Steenrod algebra

In algebraic topology, a Steenrod algebra was defined by Henri Cartan (1955) to be the algebra of stable cohomology operations for mod p {displaystyle p} cohomology. In algebraic topology, a Steenrod algebra was defined by Henri Cartan (1955) to be the algebra of stable cohomology operations for mod p {displaystyle p} cohomology. For a given prime number p {displaystyle p} , the Steenrod algebra A p {displaystyle A_{p}} is the graded Hopf algebra over the field F p {displaystyle mathbb {F} _{p}} of order p {displaystyle p} , consisting of all stable cohomology operations for mod p {displaystyle p} cohomology. It is generated by the Steenrod squares introduced by Norman Steenrod (1947) for p = 2 {displaystyle p=2} , and by the Steenrod reduced p {displaystyle p} th powers introduced in Steenrod (1953) and the Bockstein homomorphism for p > 2 {displaystyle p>2} . The term 'Steenrod algebra' is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory. A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring, the cup product squaring operation yields a family of cohomology operations: