) q=2(p-1) as usual, and define the mod q q support of K ( n ) ∗ ( X ) K(n)^*(X) by S ( X , K ( n ) ) = { m ∈ Z / q Z ∣ ⨁ d ≡ m mod q   K ( n ) d ≠ 0 } S(X,K(n)) = \{ m \in \mathbb {Z}/q\mathbb {Z} \mid \bigoplus _{d \equiv m \bmod q}\ K(n)^d \neq 0 \} for n > 0. n>0. Call K ( n ) ∗ ( X ) K(n)^*(X) sparse if there is no m ∈ Z / q Z m \in \mathbb {Z}/q\mathbb {Z} with m , m + 1 ∈ S ( X , K ( n ) ) . m, m+1 \in S(X,K(n)). Then we show the relation S ( X , K ( n ) ) ⫅ S ( X , K ( n + 1 ) ) S(X,K(n)) \subseteqq S(X,K(n+1)) for any finite type space X X with K ( n + 1 ) ∗ ( X ) K(n+1)^*(X) being sparse. As a special case, we have K ( n + 1 ) o d d ( X ) = 0 ⟹ K ( n ) o d d ( X ) = 0 , K(n+1)^{odd}(X) = 0 \Longrightarrow K(n)^{odd}(X) = 0, and the main theorem of Ravenel, Wilson and Yagita is also generalized in terms of the mod q q support.