Hopf orders in Hopf algebras with trivial Verschiebung Alan Koch - PowerPoint PPT Presentation

Hopf orders in Hopf algebras with trivial Verschiebung Alan Koch Agnes Scott College June, 2015 Alan Koch (Agnes Scott College) 1 / 47

Overview, Assumptions Let R be a complete discrete valuation ring of (equal) characteristic p , K = Frac R . We describe R -Hopf orders in a class of K -Hopf algebras H which are generated as K -algebras by their primitive elements P ( H ) . [Some of this will work for R = F q [ t ] , K = F q ( t ) , q a power of p .] These include orders in: K [ t ] / ( t p n ) , the monogenic local-local Hopf algebra of rank p n ( K Γ ) ∗ , Γ an elementary abelian p -group Assumptions All group schemes are affine, flat, commutative, p -power rank. All Hopf algebras are abelian (commutative, cocommutative), free over its base ring, and of p -power rank. Alan Koch (Agnes Scott College) 2 / 47

Outline Dieudonné Module Theory 1 More Linear Algebra 2 Hopf Orders 3 Rank p Hopf orders 4 Rank p n , n usually 2 5 What to do now 6 Alan Koch (Agnes Scott College) 3 / 47

Geometric Interpretation For now, let R be any F p -algebra, S = Spec ( R ) . Let G be an S -group scheme. Then G is equipped with: The relative Frobenius morphism F : G ! G ( p ) := G ⇥ S , Frob S . The Verschiebung morphism V : G ( p ) ! G , most easily defined as V = ( F G ∨ ) ∨ where ∨ indicates Cartier duality. Note that VF = p · id G and FV = p · id G ( p ) . Alan Koch (Agnes Scott College) 4 / 47

Let G a , R be the additive group scheme over R . When R is understood, denote it G a . Then End R -gp ( G a ) ⇠ = R [ F ] , where Fa = a p F for all a 2 R . Given a finite group scheme G , define D ∗ ( G ) = Hom R -gp ( G , G a ) . The ring R [ F ] acts on D ∗ ( G ) through its action on G a . This gives a contravariant functor { finite R -group schemes } � ! { finite R [ F ] -modules } which is not an anti-equivalence. Alan Koch (Agnes Scott College) 5 / 47

D ⇤ ( G ) = Hom R -gp ( G , G a ) However, the restricted functor ⇢ R -group schemes ⇢ finite R [ F ] -modules, � � � ! killed by V R -free, killed by V is an anti-equivalence; furthermore, rk ( G ) = p rk R ( D ∗ ( G )) , and this is compatible with base change. We will call finite, R -free R [ F ] -modules Dieudonné modules . (It is the only type of Dieudonné module needed here.) Alan Koch (Agnes Scott College) 6 / 47

Q. Which finite group schemes are killed by V ? Finite subgroup schemes of G n a for some n . (Short rationale: G a = ker V : W ! W .) Group schemes killed by p include: α p n = ker F n : G a ! G a . ( n th Frobenius kernel of G a ) Z / p Z = ker ( F � id ) : G a ! G a . (constant group scheme) Finite products of the group schemes above. This is not an exhaustive list. Alan Koch (Agnes Scott College) 7 / 47

Algebraic Interpretation G a = Spec ( R [ t ]) with t primitive. Let G be a group scheme, G = Spec ( H ) . Then D ∗ ( G ) = Hom R -gp ( G , G a ) ⇠ = Hom R -Hopf alg ( R [ t ] , H ) . Under this identification, f 2 D ∗ ( G ) sends t to a primitive element in H , and f is completely determined by this image. Thus, we define D ∗ ( H ) = P ( H ) and obtain a categorical equivalence 8 9 finite, flat, abelian < = R-Hopf algebras ; � ! { Dieudonné modules } . “killed by V" : Alan Koch (Agnes Scott College) 8 / 47

The inverse Let M be a finite R [ F ] -module, free over R . Let { e 1 , e 2 , . . . , e n } be an R -basis for M . Let a i , j , 1  i , j  n be given by n X Fe i = a j , i e j . j = 1 Then D ∗ ( H ) = M , where n H = R [ t 1 , . . . , t n ] / ( { t p X i � a j , i t j } ) , { t i } ⇢ P ( H ) j = 1 By writing M = R n and using e i as a standard basis vector, we have Fe i = Ae i where A = ( a i , j ) 2 M n ( R ) . Alan Koch (Agnes Scott College) 9 / 47

