aa r X i v : . [ m a t h . R A ] N ov HOPF ORDERS IN K [ C p ] IN CHARACTERISTIC p ROBERT UNDERWOOD
Abstract.
Let p be prime, let K be a non-archimedean local field of characteristic p andlet C p denote the elementary abelian group of order p . We give a complete classificationof Hopf orders in the K -Hopf algebra ( K [ C p ]) ∗ and under a mild condition compute theirdual Hopf orders in the group ring K [ C p ]. Introduction
Let K be a non-archimedean local field of characteristic p . Equivalently, K is a field ofcharacteristic p that is complete with respect to a discrete valuation ν : K → Z ∪ {∞} withuniformizing parameter π , ν ( π ) = 1. The valuation ring is R = { x ∈ K | ν ( x ) ≥ } withunique maximal ideal p = { x ∈ R | ν ( x ) ≥ } and units U ( R ) = { x ∈ R | ν ( x ) = 0 } .Let B be a finite dimensional Hopf algebra over K . An R -Hopf order in B is an R -Hopfalgebra H which satisfies the following additional conditions: (i) H is a finitely generated,projective (hence free) R -submodule of B and (ii) K ⊗ R H ∼ = B as K -Hopf algebras. Nec-essarily, the comultiplicaton, counit and coinverse maps of H are induced from those of B .If H is an R -Hopf order in B , then its linear dual H ∗ is an R -Hopf order in B ∗ .This paper concerns the construction of Hopf orders in B in the case that B is the groupring K [ C np ] or its linear dual ( K [ C np ]) ∗ where C np , n ≥
1, denotes the elementary abeliangroup of order p n with C np = h g , g , . . . , g n i , g pi = 1, 1 ≤ i ≤ n .The problem of classifying R -Hopf orders in K [ C np ], n ≥
1, is difficult. The classificationis complete for the cases n = 1 , n ≥ R -Hopf orders in ( K [ C np ]) ∗ , n ≥
1. From Koch’s work we conclude that Hopf orders in( K [ C np ]) ∗ are determined by n ( n + 1) / K [ C np ] by first computingthe Hopf orders in ( K [ C np ]) ∗ and then taking their duals to obtain all Hopf orders in K [ C np ]but difficulties arise in computing the duals in K [ C np ]. At the very least we expect that Hopforders in K [ C np ] can be classified using n ( n + 1) / K [ C np ]) ∗ require exactly n ( n + 1) / n = 3 case and give a classification (under a mild condition)of R -Hopf orders in K [ C p ]. We achieve this by first computing all Hopf orders in ( K [ C p ]) ∗ using a cohomological argument employed by C. Greither in the n = 2, characteristic 0 case Date : December 14, 2020. [5, Part I]. We find that an R -Hopf order in ( K [ C p ]) ∗ appears as J = R [ π i ( ξ , , − µξ , , − αξ , , ) , π i ( ξ , , − βξ , , ) , π i ξ , , ] , constructed using 6 parameters, i , i , i , µ, α, β , which satisfy certain conditions.Under a mild restriction we then compute their duals in K [ C p ], finding that the dualsrequire 6 parameters as well. A dual Hopf order in K [ C p ] can be written as a “truncatedexponential” Hopf order H = R " g − π i , g g [ µ ]1 − π i , g g [ α ]1 ( g g [ µ ]1 ) [ β ] − π i using the same parameters, i , i , i , µ, α, β . Hopf orders of this form have appeared in thework of N. Byott and G. Elder as realizable Hopf orders [2, § Koch’s classification of Hopf orders in ( K [ C np ]) ∗ Let H be a Hopf algebra over a commutative ring with unity D . An element h ∈ H is primitive if ∆( h ) = 1 ⊗ h + h ⊗
1. The Hopf algebra H is primitively generated if it isgenerated as a D -algebra by primitive elements. The K -Hopf algebra ( K [ C np ]) ∗ is primitivelygenerated. We have ( K [ C np ]) ∗ = K [ t , t , . . . , t n ] , with ∆( t i ) = 1 ⊗ t i + t i ⊗ t pi = t i for 1 ≤ i ≤ n , see [8, Example 2.6].In [8] A. Koch has completely determined all R -Hopf orders in ( K [ C np ]) ∗ using the cate-gorical equivalence between the category of primitively generated R -Hopf algebras and thecategory of finite R [ F ]-modules that are free as R -modules, i.e., the category of Dieudonn´emodules in the sense of [7, § F is an indeterminate in the non-commutative polyno-mial ring R [ F ]; F satisfies the condition F a = a p F for all a ∈ R .We review Koch’s result. Let J be a primitively generated R -Hopf algebra. There existsa matrix A = ( a i,j ) ∈ Mat n ( R ) for which J = R [ u , u , . . . , u n ] /I where I is the ideal in R [ u , u , . . . , u n ] generated by { u pi − P nj =1 a j,i u j } ≤ i ≤ n , with u i primitivefor 1 ≤ i ≤ n ; A is the matrix associated to J . For a matrix M = ( m i,j ) ∈ Mat n ( K ), let M ( p ) be the matrix whose i, j th entry is m pi,j . Theorem 2.1 (Koch) . (i) Let n ≥ , let Θ = ( θ i,j ) be a lower triangular matrix in GL n ( K ) and let A = Θ − Θ ( p ) .Suppose that A = ( a i,j ) ∈ Mat n ( R ) . Let J = R [ u , u , . . . , u n ] /I OPF ORDERS IN K [ C p ] IN CHARACTERISTIC p where I is the ideal in R [ u , u , . . . , u n ] generated by { u pi − P nj =1 a j,i u j } ≤ i ≤ n , with u i primitivefor ≤ i ≤ n . There is a Hopf embedding J → ( K [ C np ]) ∗ given as u i n X j =1 θ j,i t j ,t pj = t j , t j primitive. The image of J under this embedding is H Θ = R [ { n X j =1 θ j,i t j , ≤ i ≤ n } ] , which is an R -Hopf order in ( K [ C np ]) ∗ .