Bounds on the number of ideals in finite commutative nilpotent F p -algebras
aa r X i v : . [ m a t h . R A ] J un BOUNDS ON THE NUMBER OF IDEALS IN FINITECOMMUTATIVE NILPOTENT F p -ALGEBRAS LINDSAY N. CHILDS AND CORNELIUS GREITHER
Abstract.
Let A be a finite commutative nilpotent F p -algebrastructure on G , an elementary abelian group of order p n . If K/k isa Galois extension of fields with Galois group G and A p = 0, thencorresponding to A is an H -Hopf Galois structure on K/k of type G . For that Hopf Galois structure we may study the image of theGalois correspondence from k -subHopf algebras of H to subfields of K containing k by utilizing the fact that the intermediate subfieldscorrespond to the F p -subspaces of A , while the subHopf algebrasof H correspond to the ideals of A . We obtain upper and lowerbounds on the proportion of subspaces of A that are ideals of A ,and test the bounds on some examples. Introduction
The motivation for this work is to understand the Galois correspon-dence for certain Hopf Galois structures on field extensions.Let
K/k be a Galois extension of fields with Galois group G . Thenthe Galois correspondence sending subgroups G ′ of G to subfields K G ′ of K containing k is, by the Fundamental Theorem of Galois Theory,a bijective correspondence from subgroups of G onto the intermediatefields between k and K .In 1969 S. Chase and M. Sweedler [CS69] defined the concept ofa Hopf Galois extension of fields for a field extension K/k and H a k -Hopf algebra acting on K as an H -module algebra. They proveda weak version of the FTGT, namely, that there is an injective Ga-lois correspondence from k -subHopf algebras H ′ of H to intermediatefields, given by H ′ K H ′ , the subfield of elements fixed under theaction of H ′ . But surjectivity was not obtained. Greither and Pareigis[GP87] defined a class of non-classical Hopf Galois structures, the ”al-most classical” structures, for which surjectivity holds, but also gave anexample where it fails. Recent work of Crespo, Rio and Vela ([CRV15] Date : September 24, 2018. and especially [CRV16]) studied the image of the Galois correspondencefor Hopf Galois structures on separable extensions
K/k with normalclosure ˜ K and found numerous examples where surjectivity fails. Innearly all of the cases examined in [CRV16] the Galois group of ˜ K/K is non-abelian.In this paper we seek to quantify the failure of the FTGT for HopfGalois structures of the following type.Let
K/k be a Galois extension of fields with Galois group G , an el-ementary abelian p -group of order p n . Suppose H is a k -Hopf algebraof type G (that means, K ⊗ k H ∼ = KG ), and K/k is a H -Hopf Galoisextension. As shown in [Ch15], [Ch16], [Ch17], building on work of[CDVS06] and [FCC12], every H -Hopf Galois structure of type G ona Galois extension of fields K/k with Galois group G , an elementaryabelian p -group, arises from a commutative nilpotent F p -algebra struc-ture A on the additive group G with A p = 0. In [Ch17], it was shownthat the sub- K -Hopf algebras of H correspond to ideals of A . For aGalois extension K/k whose Galois group is an elementary abelian p -group (or equivalently, an F p -vector space), the classical FTGT givesa bijection between F p -subspaces of G and intermediate fields. So let i ( A ) denote the number of ideals of A , and s ( A ) the number of F p -subspaces of A . Then the proportion of intermediate fields k ⊆ E ⊆ K that are in the image of the Galois correspondence for a H -Hopf Galoisstructure on K/k arising from A is equal to i ( A ) /s ( A ).As observed in [Ch17], that comparison implies immediately that if A = 0, then there are subspaces of A that are not ideals, and hencethe Galois correspondence cannot be surjective.Let e be the unique integer such that A e = 0 and A e +1 = 0; weassume throughout that e > A is not zero) and e < p .To quantify the failure of surjectivity of the FTGT for a Hopf Galoisstructure corresponding to A , we obtain in section 2 of this paper ageneral upper bound, depending only on e , on the ratio i ( A ) /s ( A ).The upper bound implies, for example, that for e ≥ p ≥ i ( A ) /s ( A ) < . A ,we obtain in section 3 a lower bound on i ( A ).The upper bound is based on a lower bound on the fibers of the“ideal generated by” function G from subspaces of A to ideals of A . Inthe final section we examine that lower bound on fibers of G , and theinequalities of sections 2 and 3, for some examples. OUNDS ON IDEALS 3
