# Steenrod algebra

The graded algebra $A _ {p}$ over the field $\mathbf Z _ {p}$ of all stable cohomology operations (cf. Cohomology operation) modulo $p$. For any space (spectrum of spaces) $X$, the group $H ^ \star ( X; \mathbf Z _ {p} )$ is a module over the Steenrod algebra $A _ {p}$.

The Steenrod algebra is multiplicatively generated by the Steenrod operations (cf. Steenrod operation). Thus, the Steenrod algebra $A _ {2}$ is a graded associative algebra, multiplicatively generated by the symbols $Sq ^ {i}$ with $\mathop{\rm deg} Sq ^ {i} = i$, which satisfy the Adem relation:

$$Sq ^ {a} Sq ^ {b} = \ \sum _ { t } \left ( \begin{array}{c} b- t- 1 \\ a- 2t \end{array} \right ) Sq ^ {a+} b- t Sq ^ {t} ,$$

$a < 2b$, so that an additive basis (over $\mathbf Z _ {2}$) of the Steenrod algebra $A _ {2}$ consists of the operations $Sq ^ {i _ {1} } \dots Sq ^ {i _ {r} }$, $i _ {k} \geq 2i _ {k+} 1$( the so-called Cartan–Serre basis). Similar results are true for $A _ {p}$ with $p > 2$. Furthermore,

$$( A _ {p} ) ^ {i} \cong H ^ {i+} n ( K( \mathbf Z _ {p} , n); \ \mathbf Z _ {p} ) ,\ \ n \textrm{ large } ,$$

where $K( \mathbf Z _ {p} , n)$ is an Eilenberg–MacLane space. The multiplication

$$K( \mathbf Z _ {p} , m) \wedge K( \mathbf Z _ {p} , n ) \rightarrow K( \mathbf Z _ {p} , m+ n)$$

induces the diagonal $\Delta : A _ {p} \rightarrow A _ {p} \otimes A _ {p}$ in $A _ {p}$, which is a homomorphism of algebras, and, consequently, turns $A _ {p}$ 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)

The analogue of the Steenrod algebra for a cohomology theory defined by a spectrum $E$ is $E _ \star ( E)$; cf. Generalized cohomology theories and Spectrum of spaces. The $E _ {2}$- term of the Adams spectral sequence, cf. Spectral sequence, $\mathop{\rm Ext} _ {E _ \star ( E) } ^ {s, t } ( E _ \star ( X), E _ \star ( Y) \Rightarrow [ X, Y] _ \star )$ is a purely (homological) algebra construct obtained by regarding the homology groups as modules over the Hopf algebra $E _ \star ( E)$.