(ii) Let J be an R -Hopf order in ( K [ C np ]) ∗ . Then J is a primitively generated R -Hopfalgebra and there exists A = ( a i,j ) ∈ Mat n ( R ) with J = R [ u , u , . . . , u n ] /I where I is the ideal in R [ u , u , . . . , u n ] generated by { u pi − P nj =1 a j,i u j } ≤ i ≤ n , with u i primitivefor ≤ i ≤ n . Moreover, there exists a lower triangular matrix Θ = ( θ i,j ) ∈ GL n ( K ) so that J ∼ = H Θ = R [ { n X j =1 θ j,i t j , ≤ i ≤ n } ] ⊆ ( K [ C np ]) ∗ ,t pj = t j , t j primitive. Since Θ is lower triangular, J is determined by n ( n + 1) / parameters.Proof. (Sketch)(i): Since A = Θ − Θ ( p ) is in Mat n ( R ), [8, Theorem 4.2] applies to show that H Θ is an R -Hopf order in ( K [ C np ]) ∗ .For (ii): Since ( K [ C np ]) ∗ is primitively generated, J is a primitively generated R -Hopfalgebra. Let A = ( a i,j ) ∈ Mat n ( R ) be the matrix associated to J , i.e., J = R [ u , u , . . . , u n ] /I where I is the ideal in R [ u , u , . . . , u n ] generated by { u pi − P nj =1 a j,i u j } ≤ i ≤ n , with u i primitivefor 1 ≤ i ≤ n . Note that J is generically isomorphic to ( R [ C np ]) ∗ , i.e., K ⊗ R J ∼ = K ⊗ R ( R [ C np ]) ∗ ∼ = ( K [ C np ]) ∗ as K -Hopf algebras. Thus by [8, Corollary 4.2], there exists a matrix Θ = ( θ i,j ) ∈ GL n ( K )with Θ A = Θ ( p ) , hence A = Θ − Θ ( p ) . Let H Θ = R [ { n X j =1 θ j,i t j , ≤ i ≤ n } ] ⊆ ( K [ C np ]) ∗ , ROBERT UNDERWOOD t pi = t i , t i primitive. By [8, Theorem 4.3], Θ determines an isomorphism J → H Θ (anembedding of J into ( K [ C np ]) ∗ ) defined by u i n X j =1 θ j,i t j for 1 ≤ i ≤ n .From [8, Theorem 6.1], we can assume that Θ is lower triangular thus H Θ is determinedby n ( n + 1) / (cid:3) In the examples that follow we illustrate how to use Theorem 2.1(i) to construct Hopforders in the cases n = 1 , Example 2.2. n = 1 . In this case we choose Θ = (Θ , ) ∈ GL ( K ) ∼ = K × , with Θ , = π i for some integer i . To construct an R -Hopf order in ( K [ C p ]) ∗ , we require that A = Θ − Θ ( p ) = ( π − i )( π pi ) = ( π ( p − i ) is in Mat ( R ) . Thus we require i ≥ . Let J = R [ u ] / ( u p − π ( p − i u ) where u is primitive. Then there is a Hopf embedding J → ( K [ C p ]) ∗ = K [ t ] defined as u π i t . The image of J under this embedding is H i = R [ π i t ] ⊆ ( K [ C p ]) ∗ ; H i is a Tate-Oort
Hopf order in ( K [ C p ]) ∗ [9] , [4, Theorem 2.3] . Example 2.3. n = 2 . In this case we choose Θ = (cid:18) π i θ π j (cid:19) ∈ GL ( K ) , where i, j areintegers. To construct an R -Hopf order in ( K [ C p ]) ∗ , we require that A = Θ − Θ ( p ) = (cid:18) π ( p − i π − j θ p − π ( p − i − j θ π ( p − j (cid:19) ∈ Mat ( R ) . Thus we require i, j ≥ and ν ( π − j θ p − π ( p − i − j θ ) ≥ . (The condition A ∈ Mat ( R ) is equivalent to the conditions i, j ≥ , ν ( π − j θ p − π ( p − i − j θ ) ≥ .)Let J = R [ u , u ] /I with I = ( u p − π ( p − i u − ( π − j θ p − π ( p − i − j θ ) u , u p − π ( p − j u ) ,u , u primitive. Then there is a Hopf embedding J → ( K [ C p ]) ∗ = K [ t , t ] defined by u π i t + θt , u π j t . OPF ORDERS IN K [ C p ] IN CHARACTERISTIC p The image of J under this embedding is H i,j,θ = R [ π i t + θt , π j t ]; H i,j,θ is an R -Hopf order in ( K [ C p ]) ∗ . Remark 2.4.
Koch’s classification applies more broadly to construct all of the R -Hopf ordersin any primitively generated K -Hopf algebra, for instance K [ t ] / ( t p n ) , which represents thegroup scheme α p n . Classification of Hopf orders in ( K [ C p ]) ∗ using Cohomology In [4] G. Elder and this author use Greither’s cohomological method of [5] to classify Hopforders in ( K [ C p ]) ∗ and hence recover case n = 2 of Theorem 2.1. Let C p = h g , g i and let ξ , , ξ , be elements of ( K [ C p ]) ∗ defined by h ξ , , ( g − j ( g − k i = δ ,j δ ,k , h ξ , , ( g − j ( g − k i = δ ,j δ ,k , ≤ j, k ≤ p −
