The Valuative Condition and R-Hopf Algebra Orders in$\mathbf{\mathit{KC}}_{p^{3}}$

$Page 1: The Valuative Condition and R-Hopf Algebra Orders in$\mathbf{\mathit{KC}}_{p^{3}}$$
The Valuative Condition and R-Hopf Algebra Orders in Author(s): Robert UnderwoodSource: American Journal of Mathematics, Vol. 118, No. 4 (Aug., 1996), pp. 701-743Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/25098489 .

Accessed: 17/12/2014 21:18

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Mathematics.

http://www.jstor.org

This content downloaded from 131.230.73.202 on Wed, 17 Dec 2014 21:18:34 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=jhup

http://www.jstor.org/stable/25098489?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


THE VALUATIVE CONDITION AND R-HOPF ALGEBRA ORDERS IN KCpi

By Robert Underwood

Abstract. Let K be a finite extension of Qp endowed with the p-adic valuation and let R be its ring of integers. In this paper we give a complete classification of R-Hopf algebra orders in the group

ring KCpi. For an arbitrary Hopf order H, we show that either the group scheme SpH or the group

scheme SpH*, with H* the linear dual, can be viewed as the middle term of the Baer product of

a distinguished extension with a generically trivial extension. Using this characterization we then

compute algebra generators for either H or H*.

1. Introduction. The purpose of this paper is to investigate the structure

of R-Hopf algebra orders in KCp3

where Cy

denotes the cyclic group of order

/?3, and R is a local ring, the ring of integers of a finite extension K/Qp endowed

with the p-adic valuation. In general, knowing the structure of R-Hopf algebra orders in KCpn is important for the following reason, due to Childs [C]: Suppose

L/K is a finite extension with Galois group Cy, and ring of integers Ol, and

suppose the Leopoldt order:

A = {a e KCpn | aOL Ol}

is an R-Hopf order in KCpn. Then the ring of integers Ol is isomorphic to A as an

A-module (cf. [C] theorem 2.1). For example, when L/K is a Kummer extension

of prime degree, Childs in [C], has determined when A is a Tate/Oort Hopf

algebra, and hence Ol for these extensions is characterized. Similarly, Greither in

[G] has constructed a large class of R-Hopf orders in KCpi, referred to as Greither

orders, and has determined necessary and sufficient conditions for the existence

of a cyclic extension L/K of degree p2 so that the corresponding Leopoldt order

is a given Greither order (cf. [G], introduction and part II, theorem 3.2). Greither orders are significant because this author, [U2], has shown that all

orders in KCpi

are obtained from Greither's class of orders in a natural way: An

arbitrary R-Hopf order in KCpi

is either a Greither order Hv(s, r) = R[%-^-, ^^J,

(g) =

Cy, or the linear dual of a Greither order Hv(s,r)*. Here xs, xr denote

elements in R of value s, r, pr < s, and av is a certain element of Hv(s, r) (cf.

[U2], theorem 2.6).

Manuscript received July 15, 1994; revised May 8, 1995.

American Journal of Mathematics 118 (1996), 701-743.

701



702 ROBERT UNDERWOOD

This classification scheme is a two-fold process. First, this author shows that

the Valuative Condition for n = 2 holds: Let H(s) =

/?[*^] denote an arbitrary

R-Hopf order in KCP, with H(s)* =

H(s'), s' = ^

- s. Then we have the

following:

Valuative Condition for n = 2. Suppose an arbitrary R-Hopf order H in

KCp2 induces the short exact sequence H(s)

? //?> H(r) and the short exact

sequence of duals //(/) ? H* ?? H(s'). Then either pr < s, orps' < r7 (cf. [U2],

theorems 2.4 and 2.5).

Next, the above result is combined with Greither's classification of all short

exact sequences of the form H(s) ? H ?> H(r), pr < s, [G] corollary 3.6,

allowing us to give the mentioned classification scheme for R-Hopf orders in

KCp2.

Since there is to my knowledge no known classification of R-Hopf orders

in KCpn, n > 2, in this paper we generalize the above scheme to obtain a char

acterization of all R-Hopf orders in KCpi.

