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:
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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
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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:
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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 ![]() |
[a6] | T. Kashiwabara, N.P. Strickland, P.R. Turner, "Morava ![]() ![]() |
[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 ![]() |
[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 ![]() |
[a11] | N. Strickland, "Bott periodicity and Hopf rings" Ph.D. Thesis, Univ. Manchester (1992) |
[a12] | P.R. Turner, "Dickson coinvariants and the homology of ![]() |
[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 ![]() |
Hopf ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_ring&oldid=47272