1, where h− , −i : ( K [ C p ]) ∗ × K [ C p ] → K is the duality map. Then ξ , and ξ , are primitive elements in ( K [ C p ]) ∗ with ξ p , = ξ , , ξ p , = ξ , . The K -Hopf algebra( K [ C p ]) ∗ is primitively generated by ξ , , ξ , ,( K [ C p ]) ∗ = K [ ξ , , ξ , ] . (The ξ , , ξ , play the role of t , t in Koch’s construction given above.)For x ∈ K , let ℘ ( x ) = x p − x . Proposition 3.1 (Elder-U.) . (i) Let i , i ≥ be integers, let µ be an element of K satisfying ν ( ℘ ( µ )) ≥ i − pi and let H ∗ i ,i ,µ = R [ π i ( ξ , − µξ , ) , π i ξ , ] . Then H ∗ i ,i ,µ is an R -Hopf order in ( K [ C p ]) ∗ .(ii) Let J be an R -Hopf order in ( K [ C p ]) ∗ . Then J ∼ = H ∗ i ,i ,µ = R [ π i ( ξ , − µξ , ) , π i ξ , ] , where i , i ≥ are integers and µ is an element of K satisfying ν ( ℘ ( µ )) ≥ i − pi .Proof. (Sketch)(i): We have( π i ( ξ , − µξ , )) p = π pi ( ξ , − µ p ξ , )= π ( p − i π i ( ξ , − µξ , ) + π pi µξ , − π pi µ p ξ , = π ( p − i π i ( ξ , − µξ , ) − ℘ ( µ ) π pi − i π i ξ , . ROBERT UNDERWOOD
Since i ≥ ν ( ℘ ( µ )) ≥ i − pi , π i ( ξ , − µξ , ) satisfies a monic polynomial withcoefficients in R [ π i ξ , ]. Moreover,( π i ξ , ) p = π ( p − i π i ξ , , and so, i ≥ π i ξ , satisfies a monic polynomial with coefficients in R .Thus the R -algebra H ∗ i ,i ,µ is a free R -submodule of ( K [ C p ]) ∗ .The R -algebra H ∗ i ,i ,µ is an R -Hopf algebra with comultiplication map defined by∆( π i ( ξ , − µξ , )) = 1 ⊗ ( π i ( ξ , − µξ , )) + ( π i ( ξ , − µξ , )) ⊗ , ∆( π i ξ , ) = 1 ⊗ π i ξ , + π i ξ , ⊗ , counit map defined as ε ( π i ( ξ , − µξ , )) = 0 , ε ( π i ξ , ) = 0 , and coinverse defined by S ( π i ( ξ , − µξ , )) = − π i ( ξ , − µξ , ) , S ( π i ξ , ) = − π i ξ , . In addition, K ⊗ R H ∗ i ,i ,µ ∼ = ( K [ C p ]) ∗ , as K -Hopf algebras. Thus H ∗ i ,i ,µ is an R -Hopf orderin ( K [ C p ]) ∗ .For (ii): The R -Hopf order J induces a short exact sequence of R -Hopf algebras R → H → J → H ′ → R (1)where H ′ , H are Hopf orders in ( K [ C p ]) ∗ . By [4, Theorem 2.3], H ′ and H are Tate-OortHopf orders in ( K [ C p ]) ∗ , i.e., H ′ = H ∗ i ∼ = R [ π i ξ , ] and H = H ∗ i ∼ = R [ π i ξ , ] for integers i , i ≥ D ∗ i = Spec H ∗ i , D ∗ i = Spec H ∗ i , and D ∗ = Spec J denote the corresponding groupschemes over R . From (1) we obtain the short exact sequence of group schemes0 −→ D ∗ i −→ D ∗ −→ D ∗ i −→ , (2)which over K appears as0 −→ Spec ( K [ C p ]) ∗ −→ Spec ( K [ C p ]) ∗ −→ Spec ( K [ C p ]) ∗ −→ . The short exact sequence (2) represents a class in Ext gt ( D ∗ i , D ∗ i ), which is the group ofgenerically trivial extensions of D ∗ i by D ∗ i . So computing Ext gt ( D ∗ i , D ∗ i ), and ultimatelythe representing algebras of the middle terms of these short exact sequences, will completelydetermine J .Let G a denote the additive group scheme represented by the R -Hopf algebra R [ x ], x indeterminate. There is a faithfully flat resolution of group schemes0 −→ D ∗ i −→ G a Ψ −→ G a −→ , Ψ( x ) = x p − π ( p − i x , and through the long exact sequence we obtain an isomorphismcoker(Ψ : Hom( D ∗ i , G a ) (cid:9) ) gt ∼ = Ext gt ( D ∗ i , D ∗ i ) . OPF ORDERS IN K [ C p ] IN CHARACTERISTIC p By [4, Proposition 4.5], coker(Ψ : Hom( D ∗ i , G ) (cid:9) ) gt is isomorphic to the additive subgroupof K/ ( F p + p i − i )represented by those elements µ ∈ K satisfying ν ( ℘ ( µ )) ≥ i − pi . We conclude that classes[ µ ] in K/ ( F p + p i − i ) correspond to equivalence classes of generically trivial short exactsequences 0 −→ D ∗ i −→ D ∗ −→ D ∗ i −→ . Using a push-out diagram, the representing Hopf algebra of D ∗ is J ∼ = H ∗ i ,i ,µ = R [ π i ( ξ , − µξ , ) , π i ξ , ] . (cid:3) Remark 3.2.
Note that H ∗ i ,i ,µ is Koch’s H i,j,θ of Example 2.3 with µ = − π − i θ , ξ , = t , ξ , = t , i = i , and i = j . With this change of notation H ∗ i ,i ,µ ∼ = R [ u , u ] /I where I is generated by u p − π ( p − i u − ( π − j θ p − π ( p − i − j θ ) u = u p − π ( p − i u + ( π − i π pi µ p − π ( p − i − i π i µ ) u = u p − π ( p − i u + ℘ ( µ ) π pi − i u and u p − π ( p − j u = u p − π ( p − i u . Moreover, Koch’s condition ν ( π − j θ p − π ( p − i − j θ ) ≥ becomes ν ( ℘ ( µ )) ≥ i − pi . Hopf orders in ( K [ C p ]) ∗ We can extend Proposition 3.1 to the case n = 3. Let C p = h g , g , g i and let ξ , , , ξ , , , ξ , , be elements of ( K [ C p ]) ∗ defined by h ξ , , , ( g − j ( g − k ( g − l i = δ ,j δ ,k δ ,l , h ξ , , , ( g − j ( g − k ( g − l i = δ ,j δ ,k δ ,l , h ξ , , , ( g − j ( g − k ( g − l i = δ ,j δ ,k δ ,l , ≤ j, k, l ≤ p −
1, where h− , −i : ( K [ C p ]) ∗ × K [ C p ] → K is the duality map. Then ξ , , , ξ , , and ξ , , are primitive elements in ( K [ C p ]) ∗ with ξ p , , = ξ , , , ξ p , , = ξ , , and ξ p , , = ξ , , . The K -Hopf algebra ( K [ C p ]) ∗ is primitively generated by ξ , , , ξ , , , ξ , , ;( K [ C p ]) ∗ = K [ ξ , , , ξ , , , ξ , , ] . ROBERT UNDERWOOD
Proposition 4.1.
Let i , i , i ≥ be integers, let µ, α, β be elements of K satisfying ν ( ℘ ( µ )) ≥ i − pi , ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi and ν ( ℘ ( β )) ≥ i − pi . Let H ∗ i ,i ,i ,µ,α,β = R [ π i ( ξ , , − µξ , , − αξ , , ) , π i ( ξ , , − βξ , , ) , π i ξ , , ] . Then H ∗ i ,i ,i ,µ,α,β is an R -Hopf order in ( K [ C p ]) ∗ .Proof. We have( π i ( ξ , , − µξ , , − αξ , , )) p = π pi ( ξ , , − µ p ξ , , − α p ξ , , )= π ( p − i π i ( ξ , , − µξ , , − αξ , , )+ π pi µξ , , + π pi αξ , , − π pi µ p ξ , , − π pi α p ξ , , = π ( p − i π i ( ξ , , − µξ , , − αξ , , ) − ℘ ( µ ) π pi ξ , , − ℘ ( α ) π pi ξ , , = π ( p − i π i ( ξ , , − µξ , , − αξ , , ) − ℘ ( µ ) π pi − i π i ξ , , + ℘ ( µ ) βπ pi − i π i ξ , , − ℘ ( µ ) βπ pi − i π i ξ , , − ℘ ( α ) π pi − i π i ξ , , = π ( p − i π i ( ξ , , − µξ , , − αξ , , ) − ℘ ( µ ) π pi − i π i ( ξ , , − βξ , , ) − ( ℘ ( α ) + ℘ ( µ ) β ) π pi − i π i ξ , , . Since i ≥ ν ( ℘ ( µ )) ≥ i − pi and ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi , the generator π i ( ξ , , − µξ , , − αξ , , )satisfies a monic polynomial with coefficients in R -Hopf order R [ π i ( ξ , , − βξ , , ) , π i ξ , , ].Thus the R -algebra H ∗ i ,i ,i ,µ,α,β is a free R -submodule of ( K [ C p ]) ∗ .The R -algebra H ∗ i ,i ,i ,µ,α,β is an R -Hopf algebra with comultiplication map defined by∆( π i ( ξ , , − µξ , , − αξ , , )) = 1 ⊗ ( π i ( ξ , , − µξ , , − αξ , , ))+( π i ( ξ , , − µξ , , − αξ , , )) ⊗ , ∆( π i ( ξ , , − βξ , , )) = 1 ⊗ ( π i ( ξ , , − βξ , , )) + ( π i ( ξ , , − βξ , , )) ⊗ , ∆( π i ξ , , ) = 1 ⊗ π i ξ , , + π i ξ , , ⊗ , counit map defined as ε ( π i ( ξ , , − µξ , , − αξ , , )) = 0 , ε ( π i ( ξ , − µξ , )) = 0 , ε ( π i ξ , ) = 0 , and coinverse defined by S ( π i ( ξ , , − µξ , , − αξ , , )) = − ( π i ( ξ , , − µξ , , − αξ , , )) ,S ( π i ( ξ , − µξ , )) = − π i ( ξ , − µξ , ) , S ( π i ξ , ) = − π i ξ , . In addition, K ⊗ R H ∗ i ,i ,i ,µ,α,β ∼ = ( K [ C p ]) ∗ , as K -Hopf algebras. Thus H ∗ i ,i ,i ,µ,α,β is an R -Hopf order in ( K [ C p ]) ∗ . (cid:3) OPF ORDERS IN K [ C p ] IN CHARACTERISTIC p We want to show that an arbitrary R -Hopf order J in ( K [ C p ]) ∗ can be written as H ∗ i ,i ,i ,µ,α,β for some parameters i , i , i , µ, α, β .An R -Hopf order J induces a short exact sequence of R -Hopf algebras R → H → J → H ′ → R (3)where H ′ is an R -Hopf order in ( K [ C p ]) ∗ and H is an R -Hopf order in ( K [ C p ]) ∗ . By Propo-sition 3.1 (ii), H ′ is of the form H ′ ∼ = H ∗ i ,i ,µ = R [ π i ( ξ , , − µξ , , ) , π i ξ , , ] , for integers i , i ≥ µ ∈ K with ν ( ℘ ( µ )) ≥ i − pi , and by [4, Theorem 2.3], H is of the form H ∼ = H ∗ i = R [ π i ξ , , ]for integer i ≥ D ∗ = Spec J , D ∗ i ,i ,µ = Spec H ∗ i ,i ,µ and D ∗ i = Spec H ∗ i . Then from (3) we obtainthe short exact sequence of R -group schemes0 −→ D ∗ i ,i ,µ −→ D ∗ −→ D ∗ i −→ , (4)which over K appears as0 −→ Spec ( K [ C p ]) ∗ −→ Spec ( K [ C p ]) ∗ −→ Spec ( K [ C p ]) ∗ −→ . The short exact sequence (4) represents a class in Ext gt ( D ∗ i , D ∗ i ,i ,µ ), which is the groupof generically trivial extensions of D ∗ i by D ∗ i ,i ,µ . So computing Ext gt ( D ∗ i , D ∗ i ,i ,µ ), andultimately the representing algebras of the middle terms of these short exact sequences, willcompletely determine J .As an R -Hopf algebra, H ∗ i ,i ,µ ∼ = R [ u , u ] /I, where I = ( u p − π ( p − i u + ℘ ( µ ) π pi − i u , u p − π ( p − i u ) . (Recall the change in notation of Remark 3.2.) Let G a be the R -group scheme representedby the R -Hopf algebra R [ x, y ], x, y indeterminates. There is a faithfully flat resolution0 −→ D ∗ i ,i ,µ −→ G a Ψ −→ G a −→ ( x, y ) , Ψ ( x, y )) withΨ ( x, y ) = x p − π ( p − i x + ℘ ( µ ) π pi − i y and Ψ ( x, y ) = y p − π ( p − i y. From (5) we obtain the long exact sequenceHom( D ∗ i , G a ) Ψ −→ Hom( D ∗ i , G a ) ω −→ Ext ( D ∗ i , D ∗ i ,i ,µ ) ι −→ Ext ( D ∗ i , G a ) , with connecting homomorphism ω , which induces the map ρ in the exact sequence −→ coker(Ψ : Hom( D ∗ i , G a ) (cid:9) ) ρ −→ Ext ( D ∗ i , D ∗ i ,i ,µ ) ι −→ Ext ( D ∗ i , G a ) . Tensoring with K and considering kernels (indicated by gt ) results in the following commu-tative diagram with exact rows and columns: ↓ ↓ ↓ −→ coker(Ψ : Hom( D ∗ i , G a ) (cid:9) ) gt ρ −→ Ext gt ( D ∗ i , D ∗ i ,i ,µ ) ι −→ Ext gt ( D ∗ i , G a ) ↓ ↓ ↓ −→ coker(Ψ : Hom( D ∗ i , G a ) (cid:9) ) ρ −→ Ext ( D ∗ i , D ∗ i ,i ,µ ) ι −→ Ext ( D ∗ i , G a ) ↓ ↓ ↓ −→ coker(Ψ : Hom( D ∗ i ,K , G a,K ) (cid:9) ) ρ −→ Ext ( D ∗ i ,K , D ∗ i ,i ,µ,K ) ι −→ Ext ( D ∗ i ,K , G a,K ) Now by [6, Lemma 4.1], Ext gt ( D ∗ i , G a ) = (Ext gt ( D ∗ i , G a )) . Thus Ext gt ( D ∗ i , G a ) = 0since Ext gt ( D ∗ i , G a ) = 0 by [4, Proposition 4.3].We have shown the following. Proposition 4.2.
There is an isomorphism ρ : coker(Ψ : Hom( D ∗ i , G a ) (cid:9) ) gt → Ext gt ( D ∗ i , D ∗ i ,i ,µ ) . Proof.
We have Ext gt ( D ∗ i , G a ) = 0 in the first row of the commutative diagram above. (cid:3) The isomorphism of Proposition 4.2 classifies all short exact sequences0 −→ D ∗ i ,i ,µ −→ D ∗ −→ D ∗ i −→ , which over K appear as0 −→ Spec ( K [ C p ]) ∗ −→ Spec ( K [ C p ]) ∗ −→ Spec ( K [ C p ]) ∗ −→ . We seek to compute J , which is the representing algebra of D ∗ . To do this we first need tocompute coker(Ψ : Hom( D ∗ i , G a ) (cid:9) ) gt . Let F p ( µ, −
1) = { ( α, β ) ∈ K | α = mµ, β = − m, m ∈ F p } . Proposition 4.3.
