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Hopf Rings, Dieudonn E Modules, and E Contemporary Mathematics Hopf Rings Dieudonne Mo dules and E S Paul G Go erss This paper is dedicated J Michael Boardman Abstract The category of graded bicommutative Hopf algebras over the prime eld with p elements is an ab elian category which is equivalent by work of Schoeller to a category of graded mo dules known as Dieudonne mo dules Graded ring ob jects in Hopf algebras are called Hopf rings and they arise in the study of unstable cohomology op erations for extraordinary cohomology theories The central p oint of this pap er is that Hopf rings can b e studied by lo oking at the asso ciated ring ob ject in Dieudonne mo dules They can also b e computed there and b ecause of the relationship b etween BrownGitler sp ectra and Dieudonne mo dules calculating the Hopf ring for a homology theory E comes down to computing E S which Ravenel has done for E BP From this one recovers the work of Hopkins Hunton and Turner on the Hopf rings of Landweber exact cohomology theories The are two ma jor algebraic diculties encountered in this approach The rst is to decide what a ring ob ject is in the category of Dieudonne mo dules as there is no obvious symmetric monoidal pairing asso ciated to a tensor pro duct of mo dules The second is to show that Hopf rings pass to rings in Dieudonne mo dules This involves studying universal examples and here we pick up an idea suggested by Bouseld torsionfree Hopf algebras over the padic integers with some additional structure such as a selfHopfalgebra map that reduces to the Verschiebung can b e easily classied An ab elian category A with a set of small pro jective generators is equivalent to a category M of mo dules over some ring R In addition if A and M are symmetric monoidal categories and the equivalence of categories A M resp ects the monoidal structure one can study the ring ob jects in A by studying the ring ob jects in M The purp ose of this pap er is to develop this observation in the case where A is the category HA of graded bicommutative Hopf algebras over the prime eld F The graded ring ob jects in HA are called Hopf rings and they p arise naturally when studying unstable cohomology op erations for some cohomology theory E see To state some results x a prime p A slight rewording gives the results at p We will restrict attention to the sort of Hopf algebra that arises in algebraic top ology namely to Hopf algebras that are skewcommutative and so Mathematics Subject Classication Primary N S Secondary L The author was supp orted in part by the National Science Foundation c copyright holder PAUL G GOERSS that the degree part H of H is the group algebra of an ab elian group We will call the category of such Hopf algebras HA If X is a p ointed space then H X F H X HA Schoeller has essentially proved that there is p an equivalence of categories D HA D where D is the category of graded Dieudonne mo dules An ob ject M D is a nonnegatively graded ab elian group M so that M is an F vector space and n p there are homomorphisms F M M and V M M n pn n np so that FV V F p and V if n If p do es not divide n we set V k k It follows that if n p s p s then p M If H HA then the n action of F and V on D H reect the Frobenius and Verschiebung resp ectively of H The category HA is a symmetric monoidal category As with the tensor pro duct of ab elian groups the symmetric monoidal pairing arises by considering bilinear maps An example of a bilinear map in HA is supplied by considering a n ring sp ectrum E The functor X E X is representable in the homotopy category n of spaces indeed if E n E then for all CW complexes X one has n X E n E X n m nm The cuppro duct pairing E X E X E X is induced by a map of spaces E n E m E n m and the resulting map of coalgebras H E n H E m H E n m is a bilinear map of Hopf algebras One can axiomatize this situation see x or and following Hunton and Turner we prove in x that given H and K in HA there is a universal bilinear map H K H K The pairing HA H A HA is symmetric monoidal the unit is the group ring F Z p Next one would like to calculate D H K If H H K is a bilinear pairing of Hopf algebras one obtains a bilinear pairing of Dieudonne mo dules D D H D H D K in the sense that D is a bilinear map of graded ab elian groups and V D x y D V x V y and D F x y F D x V y D x F y F D V x y If M N D there is a universal such bilinear pairing M N M N D which is easy to write down see Equation Then is a symmetric monoidal D pairing and one of the main results is D H K D H D K See Theorem D This leads to eective computations 2 3 HOPF RINGS DIEUDONNE MODULES AND E S We then employ this to study Hopf rings If E is a ring sp ectrum then H E fH E ng is a Zgraded ring ob ject in HA and hence D H E is a Zgraded ring ob ject in D This last means that given x D H E j y D H E k m n there is a pro duct x y D H E j k so that mn V x y V x V y F x y F x V y x F y F V x y Such an ob ject will b e called a Dieudonne ring If the ring sp ectrum is homotopy commutative then this pro duct satises the following skew commutativity formula nmj k x y y x Since k D H E k E k E D H E is actually an E algebra so D H E is an E Dieudonne algebra To compute this ob ject we use the fact the functor on sp ectra X D H X n which assigns to a space X the degree n part of of D H X is actually part of a homology theory if n mo d p In fact by if B n is the nth BrownGitler sp ectrum there is a natural surjection B n X D H X n n which is an isomorphism if n mo d p Thus for a ring sp ectrum E one obtains a surjection E B n D H E n k k n and if B fB ng is the graded BrownGitler sp ectrum one gets a degree n shearing surjection E B D H E of bigraded groups In fact this can b e made into a morphism of E Dieudonne algebras see x In many cases the kernel of this map can b e analyzed Fur thermore the Snaith splitting of S and the analysis of the summands done by a variety of authors shows that there is a ltration on E S so that the asso ciated graded ob ject is E B Since Ravenel has eectively calculated BP S one can deduce a great deal ab out D H E for Landweber exact theories In particular one can recover the HopkinsHuntonTurner results which in this form have a pleasing statement one that in fact succinctly enco des the RavenelWilson relation of complex oriented theories See Theorem This pap er is divided into three sections The rst two are devoted to the algebra of Hopf algebras their asso ciated Dieudonne mo dules and the appropriate bilinear pairings We sp end most of our energy discussing graded commutative as opp osed to skew commutative Hopf algebras moving on to skew commutative Hopf algebras and the top ological applications cited ab ove only in the third section Because of the splitting principle for skewcommutative Hopf algebras see and Prop osition the passage from commutative to skewcommutative is easy As ab ove we denote graded bicommutative Hopf algebras by HA PAUL G GOERSS In order to come to grips with some of the algebra involved we sp end a great deal of time working with universal examples The pro jective generators of HA include the Hopf algebras H n F x x x p k k i where n p s p s degx p s all with Witt vector diagonal This is i the reduction mo dule p of a Hopf algebra over the padic integers Z p CW k Z x x x s p k with the unique diagonal so that the Witt p olynomials i1 i p p i p x px w x i i are primitive The CW stands for coWitt The Z Hopf algebra CW k comes p s equipp ed with a lift of the Verschiebung that is there is a degree lowering Hopf algebra map CW k CW k s s which reduces to the Verschiebung H n H n Picking up a thread suggested by Bouseld it turns out that torsionfree graded connected Hopf algebras over Z with a lift of the Verschiebung are completely classied by their indecomp os p H can b e simply ables Furthermore if H is such a Hopf algebra then D F p Z p computed in terms of QH See Theorem This and other related topics o ccupy the rst four sections This is the rst part of the pap er The second part of the pap er is devoted to bilinear pairings developing the formulas cited ab ove and proving the isomorphism D H K D H D K D There is a table of contents at the end of this introduction and a glossary of symbols b efore the references This pro ject has it ro ots in a conversation with Bill Dwyer who noted that the work of Mo ore and Smith shows that the functor Z H Z has excellent exactness prop erties when Z is a lo op space Furthermore Dwyer suggested this fact could b e used to study Hopf rings I knew these exactness prop erties as the statement that on sp ectra X D H X was part of a ho mology theory Much of what is done here can b e greatly generalized The restriction to the prime eld is dictated by homotopy theory not by algebra and all of the algebraic results pass to any p erfect eld of characteristic p The internal grading on the Hopf algebras can probably also b e dropp ed although some care must b e taken to deal with those Hopf algebras that are neither group rings nor connected in the sense of 2 3 HOPF RINGS DIEUDONNE MODULES AND E S Contents Part I Classifying Hopf Algebras The Dwork Lemma and algebras with a lift of the Frobenius Hopf algebras with a lift of the Verschiebung The Implications of a lift of the Frobenius
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