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 Douglas Ravenel (2003, Chapter 4). Its representing spectrum is denoted by BP.
Complex cobordism and Quillen's idempotent
Brown–Peterson cohomology BP is a summand of MU(p), which is complex cobordism MU localized at a prime p. In fact MU(p) is a wedge product of suspensions of BP.
For each prime p, Daniel 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 \( {\displaystyle \pi _{*}({\text{BP}})} \) is a polynomial algebra over \( {\displaystyle \mathbb {Z} _{(p)}} \) on generators \( v_{n} \) in degrees \( {\displaystyle 2(p^{n}-1)} \) for \( {\displaystyle n\geq 1}. \)
\( {\displaystyle {\text{BP}}_{*}({\text{BP}})} \) is isomorphic to the polynomial ring \( {\displaystyle \pi _{*}({\text{BP}})[t_{1},t_{2},\ldots ]} \) over \( {\displaystyle \pi _{*}({\text{BP}})} \) with generators \( t_{i} \) in BP \( {\displaystyle {\text{BP}}_{2(p^{i}-1)}({\text{BP}})} of degrees 2 ( p i − 1 ) {\displaystyle 2(p^{i}-1)} {\displaystyle 2(p^{i}-1)} \).
The cohomology of the Hopf algebroid ( \( {\displaystyle (\pi _{*}({\text{BP}}),{\text{BP}}_{*}({\text{BP}}))} \) is the initial term of the Adams–Novikov spectral sequence for calculating p-local homotopy groups of spheres.
BP is the universal example of a complex oriented cohomology theory whose associated formal group law is p-typical.
See also
List of cohomology theories#Brown–Peterson cohomology
References
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), Bulletin of the American Mathematical Society, 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 978-0-8218-2967-7
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 0655040
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