Some Examples Throughout, we also use F to denote the Frobenius morphism on Hopf algebras. In each example, the explicit algebra generators are primitive. Example p , H = R [ t 1 , . . . , t n ] / ( t p 1 , . . . , t p G = α n n ) . P ( H ) = Span R { t 1 , . . . , t n } . F ( t p i ) = 0 , 1  i  n . So D ∗ ( H ) is R -free on e 1 , . . . , e n with Fe i = 0 . In this case, A = 0 ( Fe i = Ae i = 0). Alan Koch (Agnes Scott College) 10 / 47

Example G = α p n , H = R [ t ] / ( t p n ) = R [ t 1 , . . . , t n ] / ( t p 1 , t p 2 � t 1 , . . . , t p n � t n − 1 ) P ( H ) = Span R { t , t p , . . . , t p n − 1 } . F ( t p i ) = t p i + 1 , 0  i  n � 1 . So D ∗ ( H ) is R -free on e 1 , . . . , e n with ⇢ e i − 1 i > 1 Fe i = i = 1 . 0 In this case, 0 1 0 1 0 · · · 0 0 0 1 · · · 0 B C B . . . C ... ... . . . B C A = . . . . B C B C ... B C 0 0 0 1 @ A 0 0 0 · · · 0 Alan Koch (Agnes Scott College) 11 / 47

Example � ∗ = R [ t 1 , . . . , t n ] / ( t p G = ( Z / p Z ) n , H = 1 � t 1 , . . . , t p RC n � n � t n ) p P ( H ) = Span R { t 1 , . . . , t n } . F ( t i ) = t p i = t i , 1  i  n . So D ∗ ( H ) is R -free on e , . . . , e n with Fe i = e i for all i . Clearly, A = I . Alan Koch (Agnes Scott College) 12 / 47

An example in the other direction Example Let A be the cyclic permutation matrix 0 1 0 1 0 · · · 0 0 0 1 · · · 0 B C B . . . C ... ... . . . B C A = . . . , D ( H ) = M A . B C B ... C B C 0 0 0 1 @ A 1 0 0 · · · 0 Then H is generated by primitive elements t 1 , . . . , t n with ⇢ t i − 1 i > 0 t p i = t n i = 0 If we set t = t n then H = R [ t ] / ( t p n � t ) , a monogenic Hopf algebra of rank p n . Alan Koch (Agnes Scott College) 13 / 47

Outline Dieudonné Module Theory 1 More Linear Algebra 2 Hopf Orders 3 Rank p Hopf orders 4 Rank p n , n usually 2 5 What to do now 6 Alan Koch (Agnes Scott College) 14 / 47

Maps Let A , B 2 M n ( R ) . Let M A , M B be free R -modules of rank n which are also R [ F ] -modules via Fe i = Ae i and Fe i = Be i respectively. A morphism of Dieudonné modules is an R -linear map M A ! M B which respects the actions of F . Alan Koch (Agnes Scott College) 15 / 47

Let Θ 2 M n ( R ) represent (and be) an R -linear map M A ! M B . Let n = 2 and write ✓ a 1 ✓ b 1 ✓ θ 1 ◆ ◆ ◆ a 2 b 2 θ 2 A = , B = , Θ = . a 3 a 4 b 3 b 4 θ 3 θ 4 Then: F ( Θ ( e 1 )) = F ( θ 1 e 1 + θ 3 e 2 ) = θ p 1 ( b 1 e 1 + b 3 e 2 ) + θ p 3 ( b 2 e 1 + b 4 e 2 ) = ( θ p 1 b 1 + θ p 3 b 2 ) e 1 + ( θ p 1 b 3 + θ p 3 b 4 ) e 2 Θ ( Fe 1 ) = Θ ( a 1 e 1 + a 3 e 2 ) = a 1 ( θ 1 e 1 + θ 3 e 2 ) + a 3 ( θ 2 e 1 + θ 4 e 2 ) = ( a 1 θ 1 + a 3 θ 2 ) e 1 + ( a 1 θ 3 + a 3 θ 4 ) e 2 . Alan Koch (Agnes Scott College) 16 / 47