Let s ( n ) denote the number of subspaces of an F p -vector space ofdimension n . Then s ( n ) is a sum of Gaussian binomial coefficients, alsocalled q -binomial coefficients (where q = p ). The first section of thepaper describes properties of these coefficients and obtains inequalitiesrelating s ( m ) and s ( n ) for m < n .Throughout the paper, we assume that A has dimension n and that A p = 0. Recall that e is the largest number so that A e = 0 (so A e +1 =0). All vector spaces are over F p .Our thanks go to the University of Nebraska at Omaha and to GriffElder for their hospitality and support.1. Gaussian binomial coefficients
To compare the number of ideals of a commutative nilpotent F p -algebra A with the number of subspaces of A , we need to collect someinformation concerning the number of subspaces of dimension k of an F p -vector space of dimension n . So we begin with Gaussian binomialcoefficients.The Gaussian binomial coefficient, or q -binomial coefficient (here q = p ), is defined as (cid:20) nk (cid:21) = ( p n − p n − p ) · · · ( p n − p k − )( p k − p k − p ) · · · ( p k − p k − )= ( p n − p n − − · · · ( p n − ( k − − p − p − · · · ( p k − . It counts the number of k -dimensional subspaces of F np . So (cid:20) nk (cid:21) = (cid:20) nn − k (cid:21) for all k, (cid:2) n (cid:3) = (cid:2) nn (cid:3) = 1, and (cid:2) nk (cid:3) = 0 for k > n . Then s ( n ) = n X k =0 (cid:20) nk (cid:21) is the total number of subspaces of F np . Note that it suffices to replacethe factors ( p n − p r ) / ( p k − p r ) by p n /p k in order to see that (cid:20) nk (cid:21) ≥ p k ( n − k ) , and that (cid:2) nk (cid:3) has order of magnitude p k ( n − k ) for large p . LINDSAY N. CHILDS AND CORNELIUS GREITHER (In fact, the rational function (cid:20) nk (cid:21) x = ( x n − x n − x ) · · · ( x n − x k − )( x k − x k − x ) · · · ( x k − x k − )is a polynomial of degree ( n − k ) k in Z [ x ]. For let b ( x ) , a ( x ) be thenumerator and denominator of (cid:2) nk (cid:3) x . Both are monic polynomials in Z [ x ]. Dividing b ( x ) by a ( x ) in Q [ x ] gives b ( x ) = a ( x ) q ( x ) + r ( x ) , where deg( r ( x )) < deg( a ( x )). Since a ( x ) is monic, q ( x ) and r ( x ) arein Z [ x ]. Now b ( p ) /a ( p ) = (cid:2) nk (cid:3) p is a positive integer for every prime p ,so the rational function r ( p ) /a ( p ) is also an integer for every prime p .But lim p →∞ r ( p ) a ( p ) = 0 . So r ( p ) = 0 for all primes greater than some fixed bound, and hence r ( x ) = 0. So b ( x ) /a ( x ) = q ( x ) is in Z [ x ].)The Gaussian binomial coefficients satisfy two recursive formulas,analogous to that satisfied by the usual binomial coefficients: (cid:20) nk (cid:21) = (cid:20) n − k − (cid:21) + p k (cid:20) n − k (cid:21) = (cid:20) n − k (cid:21) + p n − k (cid:20) n − k − (cid:21) . Using properties of the Gaussian binomial coefficients, we will nowobtain some inequalities relating the number of subspaces of F p -vectorspaces of dimensions n, n − n − n .Let δ ( n ) = ⌊ n ⌋ = ( n / n is even( n − / n is odd . Lemma 1.1. a) For all n ≥ we have s ( n ) ≥ p n − s ( n − . b) If n > is even, then s ( n ) ≥ p n/ s ( n − . c) If n > is odd, then s ( n ) ≥ p ( n − / s ( n − . d) For n ≥ m ≥ arbitrary, we have s ( n ) ≥ p δ ( n ) − δ ( m ) s ( m ) . Thefactor / may be omitted if m and n have the same parity orif n is even. OUNDS ON IDEALS 5
Proof. a) Using the two recursion formulas for (cid:2) nd (cid:3) in turn we find: (cid:20) nd (cid:21) = (cid:20) n − d − (cid:21) + p d (cid:20) n − d (cid:21) = (cid:20) n − d − (cid:21) + p d ( p n − − d (cid:20) n − d − (cid:21) + (cid:20) n − d (cid:21) ) ≥ p n − (cid:20) n − d − (cid:21) . Summing these for d = 1 , . . . , n − n = 2 k . We may calculate as follows: s ( n ) ≥ (cid:20) n (cid:21) + . . . + (cid:20) nk (cid:21) = ( p n − (cid:20) n − (cid:21) + (cid:20) n − (cid:21) ) + ( p n − (cid:20) n − (cid:21) + (cid:20) n − (cid:21) ) + . . .. . . + ( p k (cid:20) n − k − (cid:21) + (cid:20) n − k (cid:21) ) ≥ p k (cid:20) n − (cid:21) + p k (cid:20) n − (cid:21) + . . . p k (cid:20) n − k − (cid:21) = p k s ( n − / . c) Let n = 2 k + 1. Using one recursive formula, then the other, weget: (cid:20) n (cid:21) ≥ p n − (cid:20) n − (cid:21) ≥ p k (cid:20) n − (cid:21) ; (cid:20) n (cid:21) ≥ p n − (cid:20) n − (cid:21) ≥ p k (cid:20) n − (cid:21) ;... (cid:20) nk − (cid:21) ≥ p n − ( k − (cid:20) n − k − (cid:21) ≥ p k (cid:20) n − k − (cid:21) ; LINDSAY N. CHILDS AND CORNELIUS GREITHER (now we switch to the other recursive formula) (cid:20) nk (cid:21) ≥ p k (cid:20) n − k (cid:21) ; (cid:20) nk + 1 (cid:21) ≥ p k +1 (cid:20) n − k + 1 (cid:21) ; (cid:20) nk + 2 (cid:21) ≥ p k +2 (cid:20) n − k + 2 (cid:21) ≥ p k (cid:20) n − k + 2 (cid:21) ;... (cid:20) nn − (cid:21) ≥ p n − (cid:20) n − n − (cid:21) ≥ p k (cid:20) n − n − (cid:21) . Now observe that (cid:20) n − k + 1 (cid:21) = (cid:20) kk + 1 (cid:21) = (cid:20) kk − (cid:21) . Therefore p k +1 (cid:20) n − k + 1 (cid:21) = p k (2 (cid:20) n − k + 1 (cid:21) ) = p k ( (cid:20) n − k + 1 (cid:21) + (cid:20) n − k − (cid:21) ) . Thus s ( n ) is at least as large as the sum of the left sides of the inequal-ities, which is at least the sum of the right sides of the inequalities, andin view of the last observation, the sum of the right sides is at least p k s ( n − δ ( n ) − δ ( n −
2) = n −
1, so by a), s ( n ) ≥ p n − s ( n −
2) = p δ ( n ) − δ ( n − s ( n − . Iterating this shows that if n > m and n ≡ m (mod 2) then s ( n ) ≥ p δ ( n ) − δ ( m ) s ( m ) . If n is even and m is odd, then n/ δ ( n ) − δ ( n −
1) and by b) wefind s ( n ) ≥ p n s ( n −
1) = 12 p δ ( n ) − δ ( n − s ( n − , so s ( n ) ≥ p δ ( n ) − δ ( m ) s ( m ) . If n is odd and m is even, then ( n − / δ ( n ) − δ ( n − s ( n ) ≥ p n − s ( n −
1) = p δ ( n ) − δ ( n − s ( n − , hence s ( n ) ≥ p δ ( n ) − δ ( m ) s ( m ) . (cid:3) OUNDS ON IDEALS 7 An upper bound on the number of ideals of A In this section we obtain a general upper bound for the ratio i ( A ) /s ( A )of the number of ideals of A to the number of subspaces of A , for A anarbitrary commutative nilpotent F p -algebra of dimension n . To do so,we consider the function G from subspaces of A to ideals of A whichassociates to each subspace U the ideal G ( U ) = U + AU generated by U , and we establish a lower bound on the cardinality of the fiber ofeach ideal under this map (which is obviously surjective). But first, weneed to count subspaces with certain properties.Recall that δ ( t ) = ⌊ t / ⌋ . All vector spaces are over F p , and the num-ber of k -dimensional subspaces of W , an F p -vector space of dimension d , is (cid:2) dk (cid:3) .We show: Proposition 2.1.
Let dim( W ) = d and let W be a fixed subspace of W of dimension r . For k ≤ d − r , the number s ( d, r ; k ) of k -dimensionalsubspaces U of W with U ∩ W = (0) is equal to p rk times the numberof k -dimensional subspaces of W/W : s ( d, r ; k ) = p rk (cid:20) d − rk (cid:21) . Proof.
Let V be a complementary subspace to W , so that V ⊕ W = W .For k ≤ d − r , let U be a k -dimensional subspace of V , with basis( z , . . . , z k ). For each choice a = ( a , . . . , a k ) of elements of W , thesubspace U a of W generated by ( z + a , . . . , z k + a k ) is k -dimensionaland has trivial intersection with W . For suppose c ( z + a ) + . . . c k ( z k + a k ) = a in W for some c , . . . , c k in F p . Then, since V ⊕ W is a direct sum of F p -vector spaces, c z + . . . c k z k = 0 . Since z , . . . z k are linearly independent, c , . . . c k = 0, hence a = 0.The same argument with a = 0 shows that ( z + a , . . . , z k + a k ) is alinearly independent set.Finally, each choice of elements a = ( a , . . . , a k ) of W gives a differ-ent subspace U a of A . For suppose z i + b i is in the space U a . Then z i + b i = c ( z + a ) + . . . + c i ( z i + a i ) + . . . + c k ( z k + a k ) . So 0 = c ( z + a ) + . . . + (( c i − z i + c i a i − b i ) + . . . + c k ( z k + a k ) . LINDSAY N. CHILDS AND CORNELIUS GREITHER
But then 0 = c z + . . . + ( c i − z i + . . . + c k z k . So c i = 1, all other c j = 0, and the equation reduces to a i − b i = 0 . Thus for each k -dimensional subspace U of W , we obtain p rk k -dimensionalsubspaces U a of W with W ∩ W = (0). (cid:3) Corollary 2.2.
Let W be a t -dimensional space and W ⊂ W a sub-space of codimension 1. Then the number of subspaces of W not con-tained in W is at least p δ ( t ) .Proof. First we remark that via a duality argument, the number of sub-spaces of dimension k not contained in a fixed subspace of codimension1 is the same as the number of subspaces of dimension t − k intersectinga fixed subspace of dimension 1 trivially. Hence the preceding proposi-tion is applicable; summing over all possible dimensions of U , we findthat the number of subspaces U ⊂ W not contained in W is s ( t, t − X k =0 s ( t, k )= t − X k =0 p k (cid:20) t − k (cid:21) ≥ t − X k =0 p k p k ( t − − k ) = t − X k =0 p k ( t − k ) ≥ p ⌊ t ⌋ = p δ ( t ) . (cid:3) Recall that G is the map from subspaces of A to ideals of A definedby G ( V ) = V + AV.