The group coker(Ψ : Hom( D ∗ i , G a ) (cid:9) ) gt is isomorphic to the additivesubgroup of K / ( F p ( µ, −
1) + ( F p + p i − i , p i − i )) represented by pairs ( α, β ) which satisfy ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi and ν ( ℘ ( β )) ≥ i − pi . OPF ORDERS IN K [ C p ] IN CHARACTERISTIC p Proof.
Let P = Prim( H ∗ i ) = Rπ i ξ , , denote the set of primitive elements of H ∗ i . Thenelements of coker(Ψ : Hom( D ∗ i , G a ) (cid:9) ) gt correspond to the quotient(Ψ( K ⊗ R P ) ∩ P ) / Ψ( P ) . We give a description of this quotient. Elements in K ⊗ R P can be written as ( απ i ξ , , , βπ i ξ , , )for some α, β ∈ K . Thus an element in Ψ( K ⊗ R P ) appears asΨ( απ i ξ , , , βπ i ξ , , ) = (Ψ ( απ i ξ , , , βπ i ξ , , ) , Ψ ( απ i ξ , , , βπ i ξ , , ))= ( ℘ ( α ) π pi ξ , , + ℘ ( µ ) βπ pi ξ , , , ℘ ( β ) π pi ξ , , )= (( ℘ ( α ) + ℘ ( µ ) β ) π pi ξ , , , ℘ ( β ) π pi ξ , , ) . The image Ψ( απ i ξ , , , βπ i ξ , , ) is in P precisely when ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi and ν ( ℘ ( β )) ≥ i − pi , and Ψ( απ i ξ , , , βπ i ξ , , ) is in Ψ( P ) if and only if ( α, β ) is in thesubgroup F p ( µ, −
1) + ( F p + p i − i , p i − i ) . (cid:3) Proposition 4.4.
The representing algebra of D ∗ can be written in the form J ∼ = H ∗ i ,i ,i ,µ,α,β = R [ π i ( ξ , , − µξ , , − αξ , , ) , π i ( ξ , , − βξ , , ) , π i ξ , , ] , where µ ∈ K represents a coset in K/ ( F p + p i − i ) with ν ( ℘ ( µ )) ≥ i − pi and the pair ( α, β ) ∈ K represents a coset in K / ( F p ( µ, −
1) + ( F p + p i − i , p i − i )) with ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi , ν ( ℘ ( β )) ≥ i − pi .Proof. Let [ E ] be a class in Ext gt ( D ∗ i , D ∗ i ,i ,µ ). In view of the isomorphism of Proposition4.2, we have [ h ] = ρ − ([ E ]) for some homomorphism h : D ∗ i → G a . Applying [4, Remark 4.1]to the resolution (5) we conclude that the image of h under the connecting homomorphism ω (inducing ρ ) is the class [ D ∗ h ] of the pull-back D ∗ h = { ( w, ( x, y )) ∈ D ∗ i × G a : h ( w ) = Ψ( x, y ) } . In view of Proposition 4.3, the homomorphism h : D ∗ i → G a is determined by a Hopfalgebra map f : R [ x, y ] → H ∗ i given as x ( ℘ ( α ) + ℘ ( µ ) β ) π pi ξ , , , y ℘ ( β ) π pi ξ , , for some α, β ∈ K with ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi , ν ( ℘ ( β )) ≥ i − pi .The representing Hopf algebra H ∗ h of D ∗ h arises from the push-out diagram H ∗ h ← R [ x, y ] ↑ Ψ ↑ H ∗ i f ← R [ x, y ] . Thus, H ∗ h = R [ π i ξ , , ] ⊗ R R [ x, y ](Ψ( απ i ξ , , , βπ i ξ , , ) ⊗ ⊗ Ψ( x, y )) ∼ = R [ π i ξ , , ][ x, y ](Ψ( x, y ) + Ψ( απ i ξ , , , βπ i ξ , , )) ∼ = R [ π i ξ , , ][ x, y ](Ψ( x + απ i ξ , , , y + βπ i ξ , , )) ∼ = R [ π i ( ξ , , − µξ , , − αξ , , ) , π i ( ξ , , − βξ , , ) , π i ξ , , ] , since H ∗ i ,i ,µ ∼ = R [ x, y ] / (Ψ ( x, y ) , Ψ ( x, y )) . (cid:3) Remark 4.5.
Let H ∗ i ,i ,i ,µ,α,β be an R -Hopf order in ( K [ C p ]) ∗ . Assume [4, Convention 1] ,i.e., if µ ∈ F p + p i − i , then µ = 0 . Then it is natural to assume that ν ( µ ) ≥ i − i and i ≥ i . The motivation behind this assumption is as follows. The pair ( α, β ) = (0 , corresponds tothe Hopf order H ∗ i ,i ,i ,µ, , = R [ π i ( ξ , , − µξ , , ) , π i ξ , , , π i ξ , , ] and the pair ( α, β ) = ( mµ, − m ) , m ∈ F p , in F p ( µ, − corresponds to the Hopf order H ∗ i ,i ,i ,µ,mµ, − m = R [ π i ( ξ , , − µξ , , − mµξ , , ) , π i ( ξ , , + mξ , , ) , π i ξ , , ]= R [ π i ( ξ , , − µ ( ξ , , + mξ , , )) , π i ( ξ , , + mξ , , ) , π i ξ , , ] . Since ( ξ , , + mξ , , ) p = ξ , , + mξ , , , and ξ , , + mξ , , is primitive, H ∗ i ,i ,i ,µ, , ∼ = H ∗ i ,i ,i ,µ,mµ, − m as R -Hopf algebras. Thus the pair ( α, β ) = ( µm, − m ) corresponds to the trivial extensionand so we assume that ( µm, − m ) ∈ ( F p + p i − i , p i − i ) , Hence, ν ( µ ) ≥ i − i and i ≥ i . OPF ORDERS IN K [ C p ] IN CHARACTERISTIC p Hopf orders in K [ C np ] , n = 1 , , J in ( K [ C np ]) ∗ for the cases n =1 , ,
3. To construct Hopf orders in K [ C np ], n = 1 , ,
3, we need to compute the linear duals J ∗ .5.1. The case n = 1 . Let C p = h g i , let i ≥ E ( i ) = R (cid:20) g − π i (cid:21) . Then it is well-known that E ( i ) is an R -Hopf order in K [ C p ] (see for instance [4, § Proposition 5.1 (Elder-U.) . Let i ≥ be an integer and let H ∗ i = R [ π i ξ ] , ξ primitive, ξ p = ξ , be an R -Hopf order in ( K [ C p ]) ∗ . Then ( H ∗ i ) ∗ = E ( i ) . Proof.