We first need to establish an analog for the Valuative Condition for n = 2. We begin by showing that an arbitrary

R-Hopf order in KCpi

can be written Av(s, r) = R[^~, ^H (g)

= Cpi

for the

appropriate s, r, av (?3.1). Note that if pr < s then Av(s, r) is the Greither order

Hv(s,r). We then show that an R-Hopf order H in KCpi

with (g) =

Cp3 gives rise to the s.e.s.

Av(s,r)^H^H(q),

which yields an associated s.e.s. of duals:

Aw(r,qf -> //* - H(sT (cf. ?4.0).

At this point we introduce the quantity E(Av(s, r))(gp), where E is a mapping

S: {/?-Hopf orders in KCpi}

? {p-adic order bounded group valuations on

Cpi),

where "p-adic order bounded group valuations" are certain functions from Cpi

to

Q U oo (?2.1). We have that E(Av(s, r)) evaluated at gp, denoted E(Av(s, r))(gp),

is equal to

v(l~px), where lgP =

{x G K | x(f -

1) G Av(s, r)}

(cf. [L3] ?3). Now our valuative condition for n - 3 can be stated as follows:

Valuative Condition for n = 3. Suppose an arbitrary R-Hopf order H in

KCp3 induces the short exact sequence Av(s, r) ?> H ?>

H(q)y and the associated



VALUATIVE CONDITION AND R-HOPF ALGEBRA ORDERS IN KCpi 703

short exact sequence of duals Aw(r, q)* ? H* ?> H(s)*. Then we must have either

pq < E(Av(s, r)){gP) orps' < E(Aw(r, qT)(gP), with s' = ^

- s.

One notes that the above is a proper generalization of the Condition for n = 2;

just replace Av(s, r) with H(s) and H(q) with //(r), then the Condition for n = 3

implies the Condition for n = 2.

Now to complete the classification of R-Hopf orders in KCp3,

we need a gen eralization of Greither's class of orders, i.e., we desire to classify all short exact

sequences of the form Av(s, r) ?> H ?> H{q), where pq < E(Av(s, r))(gp). To this

end we generalize Greither's construction of all elements in Ext1 (H(r), H(s)),

pr < s, which over K appear as KCP ?

KCpi ?

KCP (cf. [G], part I, theo

rem 3.6). We obtain all short exact sequences of the form H((c))* ?> H ?> H(q) for which pq < E(H((c))*))(gp\ where H((c))* is the dual of a one-parameter Larson order in

KCpn-\ and H(q) is a Tate/Oort order in A'Cp (?4.1). When

ai = 3 we obtain, for example, all elements of Ext1 (H(q),H(pc,c)*), pq <

E(H(pc,cf)(gp\ which over K appear as KCp2

-> KCp3

- KCP. The middle

terms of these s.e.s.'s are R-Hopf orders in KCp3

of the form

l// x* \g ~

bA a [^ " l 8P ~<*C g~bw

H(pc,c) - =

Ac(c,(/?c)) - = /_ -,-*-,- ,

since _?(/?c,c)* =

A^c', (/?c)') by ?3.1. Here vv is a representative in a certain

quotient of units groups;

Uce+(qe/p2) ^

U(ce/p)+qe/Uce+qe

where Un = {x R \ x ? l+Rpn/ }, for any positive integer n, e is the ramification

index of p in R and ? denotes a primitive p2nd root of unity. The quantity bw is

an element in the Larson dual H(pc, c)* of the form

bw =

eo + vvî + + w^ "^^.j.

The ?/ are the idempotents for the maximal integral order in KCpi.

We then show that elements of the more general groups Ext1 (H(q),H(s, r)*),

pq < E(H(s,r)*)(gP) and Ext1 (//(?), A^s, r)), M < E(A0(j,r))(^), which over # appear as

KCpi ?>

/fC^ ?>

A'Cp, can also be described as quotients of units

groups (?4.2, ?4.3).

Using these descriptions and the Valuative Condition for n = 3, we are able

to give a complete classification of R-Hopf orders in KCp3:

Given an arbitrary

R-Hopf order H in tfCy,

we find that either H or H* is of the form A^(s, r)[*=^] for some appropriate parameters s, r, <? and </>, fc.

We begin our work with some general information regarding R-Hopf orders

in KCpn.