To simplify notation, we write G ( S ) instead of G ( h S i F p ) for any subset S of A . To get a sense of the relationship between the number ofsubspaces of A and the number of ideals of A , we will count the numberof elements in the fibers of G . OUNDS ON IDEALS 9
Assume e > A e +1 = 0. (The zero algebra A = 0can be safely excluded from our study.) Consider the chain N ⊂ N ⊂ . . . ⊂ N e = A of annihilator ideals defined by N k := Ann k ( A ) = { a ∈ A | x x · · · x k a = 0 for all x , . . . x k in A } . Let dim F p ( N k ) = d k . Then the sequence ( d k ) k is obviously increasing,and a little argument shows that 0 < d < d < . . . < d e = n .The strategy for bounding the number of ideals of A begins with thefollowing idea. Let J t be the set of ideals J of A contained in N t butnot contained in N t − . Then, since N t is an ideal of A for all t , we have X J ∈J t | G − ( J ) | = s ( N t ) − s ( N t − ) . The next lemma will help us find a lower bound on | G − ( J ) | . Lemma 2.3.
Let W = G ( { x } ) = F p x + Ax , W = Ax as above. Let U be a subspace of W , not contained in W . Then G ( U ) = W .Proof. Let y be in U , y not in W . Then G ( { y } ) ⊆ G ( U ). Aftermultiplying y by a non-zero element of F p , we can assume that y = x − ax for some a in A . Then y + ay + a y + . . . + a e − y = x is in G ( { y } ). So W = G ( { x } ) ⊆ G ( { y } ) ⊆ G ( U ) ⊆ W. (cid:3) Let J be an ideal of A of F p -dimension d , let s ( J ) (or s ( d )) be thenumber of subspaces of J , and let i ( J ) be the number of ideals of A thatare contained in J . Lemma 2.3 enables us to prove a result relatingthe number of subspaces and the number of ideals contained in theannihilator ideal N t in A for each t . Proposition 2.4.
For each t with ≤ t ≤ e , consider the ideal map G restricted to the set of subspaces V of N t that are not contained in N t − . For each x in N t \ N t − , let q ( x ) = dim( G ( x )) , and let q t =min x ∈ N t \ N t − q ( x ) . Then for all ideals J in J t , | G − ( J ) | ≥ p δ ( q t ) . Hence p δ ( q t ) (cid:0) i ( N t ) − i ( N t − ) (cid:1) ≤ s ( N t ) − s ( N t − ) . Proof.
Let J be an ideal contained in N t , not contained in N t − . Let x be in J , x not in N t − . Let W = Ax and W = G ( { x } ) = F p x + Ax .Then W has dimension at least q t , and W has codimension 1 in W .Let Y be a complement of W in J . Then for every subspace U of W not contained in W , we have G ( U ) = W and thus G ( U + Y ) = J .Whenever U and U ′ are distinct subspaces of W not contained in W ,we have U + Y = U ′ + Y . So the number of preimages of J = G ( { x } + Y )is at least equal to the number of subspaces of W that are not containedin W . Since dim( W ) ≥ q t , that number of subspaces is ≥ p δ ( q t ) byCorollary 2.2. (cid:3) Dividing both sides of the t -th inequality of Proposition 2.4 by p δ ( q t ) and summing them over all t yields an upper bound for the number ofideals of A : Corollary 2.5. i ( A ) ≤ e − X t =1 ( p − δ ( q t ) − p δ ( q t +1 ) ) s ( N t ) + p − δ ( q e ) s ( N e ) . Omitting the negative terms and applying Lemma 1.1 d) yields thefollowing upper bound on i ( A ) in terms of s ( A ) (recall d t = dim N t ): Corollary 2.6. i ( A ) ≤ (cid:0) e − X t =1 p − δ ( q t )+ δ ( d t ) − δ ( d e ) + p − δ ( q e ) (cid:1) s ( A ) . To make it easier to apply this inequality for general A , we show thefollowing simple lower bound on the quantity q t . (Recall it was definedby q t = min x ∈ N t \ N t − q ( x ) with q ( x ) = dim( G ( x )).) Proposition 2.7.
For all t > we have q t ≥ t .Proof. This is clear for t = 1.For t > x be in N t and not in N t − . Let u , u , . . . , u t − in A sothat u u · · · u t − x = 0. Then for each k , x k = u k · · · u t − x is in N k andnot in N k − . So x , . . . , x t − , x are linearly independent in A . Thus G ( x ) = F p x + Ax has dimension at least t . (cid:3) In the next theorem we will use this lower bound on q t to get ageneral, fairly elegant upper bound on i ( A ) /s ( A ) that only depends on e , the length of the annihilator chain in A . However, in some of theexamples treated below it will be worthwhile to have a closer look at OUNDS ON IDEALS 11 q t ; we will find it to be considerably larger than t , which will enable usto sharpen the upper bound.The general bound goes as follows. Theorem 2.8.