Repeat for F ( Θ ( e 2 )) = Θ ( Fe 2 ) . We get: θ 1 a 1 + θ 2 a 3 = b 1 θ p 1 + b 2 θ p 3 θ 3 a 1 + θ 4 a 3 = b 3 θ p 1 + b 4 θ p 3 θ 1 a 2 + θ 2 a 4 = b 1 θ p 2 + b 2 θ p 4 θ 3 a 2 + θ 4 a 4 = b 3 θ p 2 + b 4 θ p 4 . In other words, Θ A = B Θ ( p ) where Θ ( p ) = ( θ p i ) for all i . Furthermore, Θ is an isomorphism if and only if Θ 2 M 2 ( R ) × . This generalizes to any n . Alan Koch (Agnes Scott College) 17 / 47

Choosing A 2 M n ( R ) gives an R -Hopf algebra, say H A . But. Different choices of A can produce the “same" Hopf algebra. Example Pick r 2 R , r 62 F p , and let ✓ 1 ◆ ✓ ◆ 0 1 0 A = and B = . r p � r 0 1 1 Then H A = R [ t 1 , t 2 ] / ( t p 1 � t 1 , t p 2 � t 2 ) 1 � u 1 � ( r p � r ) u 2 , u p H B = R [ u 1 , u 2 ] / ( u p 2 � u 2 ) Since ( t 1 + rt 2 ) p = t 1 + r p t 2 = t 1 + rt 2 + r p t 2 � rt 2 = ( t 1 + rt 2 ) + ( r p � r ) t 2 if we let u 1 = t 1 + rt 2 , u 2 = t 2 , then H A = H B . Alan Koch (Agnes Scott College) 18 / 47

Outline Dieudonné Module Theory 1 More Linear Algebra 2 Hopf Orders 3 Rank p Hopf orders 4 Rank p n , n usually 2 5 What to do now 6 Alan Koch (Agnes Scott College) 19 / 47

From now on, R = F q [[ T ]] , K = F q (( T )) . Let v K be the valuation on K with v K ( T ) = 1. We have R -Dieudonné modules and K -Dieudonné modules, compatible with base change. Pick A , B 2 M n ( R ) and construct R -Dieudonné modules M A , M B . Write M A = D ∗ ( H A ) and M B = D ∗ ( H B ) . Then D ∗ ( KH A ) is a Dieudonné module over K and D ∗ ( KH A ) ⇠ = D ∗ ( H A ) ⌦ R K . Similarly, D ∗ ( KH B ) ⇠ = D ∗ ( H B ) ⌦ R K . Alan Koch (Agnes Scott College) 20 / 47

Now KH A ⇠ = KH B if and only if there is a Θ 2 GL n ( K ) which, viewed as a K -linear isomorphism D ∗ ( KH A ) ! D ∗ ( KH B ) , respects the F -actions on the K -Dieudonné modules. Thus, H A and H B are Hopf orders in the same K -Hopf algebra iff Θ A = B Θ ( p ) for some Θ 2 GL n ( K ) . Alan Koch (Agnes Scott College) 21 / 47

Θ A = B Θ ( p ) for some Θ 2 GL n ( K ) Write A = ( a i , j ) , B = ( b i , j ) , Θ = ( θ i , j ) . Then H A is viewed as an R -Hopf algebra using A , i.e. H A = R [ u 1 , . . . u n ] / ( { u p X i � a j , i u j } ) . H B is viewed as an R -Hopf algebra using B , i.e. H B = R [ t 1 , . . . t n ] / ( { t p X i � b j , i t j } ) . KH B is viewed as a K -Hopf algebra in the obvious way. H B is viewed as an order in KH B in the obvious way. H A is viewed as an order in KH B through Θ , i.e. 2 8 9 3 n < = X 5 ⇢ KH B H A = R θ j , i t j : 1  i  n 4 : ; j = 1 (apologies for the abuses of language) Alan Koch (Agnes Scott College) 22 / 47

Outline Dieudonné Module Theory 1 More Linear Algebra 2 Hopf Orders 3 Rank p Hopf orders 4 Rank p n , n usually 2 5 What to do now 6 Alan Koch (Agnes Scott College) 23 / 47