See [4, Proposition 2.2]. (cid:3)
So every R -Hopf order in K [ C p ] can be written in the form E ( i ) = R (cid:20) g − π i (cid:21) for some integer i ≥ The case n = 2 . For x, y the truncated exponential is defined as x [ y ] = p − X m =0 (cid:18) ym (cid:19) ( x − m , where (cid:0) ym (cid:1) is the generalized binomial coefficient (cid:18) ym (cid:19) = y ( y − y − · · · ( y − m + 1) /m ! Proposition 5.2 (Elder-U.) . Let C p = h g , g i , let i , i ≥ be integers and let µ be anelement of K that satisfies ν ( ℘ ( µ )) ≥ i − pi . Let E ( i , i , µ ) = R " g − π i , g g [ µ ]1 − π i . Then E ( i , i , µ ) is an R -Hopf order in K [ C p ] .Proof. See [4, Proposition 3.4]. (cid:3)
Proposition 5.3 (Elder-U.) . Let i , i ≥ be integers and let µ be an element of K thatsatisfies ν ( ℘ ( µ )) ≥ i − pi . Let H ∗ i ,i ,µ = R [ π i ( ξ , − µξ , ) , π i ξ , ] be an R -Hopf order in ( K [ C p ]) ∗ . Then ( H ∗ i ,i ,µ ) ∗ = E ( i , i , µ ) . Proof.
See [4, Theorem 3.6]. (cid:3)
Thus an arbitrary R -Hopf order in K [ C p ] appears as E ( i , i , µ ) = R " g − π i , g g [ µ ]1 − π i for integers i , i ≥ µ ∈ K with ν ( ℘ ( µ )) ≥ i − pi .5.3. The case n = 3 . Let K [ C p ] = K h g , g , g i . Let E ( i , i , µ ) = R " g − π i , g g [ µ ]1 − π i be an R -Hopf order in K [ C p ]. Let i ≥
0, let α, β ∈ K and let E ( i , i , i , α, β, µ ) = R " g − π i , g g [ µ ]1 − π i , g g [ α ]1 ( g g [ µ ]1 ) [ β ] − π i be a “truncated exponential” algebra over R . We want to find conditions so that E ( i , i , i , α, β, µ )is an R -Hopf order in K [ C p ]. We first prove some lemmas. Lemma 5.4.
Assume the conditions ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi , ν ( ℘ ( β )) ≥ i − pi . Then g [ α ]1 ( g g [ µ ]1 ) [ β ] is a unit in E ( i , i , µ ) .Proof. By [4, (2)], ( g [ α ]1 ( g g [ µ ]1 ) [ β ] )( g [ − α ]1 ( g g [ − µ ]1 ) [ − β ] ) = 1 , so it is a matter of checking that g [ α ]1 ( g g [ µ ]1 ) [ β ] and g [ − α ]1 ( g g [ − µ ]1 ) [ − β ] are in E ( i , i , µ ). Weshow that g [ α ]1 ( g g [ µ ]1 ) [ β ] ∈ E ( i , i , µ ). Now,( g g [ µ ]1 ) [ β ] = p − X m =0 (cid:18) βm (cid:19) π mi g g [ µ ]1 − π i ! m . We claim that (cid:0) βm (cid:1) π mi ∈ R for 0 ≤ m ≤ p −
1. If ν ( β ) ≥
0, then clearly this holds.So we assume that ν ( β ) <
0, which yields ν ( (cid:0) βm (cid:1) ) ≥ − mi , 0 ≤ m ≤ p −
1. Hence( g g [ µ ]1 ) [ β ] ∈ E ( i , i , µ ). OPF ORDERS IN K [ C p ] IN CHARACTERISTIC p We next show that g [ α ]1 ∈ E ( i , i , µ ). We have g [ α ]1 = p − X m =0 (cid:18) αm (cid:19) π mi (cid:18) g − π i (cid:19) m . We claim that (cid:0) αm (cid:1) π mi ∈ R for 0 ≤ m ≤ p −
1. If ν ( α ) ≥
0, then clearly this holds.So we assume that ν ( α ) <
0, so that ν ( ℘ ( α )) = pν ( α ). If ν ( ℘ ( α )) = ν ( ℘ ( µ ) β ), then pν ( α ) ≥ ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi , thus ν ( α ) ≥ − i , which yields ν ( (cid:0) αm (cid:1) ) ≥ − mi ,0 ≤ m ≤ p −
1. So we assume that ν ( ℘ ( α )) = pν ( α ) = ν ( ℘ ( µ ) β ). If ν ( β ) ≥
0, then pν ( α ) = ν ( ℘ ( µ ) β ) = ν ( ℘ ( µ )) + ν ( β ) ≥ ν ( ℘ ( µ )) ≥ i − pi , thus ν ( (cid:0) αm (cid:1) ) ≥ − mi , 0 ≤ m ≤ p −
1. If ν ( β ) <
0, then ν ( β ) ≥ i /p − i and so pν ( α ) = ν ( ℘ ( µ )) + ν ( β ) ≥ i − pi + i /p − i = i /p − pi , thus ν ( (cid:0) αm (cid:1) ) ≥ − mi , 0 ≤ m ≤ p − g [ − α ]1 ( g g [ − µ ]1 ) [ − β ] ∈ E ( i , i , µ ). (cid:3) We have the formula from [1, Lemma 2.2]:(6) (1 + x + y + xy ) [ z ] = (1 + x ) [ z ] (1 + y ) [ z ] (1 + ℘ ( z ) Q ( x, y )) ∈ F p [ x, y, z ] / ( x p , y p ) , where Q ( x, y ) = (( x + y + xy ) p − x p − y p − ( xy ) p ) /p ∈ ( x, y ) p ⊂ Z [ x, y ] ,Q ( x, y ) ∈ ( x p , y p ). (See also [4, pp. 9-10].) We give a generalization of this formula. Lemma 5.5. ((1 + x + y + xy ) [ z ] ) [ a ] = ((1 + x ) [ z ] (1 + y ) [ z ] ) [ a ] (1 + ℘ ( z ) aQ ( x, y )) in F p [ x, y, z, a ] / ( x p , y p ) .Proof. From (6) we obtain((1 + x + y + xy ) [ z ] ) [ a ] = ((1 + x ) [ z ] (1 + y ) [ z ] (1 + ℘ ( z ) Q ( x, y ))) [ a ] in F p [ x, y, z, a ] / ( x p , y p ). Put D = (1 + x ) [ z ] (1 + y ) [ z ] , so that ((1 + x ) [ z ] (1 + y ) [ z ] (1 + ℘ ( z ) Q ( x, y ))) [ a ] = ( D (1 + ℘ ( z ) Q ( x, y ))) [ a ] . Now, by (6), ( D (1 + ℘ ( z ) Q ( x, y ))) [ a ] = ((1 + ( D − ℘ ( z ) Q ( x, y ))) [ a ] = (1 + ( D − [ a ] (1 + ℘ ( z ) Q ( x, y )) [ a ] · (1 + ℘ ( a ) Q ( D − , ℘ ( z ) Q ( x, y ))) . Now Q ( D − , ℘ ( z ) Q ( x, y )) = 0 since x p = y p = 0 and Q ( x, y ) = 0. Thus( D (1 + ℘ ( z ) Q ( x, y ))) [ a ] = (1 + ( D − [ a ] (1 + ℘ ( z ) Q ( x, y )) [ a ] = D [ a ] (1 + ℘ ( z ) Q ( x, y )) [ a ] . Moreover, Q ( x, y ) = 0 implies that(1 + ℘ ( z ) Q ( x, y )) [ a ] = 1 + ℘ ( z ) aQ ( x, y ) . Thus ((1 + x + y + xy ) [ z ] ) [ a ] = ((1 + x ) [ z ] (1 + y ) [ z ] ) [ a ] (1 + ℘ ( z ) aQ ( x, y ))in F p [ x, y, z, a ] / ( x p , y p ). (cid:3) Proposition 5.6.