With the above hypotheses on A and e we have i ( A ) s ( A ) ≤ e − p δ ( e ) . Proof.
In the inequality of Corollary 2.6, replace q t by t and observethat since 1 < d < d < . . . < d e , one has δ ( d e ) − δ ( d t ) ≥ δ ( e ) − δ ( t ). Ifwe insert this into the inequality, the terms p ± δ ( t ) cancel and we obtain i ( A ) ≤ e − X t =1 p − δ ( e ) s ( N e ) + p − δ ( e ) s ( N e ) = (2 e − p − δ ( e ) s ( A ) . (cid:3) For e = 2 , i ( A ) ≤ p s ( A ) for e = 2; i ( A ) ≤ p s ( A ) for e = 3 . We can improve these bounds by some constant factors, (almost) with-out imposing further conditions on the algebra A . Recall that n =dim( A ). Proposition 2.9.
For e = 2 , we have i ( A ) ≤ p s ( A ) whenever p ≥ and n ≥ . For e = 3 , we have i ( A ) ≤ p s ( A ) whenever p ≥ , n ≥ .Proof. Case e = 2: From Corollary 2.5 with δ ( q t ) replaced by δ ( t ), wehave i ( A ) ≤ (cid:0) − p (cid:1) s ( N ) + 1 p s ( A ) . To get the claimed inequality it suffices to assume that dim N = n − (cid:0) − p (cid:1) s ( n − ≤ p s ( n ) , or ( p − s ( n − ≤ s ( n ). Using Lemma 1.1b) for n even it suffices toshow that p n/ > p − , which holds for p ≥ , n ≥
4, while for n odd, it suffices by Lemma1.1c) to show that p n/ > p − , which holds for p ≥ , n ≥ e = 3. From Corollary 2.5 we have i ( A ) ≤ (cid:0) − p (cid:1) s ( N ) + (cid:0) p − p (cid:1) s ( N ) + 1 p s ( A ) . Since s ( A ) = s ( n ), the right side is maximized when s ( A ) = s ( n ) , s ( N ) = s ( n − , s ( N ) = s ( n − ≤ p s ( A ), itsuffices to show that (cid:0) − p (cid:1) s ( n −
2) + (cid:0) p − p (cid:1) s ( n − ≤ p s ( n ) . Using Lemma 1.1b) for n even, we are reduced to showing that (cid:0) p − p (cid:1) p n/ + (cid:0) − p (cid:1) p n − ≤ p , which holds for p ≥ , n ≥
4. Using Lemma 1.1c) for n odd, we see itsuffices to show that (cid:0) − p (cid:1) p n − + (cid:0) p − p (cid:1) p ( n − / ≤ p , which holds for n ≥ p ≥ (cid:3) The bounds of Theorem 2.8 and Proposition 2.9 imply:
Corollary 2.10.
Suppose
K/k is a Galois extension with elementaryabelian p -group G and is also a H -Hopf Galois extension where H arises from a commutative nilpotent F p -algebra structure A on the ad-ditive group G , where A e = 0 , A e +1 = 0 and e < p . Then i ( A ) /s ( A ) isthe proportion of intermediate fields that are in the image of the Galoiscorrespondence from sub-Hopf algebras of H , and i ( A ) /s ( A ) < . for • e = 2 , p ≥ , • e = 3 , p ≥ , • e = 4 , p ≥ , • all e, p with ≤ e < p . OUNDS ON IDEALS 13 A lower bound on the number of ideals
We now obtain a lower bound on the number of ideals of A , byexhibiting a collection of ideals in A and estimating its size. Recallthat A e = 0 = A e +1 and that we defined N r = Ann r ( A ) = { a ∈ A | x x · · · x r a = 0 for all x , . . . , x r in A } . Then N r is an ideal of A , and(0) ⊂ N ⊂ N ⊂ . . . ⊂ N e = A, all inclusions being proper.We already defined d r = dim( N r ); let us put t r = dim F p ( N r /N r − ).For each r = 1 , . . . , e , let W r be a subspace of A so that N r = W r ⊕ N r − . (In particular, W = N .) Then t r = dim( W r ), A = W ⊕ W ⊕ . . . ⊕ W e and t + t + . . . + t e = n. Proposition 3.1. i ( A ) ≥ λ ( A ) := s ( t ) + ( s ( t ) −
1) + . . . + ( s ( t e ) − . Proof.
For each r , 1 ≤ r ≤ e , and each non-zero subspace V r of W r ,let J = N r − + V r . Then J is an ideal of A . Indeed, we have AV r ⊂ AN r ⊂ N r − , and therefore AJ ⊂ AN r − + N r − ⊂ N r − ⊂ J .The formula λ ( A ) of the proposition simply counts the number ofideals J just described. (cid:3) Since for any m , s ( m ) = m X k =0 (cid:20) mk (cid:21) and (cid:2) mk (cid:3) ≥ p ( m − k ) k , we can let t M = max t k and get a rough lowerbound for the number of ideals in A : i ( A ) ≥ p δ ( t M ) . Some classes of examples
To see how sharp the bounds on ideals are that we obtained in thelast two sections, we look at some explicit classes of algebras.
