The Steenrod algebra has to do with the Cohomology operations in singular Cohomology with Integer mod
2 Coefficients. For every and
there are natural
transformations of Functors
The existence of these cohomology operations endows the cohomology ring with the structure of a Module over the Steenrod algebra , defined to be , where is the free module functor that takes any set and sends it to the free module over that set. We think of as being a graded module, where the -th gradation is given by . This makes the tensor algebra into a Graded Algebra over . is the Ideal generated by the elements and for . This makes into a graded algebra.
By the definition of the Steenrod algebra, for any Space , is a Module over the Steenrod algebra , with multiplication induced by . With the above definitions, cohomology with Coefficients in the Ring , is a Functor from the category of pairs of Topological Spaces to graded modules over .
See also Adem Relations, Cartan Relation, Cohomology, Graded Algebra, Ideal, Module, Topological Space
© 1996-9 Eric W. Weisstein