On the classifying ring for Abel formal group laws
An Abel formal group law over a ring \(R\) is a formal group law of the form \(F(u,v)=u+v+\alpha_{1} uv+ \sum_{i>1} \alpha_{i}(uv^{i}+u^{i}v)\). In [\textit{V.M. Bukhshtaber} and \textit{A.N. Kholodov}, Math. USSR, Sb. 69, No.1, 77--97 (1991; Zbl 0716.55006)] the authors described the ring \(\Lambda\) which carries the universal Abel formal group law and established some of its algebraic properties. In particular they showed that it is a subring of \({\mathbb Q}[a,b]^{S_{2}}\), the ring of symmetric polynomials in the formal roots \(a\) and \(b\) of the quadratic function \(X^2- \alpha_1 X - 2\alpha_2\). In the paper under review the authors give an integrality condition which identifies the subring \(\Lambda\) in \({\mathbb Q}[a,b]^{S_{2}}\), namely, a polynomial \(f\in {\mathbb Q}[a,b]^{S_{2}}\) is contained in \(\Lambda\) if and only if \(f(kt,lt)\in {\mathbb Z}[t,(k-l)^{-1}]\) for any pair of distinct integers \(k\) and \(l\). The ring \(\Lambda\) can be described canonically as a specific quotient of the Lazard ring by a suitable ideal, which, however, is not regular. Thus at first Landweber's exact functor theorem does not apply to obtain a corresponding complex oriented generalized cohomology theory with coefficients in \(\Lambda\). For that reason Bukhstaber and Kholodov investigated certain localizations of \(\Lambda\) in [op. cit.] which then allow to apply the theorem and therefore do yield complex oriented generalized cohomology theories. The authors of the present paper also investigate these localizations and establish similar integrality conditions which identify these localizations of \(\Lambda\) inside corresponding localizations of the algebra \({\mathbb Q}[a,b]^{S_{2}}\). In particular their approach allows to show that one of these localizations yields a multiplicative formal group law. Finally we would like to mention that \textit{P. Busato} in [Math. Z. 239, No.~3, 527--561 (2002; Zbl 1012.55004)] managed to realize the universal Abel formal group law as the formal group law of a complex oriented generalized cohomology theory without localizing the ring \(\Lambda\). All the generalized cohomology theories mentioned here are of interest as they provide examples of so-called elliptic cohomology theories.
scientific article; zbMATH DE number 5544382