Deformations of formal groups and stable homotopy theory
Let \(F(n)\) be the standard height \(n\) formal group. The cohomology of \(F(n)\) which corresponds to the integral Morava \(K\)-theory \(K(n)\) is investigated. A complete description and representations of the classes of infinitesimals in the standard complex are presented, as well as a description of infinitesimals of the formal group corresponding to connective Morava \(K\)-theory. Using these results and Landweber's exact functor theorem, an uncountable family of deformations of cohomology theories is constructed. Several applications of topological nature are given: splitting theorems and linear approximations of the action of the \(n\)-th Morava stabilizer group of \(K^*(n)(K(\mathbb{Z},3))\). Some of the main results are formulated as follows: Theorem 1: Let \(F(x,y)\) be a formal group over a \(\mathbb{Z}_{(p)}\)-algebra \(A\) without torsion. Suppose that \(F(x,y)\) is not isomorphic to the additive formal group over \(A\). Denote by \(\tilde F(x,y)\) and \(\hat A\) the \(p\)-complement of \(F(x,y)\) and of \(A\), respectively. Then \(H^i(F(x,y))=H^i(\tilde F(x,y))\) for \(i\neq 2\) and \(H^2(F(x,y))=H^2(\tilde F(x,y))\oplus \hat A/A\). Let \(k\) be an algebraic extension of \(\mathbb{F}_p\) containing a primitive root of unity of degree \(p^n\) and let \(W(k)\) be the ring of Witt vectors over \(k\). Let \(E_n\) be the ring of noncommutative formal power series in one variable \(S\) with coefficients in \(W(k)\) subject to the relations \(S^n=p\) and \(Sa=\sigma(a)S\) for \(a\in W(k)\). Here \(\sigma\) denotes the Frobenius automorphism. Let \(S(n)\) be the \(n\)-th Morava stabilizer group. Then \(S(n)\) is identified with the group of strict automorphisms of the formal group \(F(n)\) over \(k\). So an element of \(S(n)\) is of the form \(f=\sum_{i\geq 0} a_iS^i\) where \(a_0=1\) and all other \(a_i\) are either \(0\) or roots of unity, and \(f\) corresponds to an automorphism of \(F(n)\): \(\tilde f=\sum_{F(n)} a_ix^i\). Furthermore, \(S(n)\) may be identified with the group of multiplicative operations in the theory \(K(n)^*(\cdot)\otimes_{K(n)_*} k\). One knows that \(K(n)^*(K(\mathbb{Z},3))\otimes_{K(n)_*} k=k[[u_1,u_2,\cdots, u_{n-1}]]\). Theorem 2: Let \(f=\sum_{i\geq 0} a_iS^i\in S(n)\). Then the action of \(f\) on the generators \(u_i\) of \(K(n)^*(K(\mathbb{Z},3))\otimes_{K(n)_*}k=k[[u_1,u_2,\cdots, u_{n-1}]]\) is given by the formula \(f(u_i)=\sum \sigma^i(a_k)u_{i+k}+O(u^2)\) where \(O(u^2)\) means elements divisible by \(u^2\) and \(u_{i+k}\) is understood to be \(0\) if \(i+k>n-1\).
scientific article; zbMATH DE number 1034159