The mod-two Steenrod algebra is the algebra of stable cohomology operations in singular cohomology with coefficients in 
. For every 
and 
there are natural
transformations of functors
 |
(1)
|
satisfying:
1. 
for 
.
2. 
for all 
and all pairs 
.
3. 
.
4. The 
maps commute with the coboundary maps in the long exact sequence of a pair. In other
words,
 |
(2)
|
is a degree 
transformation of cohomology theories.
5. (Cartan relation)
 |
(3)
|
6. (Adem relations) For 
,
 |
(4)
|
7. 
where 
is the cohomology suspension isomorphism.
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 
.
At an odd prime 
, the mod-
Steenrod algebra 
is instead generated by the reduced power operations 
and the Bockstein operation 
. Their degrees are 
and 1, respectively. This odd-prime
convention is used in the Adams spectral sequence
at the prime three.
