# 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)