2. R-Hopf algebra orders in KCpn.

2.0. Definitions and basic concepts. In this section we discuss the basic

properties of R-Hopf algebra orders in KCpn. We have that A' is a finite extension

of the p-adic rationals Qp, endowed with the p-adic valuation v. We assume

throughout this paper that K contains a primitive pnth root of unity denoted ?. We have found it convenient to set v(p)

= 1, so that

v(l-(pi)= ?_,. *

?_,_r for/ = 0,...,n-l. pn

i ? pn

i l

The integral closure of Zp in K, denoted R, is a local ring with maximal

ideal m. For the purposes of this paper, all Hopf algebras over R are assumed to

be commutative and co-commutative. We will denote the diagonal, counit, and

antipode maps of the R-Hopf algebra H by A//, sh and cr//, respectively. Let KCpn be the group ring over K of the cyclic group of order pn. An R

Hopf algebra order in KCpn is an R-Hopf algebra H which is finitely generated and projective as an R-module and which satisfies H ?r K =

KCpn, as K-Hopf

algebras. The group ring RCpn is an R-Hopf algebra order in KCpn via the maps

^RCpn(g) =

g?g, ?RCpn(g) = 1,

VRCpn(g) =

g"1, with Cpn =

(g). In fact, Larson

has shown that RCpn is contained in every R-Hopf order in KCpn (see [L5]). Since ( #,#*= HomR(H,R) is an R-Hopf order in KCpn

* KCpn, where

Cpn denotes the cyclic group of characters. It is not difficult to show that if A

and B are two R-Hopf orders in KCpn, then A C B if B* C A*.

As an R-algebra, the dual RC^n

can be written

p?-\

/?C> = 0 Ret

with idempotents et = ^ ?^_1 C~V> where x'U1) = Cy- If H is any R-Hopf

order in KCpn, the identification g ?-* x induces an inclusion // C /?Cy,,

and thus

/?C?, is isomorphic to the maximal integral order in jKCp*. Moreover, RC*n = Z?^

the collection of /Atuples in /? endowed with coordinatewise multiplication and

addition. The space of left integrals of an R-Hopf order H in KCpn is defined to

be the set

[ ={AeH\hA = eH(h)A}, for all h G H.

Jh

If A fH is so that fH = RA, then A is called a generating integral for H. Since

/? is a PID, every R-Hopf order has a generating integral. Moreover, the space

of left integrals is invertible as a module over R (cf. [L2] corollary 2.3.)




2.1. Larson orders. For n > 1, Larson [L3] gives a method for constructing

R-Hopf algebra orders in KCpn using order bounded group valuations. A group valuation on Cpn

= (g) is a function ?: Cp*

-> Q U oo satisfying

(1) ?l) = oo

(2) 0<e(^)<oo,for/=l,...,pw~l

(3) ?(*') = ?*-'), for all i

(4) ?(##) > mintftf), ?(s0}, for /j = 0,... ,/>" - 1.

Let ? be a group valuation on Cpn. If the range of ? is contained in the range of

v, then ? is an order bounded group valuation (o.b.g.v.) if

M^*m {?ri=,.""-'

where <t> is Euler's function. ? is a p-adic o.b.g.v. if ^(gpl) > p(?(g1)), for i =

0,...,pn-l.

For any h in Cpn, condition (4) implies that ?(/*') = ?(h) for all i relatively

prime to p. Thus a p-adic o.b.g.v. on Cpn is determined by its values ?(^') = s,

for i = 0,..., n ? 1. Let xSj be an element of R of value sj. Then (cf. [L3]), up to

isomorphism, the R-Hopf algebra determined by the o.b.g.v. ? on Cpn is

y-' _i gp"-2-I g-l~ A(0

= R --.--> >-?

xsn-\ xsn-2 xSo

The order bounded and p-adic conditions on ? imply 0 < sn-j < ,_,,! , and

psn-i-\ < sn-(. We call A(?) a Larson order in KCpn and will denote it by

H(sn-\,... ,Sn). An R-basis for the Larson order H(sn-\,... ,so) consists of the

monomials

\ xSn-\ ) \ xSn-2 J \ XSq /

where a\, a2,... ,an =

0,...,/? -

1.