Example 4.1.
First, consider the “uniserial” e -dimensional algebra A generated by x with x e +1 = 0. In this case, for every element u in N t \ N t − , the dimension q ( x ) = dim( G ( { x } ) is equal to t . We then seethat the general upper bound i ( A ) ≤ (2 e − p − δ ( e ) s ( A )is in fact close to the true number i ( A ) = e + 1 for large p , since s ( A )is a polynomial in p of degree δ ( e ).The lower bound λ ( A ) in this simple class of examples is e + 1. Example 4.2.
Let A be a “binomial” nilpotent algebra: A = h x , x , . . . x e i with x k = 0 for all k . Then dim( A ) = 2 e −
1, Ann( A ) = ( x x · · · x e )and dim( N t /N t − ) = (cid:0) et − (cid:1) .Theorem 2.6 tells us that the ratio of ideals to subspaces for A isbounded as follows: i ( A ) s ( A ) ≤ e − p δ ( e ) = 2 e − p ⌊ e ⌋ . But that inequality arose from minorizing q t = min x ∈ N t \ N t − dim( G ( x ))by t throughout. In this class of examples we can do better, having acloser look at q t . Proposition 4.3.
Let A be the binomial algebra of dimension e − .Then for every non-zero u in N t \ N t − we have dim( G ( u )) ≥ t − . Proof.
For any given u in N t \ N t − , pick a monomial summand y of u in N t \ N t − . Renumber the variables of A so that y = x x · · · x e − t +1 . Then we introduce an ordering on the set of all nonzero monomials of A so that any monomial of N k − comes after any monomial of N k \ N k − forall k , and the monomials within N k \ N k − are ordered lexicographically.Strictly speaking, this is a total ordering on the set of all monomialsup to multiplication with a nonzero scalar in F p .Call a family of monomials admissible if no two of them are equalup to a nonzero scalar. Every nonzero z ∈ A has a unique “leading”monomial m ( z ), according to the ordering. The following is easy to see: OUNDS ON IDEALS 15 if ( z i ) i ∈ I is a family of elements of A , such that the family of leadingmonomials ( m ( z i )) i is admissible, then ( z i ) i is F p -linearly independent.If w is any monomial, we have m ( zw ) = m ( z ) w .Now consider the family F of monomials that consist only of factors x e − t +2 , . . . , x e ; this family has 2 t − entries, and is of course admissible.If we multiply every element of this family by u , the leading terms justget multiplied by the monomial y , so they again are an admissible fam-ily. Hence the entries of the family uF are again linearly independent,which shows that the ideal G ( u ) generated by u has dimension at least2 t − . (cid:3) We illustrate how working with q t ≥ t − instead of the crude lowerbound q t ≥ t affects the upper bound on the ratio i ( A ) /s ( A ) of Theo-rem 2.8 for a binomial algebra.Consider the binomial algebra A = h x , x , x , x i with x i = 0. Thendim( N ) = 1 , dim ( N ) = 5 , dim( N ) = 11 , dim( N ) = 2 − . (Note N = A .) The general inequality 2.8 gives i ( A ) s ( A ) ≤ p . Let us start afresh. From Corollary 2.5 we have i ( A ) ≤ X t =1 ( p − δ ( q t ) − p δ ( q t +1 ) ) s ( N t ) + p − δ (4) s ( A ) . Omitting the negative terms gives i ( A ) ≤ p δ ( q ) s ( N ) + 1 p δ ( q ) s ( N ) + 1 p δ ( q ) s ( N ) + 1 p δ ( q ) s ( N ) . Now δ ( q ) = δ (1) = 0 and for t > δ ( q t ) ≥ δ (2 t − ) = 2 t − . So wehave i ( A ) ≤ s (1) + 1 p s (5) + 1 p s (11) + 1 p s (15) . Now we use Lemma 1.1 d): s (15) ≥ p δ (15) − δ (1) s (1) = p s (1); s (15) ≥ p δ (15) − δ (5) s (5) = p s (5); s (15) ≥ p δ (15) − δ (11) s (11) = p s (11) . So i ( A ) s ( A ) ≤ ( 1 p + 1 p + 1 p + 1 p ) ≤ p . This is a big improvement over the inequality above that comes fromthe general approach.However, the lower bound λ ( A ) on the number of ideals of A fromProposition 3.1 is a polynomial in p of degree 9, while2 p s (15) > p (cid:20) (cid:21) > p p = 2 p . So there remains a large gap between the upper and lower bounds on i ( A ).In general, the gap between the upper and lower bounds for i ( A )arises because the upper bound is based on a lower bound on the sizesof fibers of the ideal generator map G : ( subspaces of A ) → (ideals of A ) . For J an ideal of N k , not in N k − , we showed that | G − | ( J ) | ≥ p δ ( q k ) where q k is the minimum of the dimensions of principal ideals G ( x ) for x in N k \ N k − . But for many nilpotent algebras A and many ideals J of A , this lower bound greatly underestimates | G − ( J ) | . We illustratethis with two examples. Example 4.4.
