Chromatic (co)homology of finite general linear groups (Q2092743)

The paper describes Morava \(E\)-theory of general linear groups of a finite field \(F\) of characteristic different to the chromatic prime. The information is very detailed, and the packaging encodes the information extremely elegantly. More precisely, the paper studies the Morava \(E\)-theory at a prime \(p>2\) and chromatic height \(n\) of \(BGL_d(F)\) where \(F\) is a finite field with \(|F|=q\equiv 1\) mod \(p\). The authors expect similar results to hold provided \(|F|\not \equiv 0\) mod \(p\) and also for \(p=2\). Let \(r\) be the largest integer such that \(q\equiv 1\) mod \(p^r\). The authors take all \(d\) together and consider two products: \(\bullet\) (induced by the diagonal map of spaces) and \(\times\) (induced by the transfer maps from \(b_{i,j}: GL_i(R)\times GL_j(F)\rightarrow GL_{i+j}(F)\)). There is also a coproduct using restriction along \(b_{i,j}\). The starting point is the work of \textit{M. Tanabe} [Am. J. Math. 117, No. 1, 263--278 (1995; Zbl 0820.55002)], showing \[ E^0(BGL_d(F)) =E^0(BGL_d(\overline{F}))_{\phi}=E^0[[c_1, \ldots , c_d]]/(r_1, \ldots , r_d) \] where \(\overline{F}\) is the algebraic closure, \(\phi\) is induced by the \(q\)th power map of the field, which is a topological generator of the Galois group, \(c_i\) is the \(i\)th Chern class and \(r_i=\phi^*(c_i)-c_i\). The authors make this structure more explicit (by giving explicit formulas for \(\phi^*\) and other structure maps) and by combining all \(d\) they obtain attractive structure theorems. Thus \(E^0(BGL_*(F))\) is a polynomial ring under \(\times \) on the powers \(c_{p^k}^j\) for \(0\leq j<N_k\) in \(E^0(BGL_{p^k}(F))\) where \[ N_k=p^{(n-1)k+n(r-1)}(p^n-1). \] The authors show the \(\times\) indecomposables form an ideal under \(\bullet \), and identify it explicitly, showing that the coalgebra primitives for a fixed \(d\) form a rank 1 module over the indecomposables. There are also rather complete results for Morava \(K\)-theories, the associated homology theories, and dualities between them. The authors use of the language of groupoids, the Atiyah-Hirzebruch spectral sequence, the character theory of Hopkins-Kuhn-Ravenel and subsequent transchromatic developments; all of these things are done in a thorough, systematic and natural way to get the maximum benefit.