Dual Steenrod algebra

In algebraic topology, through an algebraic operation (dualization), there is an associated commutative algebra from the noncommutative Steenrod algebras called the dual Steenrod algebra. This dual algebra has a number of surprising benefits, such as being commutative and provided technical tools for computing the Adams spectral sequence in many cases (such as formula_1pg 61-62) with much ease.
Definition.
Recallpg 59 that the Steenrod algebra formula_2 (also denoted formula_3) is a graded noncommutative Hopf algebra which is cocommutative, meaning its comultiplication is cocommutative. This implies if we take the dual Hopf algebra, denoted formula_4, or just formula_5, then this gives a graded-commutative algebra which has a noncommutative comultiplication. We can summarize this duality through dualizing a commutative diagram of the Steenrod's Hopf algebra structure:formula_6If we dualize we get mapsformula_7giving the main structure maps for the dual Hopf algebra. It turns out there's a nice structure theorem for the dual Hopf algebra, separated by whether the prime is formula_8 or odd.
Case of p=2.
In this case, the dual Steenrod algebra is a graded commutative polynomial algebra formula_9 where the degree formula_10. Then, the coproduct map is given byformula_11sendingformula_12where formula_13.
General case of p > 2.
For all other prime numbers, the dual Steenrod algebra is slightly more complex and involves a graded-commutative exterior algebra in addition to a graded-commutative polynomial algebra. If we let formula_14 denote an exterior algebra over formula_15 with generators formula_16 and formula_17, then the dual Steenrod algebra has the presentationformula_18whereformula_19In addition, it has the comultiplication formula_11 defined byformula_21where again formula_13.
Rest of Hopf algebra structure in both cases.
The rest of the Hopf algebra structures can be described exactly the same in both cases. There is both a unit map formula_23 and counit map formula_24formula_25which are both isomorphisms in degree formula_26: these come from the original Steenrod algebra. In addition, there is also a conjugation map formula_27 defined recursively by the equationsformula_28In addition, we will denote formula_29 as the kernel of the counit map formula_24 which is isomorphic to formula_5 in degrees formula_32.