Let A be the “triangular” algebra A = h x, y i with A e +1 = 0. Here one sees that q u = t ( t +1)2 for u in N t \ N t − . Let us lookat the case e = 2 in detail.Let A = h x, y i with A = 0. Then A has a basis B = ( x, y, x , xy, y ),and the annihilator N = Ann( A ) has basis x , xy, y . Moreover N = A . So we have d = 3 and d = 5. Proposition 4.5.
There are p + 4 p + 6 ideals in A .Proof. The lower bound λ ( A ) from Proposition 3.1 counts ideals of N and ideals of A properly containing N : that number is λ ( A ) = s ( t ) + s ( t ) − s (3) + s (2) −
1= (2 p + 2 p + 4) + ( p + 2)= 2 p + 3 p + 6 . To determine the number of ideals of A we let ¯ A = A/N . This is thetwo-dimensional algebra spanned by ¯ x and ¯ y with zero multiplication.We classify ideals J ⊂ A by their image ¯ J in ¯ A . Those with ¯ J = 0 aresimply the subspaces of N , which we’ve already counted. One easily OUNDS ON IDEALS 17 sees that ¯ J = ¯ A only happens once, for J = A , and since that idealcontains N , it is already counted.There remains the case where ¯ J is one-dimensional. There are p + 1one-dimensional subspaces of ¯ A , but by applying suitable automor-phisms of A it suffices to count ideals with ¯ J = F p ¯ x , and multiplythat count by p + 1. All such J contain x and xy , so the question iswhether they contain y . If yes, J is simply the linear span of x and N and has been counted. If no, then J contains an element x + ay for a unique scalar a ∈ F p ; this scalar determines the ideal. So thereare p such ideals mapping onto F p ¯ x . Thus the count of ideals J with¯ J one-dimensional is p ( p + 1). Adding that number to L ( A ) gives theresult. (cid:3) We can write down all of the ideals explicitly and determine theirfibers. The notation ( m ) denotes “subspace generated by m ”. In thelist, a, b, d are arbitrary elements of F p . A = G ( x, y ) J = J ( a, d ) = G ( x + ay + dy ) = ( x + ay + dy , x − a y , xy + ay ) J = J ( a ) = G ( x + ay, y ) = ( x + ay, x , xy, y ) J = J ( b ) = G ( y + bx ) = ( y + bx , xy, y ) J = G ( y, x ) = ( y, x , xy, y )and finally all subspaces of the ideal N = ( x , xy, y ) . To describe the subspaces of A , choose the basis ( x, y, x , xy, y ) of A .Looking at row vectors of coordinates with respect to that basis yieldsa bijection between subspaces of A and row spaces of 5 × F p . Those row spaces are in bijective correspondencewith the set of 5 × × · · · · · (we omit all rows of zeros), where the five unspecified entries can be ar-bitrary elements of F p . Thus there are p subspaces of A correspondingto echelon forms with label (124). The echelon forms (3), (4), (5), (34), (35), (45), (345) define thenon-zero subspaces of N . Those subspaces are also ideals of A sincemultiplication on N = Ann ( A ) is trivial.Every echelon form that includes both 1 and 2 defines a subspace of A that generates the ideal A . It is possible to discuss all other formsin turn, finding the ideals generated by the corresponding subspacesand the exact size of the fiber of G . Since this is repetitive and space-consuming, we only write out what happens for three echelon forms.(1) has the form (1 , a ′ , b ′ , c ′ , d ′ ) and generates J ( a, d ) for a = a ′ and d = d ′ + b ′ a ′ − c ′ a ′ . So for each ( a, d ) there are p subspaces of type(1) that generate J ( a, d ).(13) has the form (cid:18) a ′ c ′ d ′ e ′ f ′ (cid:19) . If a ′ = a , d ′ − c ′ a + e ′ a = d and f ′ = e ′ a − a , then it generates J ( a, d ). In that case, for each( a, d ) there are p subspaces of type (13) that generate J ( a, d ). If a ′ = a and f ′ = e ′ a − a , then the subspace generates J ( a ). In thatcase the choices for ( c ′ , d ′ , e ′ , f ′ ) yield p ( p −
1) subspaces of type (13)that generate J ( a ).(14) has the form (cid:18) a ′ b ′ d ′ f ′ (cid:19) . If f ′ = a ′ = a then the sub-space generates J ( a, d ) for all p choices of b ′ ; otherwise for a ′ = a = f ′ there are p ( p −
1) choices of ( b ′ , d ′ , f ′ ) for subspaces of type (14) thatgenerate J ( a ).As noted, we omit the (easy) discussion of the remaining forms (15),(134), (135), (2), (23), (234), (235), (2345), (24), (25), (245).Adding up the number of subspaces that generate each ideal, we getthe tables below. Here a, b, c are arbitrary elements of F p .ideals A p + p + 2 p + p + p + 1 J ( a, d ) p p + p + 1 J ( a ) p p + p + p + 1 J ( b ) p p + p + 1 J p + p + p + 1ideal of N p + 2 p + 4 1The center column sums to the number of ideals of A .The total number of subspaces of A accounted for by fibers of idealsof each type is: OUNDS ON IDEALS 19 ideals A p + p + 2 p + p + p + 1 J ( a, c ) 2 p + p + p J ( a ) p + p + p + pJ ( b ) 2 p + p + pJ p + p + p + 1ideal of N p + 2 p + 4The right column sums to s (5) = the number of subspaces of A .Let us compare this with our more general results. We have s (5) = 2 p + 2 p + 6 p + 6 p + 6 p + 4 p + 6 . Given that i ( A ) = 3 p + 4 p + 6 by Prop. 4.5, the inequality of Propo-sition 2.9 comes out as3 p + 4 p + 6 ≤ p + 4 p + 12 p + 12 p + 12 p + 8 + 12 p − . This inequality was based on assuming that every ideal J not containedin N has dimension ≥
