Math 2.0

Algebraic topology Spectral sequences Theorem

Adams–Novikov spectral sequence

Builds on Complex cobordismSpectrumExt functorAdams spectral sequenceSphere spectrumBrown–Peterson cohomology

Statement

Let MUMU be the complex cobordism spectrum, with coefficient ring MU∗=π∗MUMU_* = \pi_* MU and MU∗MU=π∗(MU∧MU)MU_* MU = \pi_*(MU \wedge MU). Then (MU∗,MU∗MU)(MU_*, MU_* MU) is a Hopf algebroid, and for every spectrum XX the group MU∗XMU_* X is a comodule over it. Let XX be a connective spectrum. Then there is a spectral sequence (Ers,t,dr)r≥2(E_r^{s,t}, d_r)_{r \ge 2}. Its differentials are dr ⁣:Ers,t→Ers+r,t+r−1d_r \colon E_r^{s,t} \to E_r^{s+r, t+r-1}, and its second page is E2s,t=Ext⁡MU∗MUs,t(MU∗,MU∗X),E_2^{s,t} = \operatorname{Ext}^{s,t}_{MU_* MU}(MU_*, MU_* X), the Ext groups in the category of MU∗MUMU_* MU-comodules, and the spectral sequence converges to πt−s(X)\pi_{t-s}(X).

Remarks

The construction is the same as for the Adams spectral sequence, with the mod pp Eilenberg–MacLane spectrum replaced by MUMU. Since π0MU=ℤ\pi_0 MU = \Z and XX is connective, the MUMU-based tower recovers XX itself, which gives convergence to π∗(X)\pi_*(X) with no completion. After localizing at a prime pp, one may replace MUMU by the Brown–Peterson spectrum BPBP, with E2s,t=Ext⁡BP∗BPs,t(BP∗,BP∗X)E_2^{s,t} = \operatorname{Ext}^{s,t}_{BP_* BP}(BP_*, BP_* X), which is smaller and easier to compute.

Example: for X=SX = S, the sphere spectrum, the groups MU∗MU_* and MU∗MUMU_* MU are concentrated in even degrees, so E2s,t=0E_2^{s,t} = 0 for odd tt. Also E20,tE_2^{0,t} is ℤ\Z for t=0t = 0 and 00 otherwise.

Going deeper

For a student

Through Quillen's theorem, MU∗MUMU_* MU-comodules are related to formal group laws, so the E2E_2 term is a cohomology of the moduli of formal groups. This links the stable homotopy groups of spheres to the height filtration of formal groups and to chromatic homotopy theory. At odd primes the first differential in the BPBP version that can be nonzero is d2p−1d_{2p-1}, because the E2E_2 term is concentrated in internal degrees divisible by 2(p−1)2(p-1).

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Brown–Peterson cohomology, concept, no evidence levelBrown–Peterson cohomo…conceptComplex cobordism, concept, no evidence levelComplex cobordismconceptExt functor, concept, no evidence levelExt functorconceptSpectrum, concept, no evidence levelSpectrumconceptStable homotopy category, concept, no evidence levelStable homotopy categ…conceptStable homotopy groups of spheres, concept, no evidence levelStable homotopy group…conceptSteenrod algebra, concept, no evidence levelSteenrod algebraconceptSphere spectrum, object, no evidence levelSphere spectrumobjectAdams spectral sequence, statement, no evidence levelAdams spectral sequen…statementAdams–Novikov spectral sequence, statement, no evidence levelAdams–Novikov spectra…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Brown–Peterson cohomology concept
  2. Complex cobordism concept
  3. Ext functor concept
  4. Spectrum concept
  5. Stable homotopy category concept
  6. Stable homotopy groups of spheres concept
  7. Steenrod algebra concept
  8. Sphere spectrum object
  9. Adams spectral sequence statement
  10. Adams–Novikov spectral sequence statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Adams_spectral_sequence
About this page
TagD34QHZVPES
URIm2://D34QHZVPES
Kindstatement, theorem
WritingW0 stub
MSC55T15, 55N22, 55Q45
StatusThis copy of the site has no build data. Run m2 status D34QHZVPES.
AboutComplex cobordism
UsesComplex cobordism, Spectrum, Ext functor, Adams spectral sequence, Sphere spectrum, Brown–Peterson cohomology
Changed2 November 2026

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