Invariants, exponents and formal group laws
The author extends the notion of an exponent \(\tau_d\) to a more general setting, namely, in the context of algebraic oriented cohomology theories and the associated formula group laws. Then the authors use these generalized exponents to estimate the torsion part of an arbitrary algebraic oriented cohomology theories of some flag varieties. \vskip 0.1in More specifically, if \(W\) is the Weyl group of a crystallographic root system which acts by means of simple reflections on the respective weight lattice \(\Lambda\). Let \(F\) be a formal group law over a coefficient ring \(R\). The formal group algebra \(R\Lambda_F\) is the quotient of \(R[[x_{\Lambda}]]\) by the ideal \(I_F\), where \(R[[x_{\Lambda}]]\) is the \(\ker(\epsilon)\)-adic completion of \(R[x_{\Lambda}]\) and \(I_F\) is the closure of the ideal of \(R[[x_{\Lambda}]]\) generated by \(x_0\) and elements of the form \(x_{\lambda_1+\lambda_2} - (x_{\lambda_1}+_F x_{\lambda_2})\). Let \(R\Lambda_F^W\) be the subring of invariants of \(R\Lambda_F\), and \(I_F^W\) the associated ideal. Given two formal group laws \(F\) and \(F'\), there is an isomorphism \(\Phi: R\Gamma_F \to R\Gamma_{F'}\). The generalized exponent \(\tau_d^{F, F'}\) from \(F\) to \(F'\) is defined to be the exponent of the image \(\Phi(I_F^W)\) in \(I_{F'}^W\), after passing to the \(d\)-th subsequent quotients of the \(I_F-\) and \(I_{F'}-\) filtrations. When \(F\) is the multiplicative and \(F'\) is the additive formal group laws, the exponent coincide with the original definition in [\textit{S. Baek} et al., Doc. Math., J. DMV 17, 135--150 (2012; Zbl 1244.13004)].NEWLINENEWLINE The main result of the paper is the following:NEWLINENEWLINETheorem. Let \(F\) and \(F'\) be formal group laws over a ring \(R\) of characteristic different from \(2\). For the root system of type \(B_n\) with \(n \geq 3\) (resp. \(D_n\) with \(n \geq 4\)), we have \(\tau_d^{F,F'} \; | \; 2^d\) for \(d \geq 1\) (resp. \(n>d\geq 1\)). In particular, if \(1/2 \in R\), then \(\tau_d^{F,F'} = 1\) for every \(d \geq 1\). \vskip 0.1in As an application of the main result, the paper provides an estimate of the torsion part of the subsequent quotient \(\gamma^dh(B)/\gamma^{d+1}h(B)\) where \(\gamma^dh(B)\) are \(R\)-linear subgroups of the variety \(B\) of Borel subgroups of a split simple linear algebraic group \(G\) associated to the root system, generated by product of characteristic classes.
scientific article; zbMATH DE number 6335796
scientific article