Let i , i , i ≥ . Let µ, α, β ∈ K with ν ( ℘ ( µ )) ≥ i − pi , ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi , ν ( ℘ ( β )) ≥ i − pi , ν ( µ ) ≥ i − i and i ≥ i . Then E = E ( i , i , i , α, β, µ ) is an R -Hopf order in K [ C p ] .Proof. By Lemma 5.4, g [ α ]1 ( g g [ µ ]1 ) [ β ] is a unit in H i ,i ,µ . Thus g ∈ E and so E ⊗ R K = K [ C p ].Moreover, g g [ α ]1 ( g g [ µ ]1 ) [ β ] − π i ! p = 0and so E is an R -algebra which is a free R -submodule of K [ C p ].We next check that E is a coalgebra, with counit and comultiplication induced by thatof K [ C p ]. We have ε ( E ) ⊆ R . In view of [3, Proposition (31.2)], E is a coalgebra if we canshow that ∆( g [ α ]1 ( g g [ µ ]1 ) [ β ] ) ≡ g [ α ]1 ( g g [ µ ]1 ) [ β ] ⊗ g [ α ]1 ( g g [ µ ]1 ) [ β ] (7)modulo π i E ( i , i , µ ) ⊗ E ( i , i , µ ).Let X = ( g − /π i ⊗ , Y = 1 ⊗ ( g − /π i ,A = ( g g [ µ ]1 − /π i ⊗ , B = 1 ⊗ ( g g [ µ ]1 − /π i T = ( g − /π i ⊗ , V = 1 ⊗ ( g − /π i . By Lemma 5.5(((1 + π i X )(1 + π i Y )) [ µ ] ) [ β ] = ((1 + π i X ) [ µ ] (1 + π i Y ) [ µ ] ) [ β ] (1 + ℘ ( µ ) βQ ( π i X, π i Y )) . By (6),((1 + π i X )(1 + π i Y )) [ α ] = (1 + π i X ) [ α ] (1 + π i Y ) [ α ] (1 + ℘ ( α ) Q ( π i X, π i Y )) . Multiplication yields((1 + π i X )(1 + π i Y )) [ α ] (((1 + π i X )(1 + π i Y )) [ µ ] ) [ β ] OPF ORDERS IN K [ C p ] IN CHARACTERISTIC p = (1 + π i X ) [ α ] (1 + π i Y ) [ α ] ((1 + π i X ) [ µ ] (1 + π i Y ) [ µ ] ) [ β ] · (1 + ( ℘ ( α ) + ℘ ( µ ) β ) Q ( π i X, π i Y )).Now the condition ν ( ℘ ( α ) + ℘ ( µ ) β )) ≥ i − pi implies that( ℘ ( α ) + ℘ ( µ ) β ) Q ( π i X, π i Y ) ∈ π i R [ X, Y, A, B ] . Thus((1 + π i X )(1 + π i Y )) [ α ] (((1 + π i X )(1 + π i Y )) [ µ ] ) [ β ] ≡ (1 + π i X ) [ α ] (1 + π i Y ) [ α ] ((1 + π i X ) [ µ ] (1 + π i Y ) [ µ ] ) [ β ] modulo π i R [ X, Y, A, B ].Moreover, ν ( µ ) ≥ i − i and i ≥ i imply that g − ∈ π i E ( i , i , µ ) , hence ∆( g −
1) = π i T + π i V + π i T V ∈ π i R [ X, Y, A, B ] . Consequently,((1 + π i X )(1 + π i Y )) [ α ] ((1 + π i T )(1 + π i V )((1 + π i X )(1 + π i Y )) [ µ ] ) [ β ] ≡ (1 + π i X ) [ α ] (1 + π i Y ) [ α ] ((1 + π i T )(1 + π i V )(1 + π i X ) [ µ ] (1 + π i Y ) [ µ ] ) [ β ] modulo π i R [ X, Y, A, B ].By formula (6), the condition ν ( ℘ ( β )) ≥ i − pi implies ((1 + π i A )(1 + π i B )) [ β ] ≡ (1 + π i A ) [ β ] (1 + π i B ) [ β ] modulo π i R [ X, Y, A, B ].So if ν ( ℘ ( α ) + ℘ ( µ ) β )) ≥ i − pi , ν ( ℘ ( β )) ≥ i − pi and ν ( µ ) ≥ i − i hold, then((1 + π i X )(1 + π i Y )) [ α ] ((1 + π i T )(1 + π i V )((1 + π i X )(1 + π i Y )) [ µ ] ) [ β ] ≡ (1 + π i X ) [ α ] (1 + π i Y ) [ α ] ((1 + π i T )(1 + π i X ) [ µ ] ) [ β ] ((1 + π i V )(1 + π i Y ) [ µ ] ) [ β ] modulo π i R [ X, Y, A, B ], which is condition (7).Finally, we show that S ( E ) ⊆ E . But this follows from the coalgebra condition andobservation that the coinverse map is m ( I ⊗ m )( I ⊗ I ⊗ m ) · · · ( I p − ⊗ m )( I p − ⊗ ∆) · · · ( I ⊗ I ⊗ ∆)( I ⊗ ∆)∆ , where m : E ⊗ E → E denotes multiplication in E . (cid:3) We have the following: Suppose i , i , i are non-negative integers. Suppose µ, α, β ∈ K satisfy the conditions ν ( ℘ ( µ )) ≥ i − pi , ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi , ν ( ℘ ( β )) ≥ i − pi ,and ν ( µ ) ≥ i − i . Then by Proposition 4.1 H ∗ i ,i ,i ,µ,α,β = R [ π i ( ξ , , − µξ , , − αξ , , ) , π i ( ξ , , − βξ , , ) , π i ξ , , ] is an R -Hopf order in ( K [ C p ]) ∗ and by Proposition 5.6 E ( i , i , i , µ, α, β ) = R " g − π i , g g [ µ ]1 − π i , g g [ α ]1 ( g g [ µ ]1 ) [ β ] − π i is an R -Hopf order in K [ C p ].We claim that these Hopf orders are duals to each other. Proposition 5.7.
