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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
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© 1996-9 Eric W. Weisstein