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
[edit]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
[edit]The coefficient ring is a polynomial algebra over on generators in degrees for .
is isomorphic to the polynomial ring over with generators in of degrees .
![{\displaystyle \pi _{*}({\text{BP}})[t_{1},t_{2},\ldots ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96889f39af80cfaabf5b308f9ac6aca70d8f6db0)
The cohomology of the Hopf algebroid 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
[edit]References
[edit]- 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, vol. 48, Washington, D.C.: Conference Board of the Mathematical Sciences, ISBN 978-0-8219-1699-5, MR 0655040
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