With the conditions as above, ( H ∗ i ,i ,i ,µ,α,β ) ∗ = E ( i , i , i , µ, α, β ) . Proof.
To prove the result, we show that H ∗ i ,i ,i ,µ,α,β ⊆ E ( i , i , i , µ, α, β ) ∗ and disc( H ∗ i ,i ,i ,µ,α,β ) = disc( E ( i , i , i , µ, α, β ) ∗ ) . For the containment: { ρ a,b,c } , 0 ≤ a, b, c ≤ p −
1, where ρ a,b,c = ( g − a ( g g [ µ ]1 − b ( g g [ α ]1 ( g g [ µ ]1 ) [ β ] − c /π ai + bi + ci is an R -basis for E ( i , i , i , µ, α, β ). Let x = g − y = g − z = g −
1. Then π ai + bi + ci ρ a,b,c ≡ x a ( µx + y ) b ( α + βµ ) x + βy + z ) c mod ( x, y, z ) . We have h ξ , , , x a ( µx + y ) b ( α + βµ ) x + βy + z ) c i = a = 1 , b = 0 , c = 0 µ if a = 0 , b = 1 , c = 0 α + βµ if a = 0 , b = 0 , c = 10 otherwise. h ξ , , , x a ( µx + y ) b ( α + βµ ) x + βy + z ) c i = a = 1 , b = 0 , c = 01 if a = 0 , b = 1 , c = 0 β if a = 0 , b = 0 , c = 10 otherwise. h ξ , , , x a ( µx + y ) b ( α + βµ ) x + βy + z ) c i = a = 1 , b = 0 , c = 00 if a = 0 , b = 1 , c = 01 if a = 0 , b = 0 , c = 10 otherwise. Thus h π i ( ξ , , − µξ , , − αξ , , ) , ρ , , i = 1 , h π i ( ξ , , − µξ , , − αξ , , ) , ρ , , i = 0 , h π i ( ξ , , − µξ , , − αξ , , ) , ρ , , i = 0 , OPF ORDERS IN K [ C p ] IN CHARACTERISTIC p and so h π i ( ξ , , − µξ , , − αξ , , ) , ρ a,b,c i = δ a, δ b, δ c, . Moreover, h π i ( ξ , , − βξ , , ) , ρ a,b,c i = δ a, δ b, δ c, and h π i ξ , , , ρ a,b,c i = δ a, δ b, δ c, . Thus H ∗ i ,i ,i ,µ,α,β ⊆ E ( i , i , i , µ, α, β ) ∗ . Regarding the discriminant statement, there is a short exact sequence of R -Hopf orders R → H ∗ i ,i ,β → H ∗ i ,i ,i ,µ,α,β → H ∗ i → R. By [4, Proposition 2.2], disc( H ∗ i ) = ( π p ( p − i ), by [4, Proposition 3.2], disc( H ∗ i ,i ,β ) =( π p ( p − i + i ) ) and by [3, (22.17) Corollary]disc( H ∗ i ,i ,i ,µ,α,β ) = disc( H ∗ i ,i ,β ) p disc( H ∗ i ) p = ( π p ( p − i + i + i ) ) . There is a short exact sequence of Hopf orders R → E ( i , i , µ ) → E ( i , i , i , µ, α, β ) → E ( i ) → R, which dualizes as R → H ∗ i → E ( i , i , i , µ, α, β ) ∗ → H ∗ i ,i ,µ → R. By [4, Proposition 2.2], disc( H ∗ i ) = ( π p ( p − i ), by [4, Proposition 3.2], disc( H ∗ i ,i ,µ ) =( π p ( p − i + i ) ) and by [3, (22.17) Corollary]disc( E ( i , i , i , µ, α, β ) ∗ ) = disc( H ∗ i ) p disc( H ∗ i ,i ,µ ) p = ( π p ( p − i + i + i ) ) . Thus disc( H ∗ i ,i ,i ,µ,α,β ) = disc( E ( i , i , i , µ, α, β ) ∗ )which completes the proof. (cid:3) We summarize as follows. Let H be an R -Hopf order in K [ C p ]. Then H induces a shortexact sequence of R -Hopf orders R → E ( i , i , µ ) → H → E ( i ) → R where E ( i , i , µ ) is an R -Hopf order in K [ C p ] and E ( i ) is an R -Hopf order in K [ C p ].Necessarily i , i , i ≥ ν ( ℘ ( µ )) ≥ i − pi . As in Remark 4.5 we assume that ν ( µ ) ≥ i − i and i ≥ i . Proposition 5.8.
With H as above we have H = R " g − π i , g g [ µ ]1 − π i , g g [ α ]1 ( g g [ µ ]1 ) [ β ] − π i where α, β are elements of K satisfying ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi and ν ( ℘ ( β )) ≥ i − pi .Proof. Given H , its dual J = H ∗ is an R -Hopf order in ( K [ C p ]) ∗ . By Proposition 4.4, J ∼ = H ∗ i ,i ,i ,µ,α,β where α, β ∈ K satisfy ν ( ℘ ( α ) + ℘ ( µ ) β ) ≥ i − pi and ν ( ℘ ( β )) ≥ i − pi .By Proposition 5.7, H = ( H ∗ i ,i ,i ,µ,α,β ) ∗ = E ( i , i , i , µ, α, β ). (cid:3) References [1] N. Byott, G. G. Elder, New ramification breaks and additive Galois structure, J. Th´eor. NombresBordeaux (1), (2005), 87-107.[2] N. Byott, G. G. Elder, Sufficient conditions for large Galois scaffolds, J. Num. Theory , , 2018,95-130.[3] L. N. Childs, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory, AmericanMathematical Society, Mathematical Surveys and Monographs , 2000.[4] G. G. Elder, R. G. Underwood, Finite group scheme extensions, and Hopf orders in KC p over a char-acteristic p discrete valuation ring, New York J. Math. , , 2017, 11-39.[5] C. Greither, Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valu-ation ring, Math. Z. , , (1992), 37-67.[6] P.J. Hilton, U. Stammbach, A Course in Homological Algebra, Springer-Verlag, New York, 1971.[7] A. J. de Jong, Finite locally free group schemes in characteristic p and Dieudonn´e modules, Invent.Math. (1), (1993), 89-137, MR 1235021 (94j:14043).[8] A. Koch, Primitively generated Hopf orders in characteristic p , Comm. Alg. , (6), 2017, 2673-2689.[9] J. Tate, F. Oort, Group schemes of prime order, Ann. Sci. Ec. Norm. Sup. , , (1970), 1-21. Department of Mathematics and Department of Computer Science, Auburn Universityat Montgomery, Montgomery, AL, 36124 USA
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