Bases for cooperations in \(K\)-theory. (English) Zbl 0997.55021

The study of the cooperation Hopf algebroid for \(K\)-theory, \(K_*K\), reduces to that of the ring of stably numerical polynomials \( K_0K= \{f(w)\in\mathbb Q[w,w^{-1}]:\forall n\in\mathbb Z\smallsetminus\{0\},\;f(n)\in\mathbb Z[1/n] \}.\) Earlier results of Adams and Clarke on this ring have shown that it is a free abelian group of countable rank, while K. Johnson has described explicit bases. The present paper proceeds by first considering the analogous local problem of finding a basis for the free \(\mathbb Z_{(p)}\)-module \[ K_0K_{(p)}= \{f(w)\in\mathbb Q[w,w^{-1}]:\forall u\in\mathbb Z_{(p)}^\times,\;f(u)\in\mathbb Z_{(p)} \} \] where \(p\) is a prime. Then it turns out that for an odd prime \(p\), \(K_0K_{(p)}\cap\mathbb Q[w]\) has a basis consisting of the polynomials \[ f_n(w)=\prod_{i=0}^{n-1}\frac{w-q^i}{q^n-q^i}\quad(n\geq 0), \] where \(q\) is a primitive generator modulo \(p^2\). These are related to the Gaussian polynomials \[ \begin{bmatrix} k \\ j\end{bmatrix}_t =\prod_{i=0}^{j-1}\frac{1-t^{k-i}}{1-t^{j-i}} \] by the identities \[ f_n(q^k)=\begin{bmatrix} k \\ n\end{bmatrix}_q. \] As a consequence, it is shown that the Laurent polynomials \(w^{-\lfloor n/2\rfloor}f_n(w)\quad(n\geqslant 0)\) form a basis for \(K_0K_{(p)}\).
In contrast to Johnson’s, the present basis has nice multiplicative properties since a product formula for Gaussian polynomials leads to the following formula for all \(m,n\geqslant 0\): \[ f_m(w)f_n(w)= \sum_{i=0}^{\min(m,n)}q^{(m-i)(n-i)} \begin{bmatrix} m \\ i\end{bmatrix}_q\begin{bmatrix} m+n-i \\ m\end{bmatrix}_q f_{m+n-i}(w). \] Similar results are obtained at the prime \(2\). These local bases can be integrated to give explicit global bases. Related constructions also give rise to bases for \(K_*K^\chi_{(p)}\) and \(K_*K^\chi\), the subrings of invariants under the Hopf algebroid conjugation \(\chi\). These subrings have featured in recent work on \(\Gamma\)-cohomology obstruction theory for \(E_\infty\)-ring structures on ring spectra by the third author and others.


55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
19L64 Geometric applications of topological \(K\)-theory
11B65 Binomial coefficients; factorials; \(q\)-identities
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