2, and so | G − ( J ) | ≥ p δ (2) = p .In Corollary 2.6, the factor in brackets between ≤ and s ( A ) evaluatesto 2 p − + p − , using d = 3 , d = 5 and q = 1, q = 3. This assumedthat every ideal J not contained in N has dimension ≥
3, so | G − ( J ) | ≥ p δ (3) = p . Then the inequality is3 p + 4 p + 6 ≤ p + 2 p + 10 p + 10 p + 18 + r ( p ) , where r ( p ) = 12 p − + 8 p − + 18 p − + 16 p − is always positive but tendsto 0 for p → ∞ .Looking at the actual sizes of the fibers of G in this example, theinequality | G − ( J ) | ≥ p has the correct power of p for principal ideals J . But the non-principal ideals J , J ( a ) and A that are not containedin N have fibers with cardinalities of order p , p and p , respectively.This helps explain why the general upper bound on ideals is loose. Example 4.6.
Let A = h x, y, z i with x i = 0, the binomial algebra inthree variables. Then A = 0 ( e = 3) and A has a basis( x, y, z, xy, xz, yz, xyz )with N = ( xyz ) , N = ( xy, xz, yz, xyz ). The number of subspaces of A is s (7) = 2 p + 2 p + 6 p + 8 p + terms in p of lower degree . The number of ideals of A turns out to be i ( A ) = 7 p + 4 p + 8 . The lower bound on i ( A ), the number of ideals of A , is λ ( A ) = s (1) + ( s (3) −
1) + ( s (3) −
1) = 4 p + 4 p + 8 . An upper bound on i ( A ) can be obtained by using Proposition 4.3,which says that the dimension of a principal ideal of A not containedin N is at least 4. Then we get i ( A ) ≤ ( 1 p δ (7) − δ (1) + 2 p · p δ (7) − δ (4) + 1 p ) s ( A )= ( 1 p + 2 p + 1 p ) s ( A ) ≤ p s ( A ) ∼ p + 2 p + . . . . To see why this upper bound on i ( A ) is off by a factor of a constanttimes p , we can determine | G − | for the ideals of A , by methods inthe last example. We omit the details. But we observe first that thefibers of the 2 p + 3 p + 4 ideals of N account in total for s (4) = p + 3 p + 4 p + 3 p + 5 subspaces of A . So most subspaces of A generateideals not contained in N .In obtaining our upper bound, we used that for principal ideals of A not contained in N , | G − ( J ) | ≥ p . But in fact, we find that: • For the p + p + 1 principal ideals J of the form G ( x + by + cz ) with bc = 0 in F p , the dimension of J ≥ p , so | G − ( J ) | ≥ p δ (5) = p , not p . Thus this set of ideals is generated by approximately p subspacesof A . • For the p + p + 1 non-principal ideals of the form G ( x + bz, y + cz ) , G ( x + by, z ) , G ( y, z ) , each is generated by at least p subspaces of A . Thus this set of idealsis generated by approximately p subspaces of A . • Finally, for the ideal A = G ( x, y, z ) itself, every subspace of A whose reduced row echelon form has the form (123 . . . ) generates A ,and summing the number of such subspaces yields | G − ( A ) | ≥ p + p + 2 p + 2 p + . . . . Comparing that to s (7) = s ( A ) above, it is evident that the weak-ness in the upper bound we found for i ( A ) arises from the considerableunderestimation of the size of G − ( J ) for non-principal ideals not con-tained in N , and, in particular, on the size of G − ( A ): | G − ( A ) | is apolynomial in p of the same degree as s ( A ). OUNDS ON IDEALS 21
This last fact turns out to be true in general. One can show (proofomitted) that | G − ( A ) | is always a polynomial in p with the samedegree as the polynomial s ( A ), under the fairly mild assumption that A as an F p -algebra is generated by at most dim( A ) / A will require a more nuanced lookat the fibers of non-principal ideals whose F p -dimension is close to thedimension of A .However, the primary objective of this paper has been achieved. Let L/K be a Galois extension with elementary abelian Galois group anelementary abelian p group G . If L/K is a H -Hopf Galois extensionof type G corresponding to a commutative nilpotent algebra structure A on G with A p = 0, then the upper bound on i ( A ) /s ( A ) in section2, weak as it may be for some examples, still provides the first generalquantitative estimate on how far from surjective is the Galois corre-spondence for the Hopf Galois structure on L/K . References [CDVS06] A. Caranti, F. Dalla Volta, M. Sala, Abelian regular subgroups of theaffine group and radical rings,
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