In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by Edgar H. Brown and Franklin P. Peterson (1966), depending on a choice of prime p. It is described in detail by Ravenel (2003, Chapter 4). Its representing spectrum is denoted by BP.
Complex cobordism and Quillen's idempotent
For each prime p, Quillen showed there is a unique idempotent map of ring spectra ε from MUQ(p) to itself, with the property that ε([CPn]) is [CPn] if n+1 is a power of p, and 0 otherwise. The spectrum BP is the image of this idempotent ε.
Structure of BP
The coefficient ring π*(BP) is a polynomial algebra over Z(p) on generators vn of dimension 2(pn − 1) for n ≥ 1.
BP*(BP) is isomorphic to the polynomial ring π*(BP)[t1, t2, ...] over π*(BP) with generators ti in BP2(pi−1)(BP) of degrees 2(pi−1).
BP is the universal example of a complex oriented cohomology theory whose associated formal group law is p-typical.
- Adams, J. Frank (1974), Stable homotopy and generalised homology, University of Chicago Press, ISBN 978-0-226-00524-9
- Brown, Edgar H., Jr.; Peterson, Franklin P. (1966), "A spectrum whose Zp cohomology is the algebra of reduced pth powers", Topology 5 (2): 149–154, doi:10.1016/0040-9383(66)90015-2, MR 0192494.
- Quillen, Daniel (1969), "On the formal group laws of unoriented and complex cobordism theory" (PDF), Bull. Amer. Math. Soc. 75 (6): 1293–1298, doi:10.1090/S0002-9904-1969-12401-8, MR 0253350.
- Ravenel, Douglas C. (2003), Complex cobordism and stable homotopy groups of spheres (2nd ed.), AMS Chelsea, ISBN 0-8218-2967-X
- Wilson, W. Stephen (1982), Brown-Peterson homology: an introduction and sampler, CBMS Regional Conference Series in Mathematics 48, Washington, D.C.: Conference Board of the Mathematical Sciences, ISBN 978-0-8219-1699-5, MR 655040