Suppose we are given a rational c in the range of v with 0 < c < ?_\l( _ly

Then we may form the Larson order H(pn~lc,pn~2,.. .,pc,c) =

H((c)), which

we call the one-parameter Larson order in KCpn. Since

(gpi-l\P gpi+l-l -? J

=-+ terms in RCpn, \ xp'c J xpi+lc

for i = 0,..., n - 2, (g)

= t>, we conclude that H((c))

= R[*^].

When we have need to consider Larson orders in KCP or KCp2

we will simply write H(s) or H(s, r), respectively.




Now suppose H(sn-\,... ,_?o) is a Larson order in KCpn. Then a generating

integral is given by A = ^^X^-\

(<* P-3L ?4) We point out that not every R-Hopf order in KCpn is given by an o.b.g.v.

on Cpn. For example, the R-Hopf order RC*2 in KCpi

is not a Larson order (cf.

[L3], ?3). Every R-Hopf order in KCpn does contain a largest Larson order which

we compute as follows. Let H be an R-Hopf order in KCpn, and let / __,_/, for

i = 1,..., n be the fractional ideals

lgpn-i =

{xeK\x(gp-i)e H}.

Then the values ^(/~J,_,) = sn-t are so that H(sn-\,... ,so) is the maximal Larson

8P order contained in H. We denote the largest Larson order contained in a given

R-Hopf order H by L(H) (cf. [L3], ?3).

2.2. Short exact sequences. Now that we have introduced R-Hopf algebra

orders, we proceed to define the concept of a short exact sequence (s.e.s.) of

R-Hopf algebra orders. Suppose i: A ? B is an injection of R-Hopf algebra

orders, and let A+ denote the augmentation ideal of A. If there exists a surjection of R-Hopf algebra orders P: B ?>

B/i(A+)B, then P is said to be cokernel of i.

If P: B ?> C is any surjection of R-Hopf orders, then the subset

{heB\(IB?P)AB(h) = h?l}

is a sub-R-Hopf order of _9, and the injection of this sub R-Hopf order into B is

defined to be the kernel of P. A short exact sequence (s.e.s.) of R-Hopf algebra / p

orders is a sequence A ? B ?? C, where P is the cokernel of i and i is the kernel

of P. Schneider, [Sc], Lemma 1.1, has shown that any R-Hopf order in KCpn will

give rise to a short exact sequence:

Theorem 2.2.0. (Schneider) Let R be a Dedekind domain with quotient field

K, and let H be a Hopf order in KCpn. Then the s.e.s. of Hopf algebras over K: J P i

KCpn-i ?>

KCpn ?> KC iyfor i =

1,... ,n ?

1, where P: gp i? 1, induces a s.e.s.

of R-Hopf orders over R: J~\H) -^ H -^ P{H), where J~l(H) = Hf]

K^-i and

P(H) are R-Hopf orders in KC^-i,

and KCpt, respectively.

We also have the following theorem which is immediate from [Sc, ?1]:

Theorem 2.2.1. Let H be an R-Hopf order in KCpn. The s.e.s. J~l(H) -^ H -^

P(H), induces a s.e.s. of duals: P(H)* -> //* - J~l(H)*.

Note that a Larson order H(sn-\9...,so) can be involved in a variety of

s.e.s.'s: for example, the Larson order H(s2,s\,sq) in KCp3

induces that s.e.s.'s



VALUATIVE CONDITION AND R-HOPF ALGEBRA ORDERS IN KCp 707

H(s2,s\) -? H(s2,s\,s0) -^ H(s0) andH(s2) -

H(s2,s\,s0) -> H(s\,s0). Once we

have an R-Hopf order H in KCpn involved in an s.e.s., we can use the following

theorem, from Larson [L3], proposition 2.1, to compute the ideal ?//(/#).

Theorem 2.2.2. (Larson) Suppose H is an R-Hopf order in KCp2, inducing

by Theorem 2.2.0 the s.e.s. J~l(H) ^ H -^ P(H), P: gp ^ 1. Then eH(JH) =

?/-'(#)( Jy-i(//))^(//)( //>(//))

We now turn to a discussion of the objects which are naturally opposite our R

Hopf algebra orders in KCpn: R-group schemes of order pn. An R-group scheme

is a contravariant functor from the category of commutative R-algebras to the

category of abelian groups which has the form G = SpH

= HomR-aig(H, ) where

H is a commutative, co-commutative R-Hopf algebra. H is called the representing

algebra of the R-group G. For example, the group ring RCpn, and the R-algebra

R[T, r_1], T indeterminate, will give rise to the R-group scheme ppn of roots of

unity, and the multiplicative R-group scheme Gm, respectively. When we consider R-group schemes whose representing algebras are of finite

rank as R-modules, then we may define a s.e.s. of R-group schemes as a sequence G' ?> G ?

