The ring spectrum \(P(n)\) for the prime 2
In the paper under review the author studies properties of the ring spectrum \(P(n)\) for the prime \(2\). Here \(P(n)\) has the coefficient ring \(P(n)_*=BP_*/I_n\), where \(I_n\) is the ideal \((2,v_1,\dots,v_{n-1})\). It is known that \(P(n)\) is not commutative for \(p=2\), which introduces several complications. The author first shows that there are exactly two associative \(BP\)-bilinear multiplications on \(P(n)\) that have the canonical map \(S\to BP\to P(n)\) as unit. Next the author gives the Künneth formula: the cross product \(\times: P(n)_*(X)\otimes P(n)_*(Y) \to P(n)_*(X\times Y)\) is an isomorphism of \(P(n)_*\)-modules if \(P(n)_*(X)\) or \(P(n)_*(Y)\) is a free or flat \(P(n)_*\)-module. Also the Künneth formula in cohomology is given. Finally, the author supplies the necessary details of the Hopf ring \(\overline{P(n)}_*\left(\underline{P(n)}_*\right)\), the bigraded Hopf algebroid \(Q\overline{P(n)}_*\left(\underline{P(n)}_*\right)\), and the Hopf algebroid \(\overline{P(n)}_*\left(P(n)\right)\).
scientific article; zbMATH DE number 5527618