# Steenrod algebra

The graded algebra over the field of all stable cohomology operations (cf. Cohomology operation) modulo . For any space (spectrum of spaces) , the group is a module over the Steenrod algebra .

The Steenrod algebra is multiplicatively generated by the Steenrod operations (cf. Steenrod operation). Thus, the Steenrod algebra is a graded associative algebra, multiplicatively generated by the symbols with , which satisfy the Adem relation:

, so that an additive basis (over ) of the Steenrod algebra consists of the operations , (the so-called Cartan–Serre basis). Similar results are true for with . Furthermore,

where is an Eilenberg–MacLane space. The multiplication

induces the diagonal in , which is a homomorphism of algebras, and, consequently, turns into a Hopf algebra.

#### References

[1] | N.E. Steenrod, D.B.A. Epstein, "Cohomology operations" , Princeton Univ. Press (1962) |

[2] | J. Milnor, "The Steenrod algebra and its dual" Ann. of Math. , 67 (1958) pp. 150–171 |

[3] | M.K. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968) |

#### Comments

The analogue of the Steenrod algebra for a cohomology theory defined by a spectrum is ; cf. Generalized cohomology theories and Spectrum of spaces. The -term of the Adams spectral sequence, cf. Spectral sequence, is a purely (homological) algebra construct obtained by regarding the homology groups as modules over the Hopf algebra .

#### References

[a1] | J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989) |

[a2] | R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) pp. Chapts. 18–19 |

[a3] | J.F. Adams, "Stable homotopy and generalized homology" , Univ. Chicago Press (1974) pp. Part III, Chapts. 12, 15 |

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Steenrod algebra.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Steenrod_algebra&oldid=12348