# Hopf ring

A (graded) ring object in the category of (graded) co-commutative co-algebras (cf. Co-algebra). Such an object consists, first, of a sequence $\{ H _ {i} \}$ of Abelian group objects in the category. These are better known as commutative Hopf algebras with conjugation. Since they belong to the category, they have a coproduct:

$$\psi : {H _ {i} } \rightarrow {H _ {i} \otimes H _ {i} } .$$

Let $\psi ( x ) = \sum x ^ \prime \otimes x ^ {\prime \prime }$. As group objects in a category of co-algebras, the $H _ {i}$ also have a product $H _ {i} \otimes H _ {i} \rightarrow H _ {i}$. Below, this product is denoted by $*$. The "*" product should be thought of as "addition" in the ring as it is the pairing which gives the Abelian group structure. For ring "multiplication" one has

$$\circ : {H _ {i} \otimes H _ {j} } \rightarrow {H _ {i + j } } .$$

As with any ring, there must be a distributive law relating the multiplication and the addition. Chasing diagrams in the category one sees that it is:

$$x \circ ( y * z ) = \sum \pm ( x ^ \prime \circ y ) * ( x ^ \prime \circ z ) .$$

Hopf rings arise naturally in the study of the $\Omega$- spectra associated with generalized cohomology theories. Any generalized cohomology theory, $G ^ {*} ( X )$, gives rise to a sequence of spaces, $\{ {\underline{G} } _ {k} \}$, with the property that $G ^ {k} ( X ) \simeq [ X, {\underline{G} } _ {k} ]$, the homotopy classes of mappings. If $G$ is a multiplicative theory, then $\{ {\underline{G} } _ {k} \}$ is a graded ring object in the homotopy category. If $E$ represents a generalized homology theory and if there is a Künneth isomorphism for the $E$- homology of the spaces in the $\Omega$- spectra for $G$, then the sequence $\{ E _ {*} ( {\underline{G} } _ {*} ) \}$ becomes a Hopf ring. One can thus use knowledge of generalized homologies to further the understanding of generalized cohomologies by studying their classifying spaces using Hopf rings.

There are a number of Hopf rings which have been computed. Examples are $E _ {*} ( {\underline{BP } } _ {*} )$ and $E _ {*} ( {\underline{MU } } _ {*} )$, $E$ a complex orientable theory, [a9] (the basic reference for Hopf rings); $E _ {*} ( {\underline{K ( n ) } } _ {*} )$ and $E _ {*} ( {\underline{P ( n ) } } _ {*} )$, $E$ a complex orientable theory with $I _ {n} = 0$, [a14] and [a8]; $H _ {*} ( K ( \mathbf Z/ {( p ) } , * ) )$, [a13], § 8; $K ( n ) _ {*} ( - )$ for Eilenberg–MacLane spaces, [a10]; $K ( n ) _ {*} ( {\underline{k ( n ) } } _ {\ * } )$, [a5]; $H _ {*} ( {\underline{KO } } )$, [a11]; and the breakthrough description of $H _ {*} ( QS ^ {0} , \mathbf Z/ ( 2 ) )$ in [a12], and its sequel for $H _ {*} ( QS ^ {*} , \mathbf Z/ ( 2 ) )$ in [a1] followed by corresponding results for odd primes in [a7]. Other references are [a2], [a3], [a4], and [a6].

Hopf rings have a very rich algebraic structure, useful in two distinct ways: descriptive and computational. All of the above examples have their Hopf rings described with just a few generators and relations. The computations are generally carried out using Hopf ring techniques as well.

#### References

 [a1] P.J. Eccles, P.R. Turner, W.S. Wilson, "On the Hopf ring for the sphere" Math. Z. , 224 (2) (1997) pp. 229–233 [a2] M.J. Hopkins, J.R. Hunton, "The structure of spaces representing a Landweber exact cohomology theory" Topology , 34 (1) (1995) pp. 29–36 [a3] J.R. Hunton, N. Ray, "A rational approach to Hopf rings" J. Pure Appl. Algebra , 101 (3) (1995) pp. 313–333 [a4] T. Kashiwabara, "Hopf rings and unstable operations" J. Pure Appl. Algebra , 194 (1994) pp. 183–193 [a5] R. Kramer, "The periodic Hopf ring of connective Morava -theory" Ph.D. Thesis, Johns Hopkins Univ. (1990) [a6] T. Kashiwabara, N.P. Strickland, P.R. Turner, "Morava -theory Hopf ring for " C. Broto (ed.) etAAsal. (ed.) , Algebraic Topology: New Trends in Localization and Periodicity , Progress in Mathematics , 139 , Birkhäuser (1996) pp. 209–222 [a7] Y. Li, "On the Hopf ring for the sphere" Ph.D. Thesis, Johns Hopkins Univ. (1996) [a8] D.C. Ravenel, W.S. Wilson, "The Hopf ring for " Canadian J. Math. , 48 (5) (1996) pp. 1044–1063 [a9] D.C. Ravenel, W.S. Wilson, "The Hopf ring for complex cobordism" J. Pure Appl. Algebra , 9 (1977) pp. 241–280 [a10] D.C. Ravenel, W.S. Wilson, "The Morava -theories of Eilenberg–Mac Lane spaces and the Conner–Floyd conjecture" Amer. J. Math. , 102 (1980) pp. 691–748 [a11] N. Strickland, "Bott periodicity and Hopf rings" Ph.D. Thesis, Univ. Manchester (1992) [a12] P.R. Turner, "Dickson coinvariants and the homology of " Math. Z. , 224 (2) (1997) pp. 209–228 [a13] W.S. Wilson, "Brown–Peterson homology: an introduction and sampler" , CBMS , 48 , Amer. Math. Soc. (1982) [a14] W.S. Wilson, "The Hopf ring for Morava -theory" Publ. RIMS Kyoto Univ. , 20 (1984) pp. 1025–1036
How to Cite This Entry:
Hopf ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_ring&oldid=47272
This article was adapted from an original article by W.S. Wilson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article