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 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:

Let . As group objects in a category of co-algebras, the also have a product . 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

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:

Hopf rings arise naturally in the study of the -spectra associated with generalized cohomology theories. Any generalized cohomology theory, , gives rise to a sequence of spaces, , with the property that , the homotopy classes of mappings. If is a multiplicative theory, then is a graded ring object in the homotopy category. If represents a generalized homology theory and if there is a Künneth isomorphism for the -homology of the spaces in the -spectra for , then the sequence 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 and , a complex orientable theory, [a9] (the basic reference for Hopf rings); and , a complex orientable theory with , [a14] and [a8]; , [a13], § 8; for Eilenberg–MacLane spaces, [a10]; , [a5]; , [a11]; and the breakthrough description of in [a12], and its sequel for 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