G" of R-group schemes for which G'(A) is a normal subgroup of

G(A) for any commutative R-algebra A (cf. [Sc], ?1). Now suppose Gf, G, G" are

R-group schemes represented by the R-Hopf orders H', H and H", respectively. Then G' ?> G ?> G" is a s.e.s. of R-group schemes if and only if H" ?? H ?> H'

is a s.e.s. of R-Hopf orders (cf. [Sc], ?1).

3. Classification of R-Hopf orders in KCP and KCp2.

3.0. R-Hopf orders in KCP. Though not all orders in KCpn, n > 1, are

Larson orders, when n = 1 we have the following well known theorem:

Theorem 3.0.0. (Greither) Let H be an R-Hopf order in KCP. Then H is a Larson

order, i.e., H is given by a p-adic o.b.g.v. on Cp (cf. [G], lemma 1.2, remark (a)).

It follows from Theorem 3.0.0 that every R-Hopf order H is KCP is of the

form H(s) for some s, 0 < s < -^j.

To calculate s, we employ the Tate/Oort

classification of R-Hopf orders in KCP to write

H^R[x]/(xp-bx),

where up~]b = ujp is a factorization in R of a certain quantity ujp (cf. [TO]). Then

s = v(u). Since C G K, the dual order H(s)* is an R-Hopf order in KCP, and hence by

Theorem 3.0.0 we have H(s)* = //(f), for some t, 0 < t <

-^. In fact we find

that H(s)* = H(sf), where sf =

-^ - s (cf. [G], part I, lemma 3.1).




3.1. R-Hopf orders in KCpi. Greither, [G], has obtained a large class of

R-Hopf orders in KCpi of the form

R ??,?? ,(*) =

Cp2 is ?*r

where s and r are values from a p-adic o.b.g.v. on Cpi, i.e., 0 < s <

-^y, 0 < r <

p(p_\) and pr < s, and av is an element in the Larson order H(s) of the

form av = e?+ve\ + - + t?~xep-\ where e\ =

? X^1 (~piJgpj are the idempotents

of the maximal integral order contained in #Cp2.

Here ( denotes a primitive p2nd root of unity. The parameter v is a unit in

Us'e+(re/p) ^

U(s'e/p)+re

where ?/? = {jc G /? | x = 1 + /?/?"/*}, for any positive integer rc, and e is the

ramification index of p in /? (cf. [G], theorem 3.6). These R-Hopf orders can be involved in the s.e.s.

H(s)^R LJZl9LZ^lL -+H(r). %s %r

Conversely, if H induces the s.e.s. H(s) ?> H ?> //(r), with pr < s, then // has

the form R[s-^-9 ^f11]

for some appropriate v (cf. [G], theorem 3.6). An R-Hopf algebra order in

KCpi characterized as above is called a Greither

order and will be denoted

HM r) = tf[^,^". %s %r

If v = 1, then the Greither order //i(s, r) = //(s, r), hence every Larson order

is a Greither order. Suppose Hv(s, r) and Hw(s, r) are two Greither orders so that

v/w e Us'e+re, then Hv(s, r)=Hw(s, r). It follows that if Hv(s, r) is a Greither order

with v e Us>e+re, then Hv(s,r) = //Cs\r) (cf. [G], corollary 3.6b). As we have already discussed (see Introduction), this author has shown that

for any R-Hopf order H in KCpi,

either H = Hv(s,r), or H = Hv(s,r)* for

appropriate _;, 5, r. Thus, to completely determine the algebra structure of //, we

must know the structure of Hv(s, r)*. We have the following theorem.

Theorem 3.1.0. Let Hv(s,r) be an arbitrary Greither order in KCpi.

Then

Hv(s,r)* =

Avi(r',s') where Av>(rf,s') is the H^-span of the elements (^)fl,

a = 0,...,/?

- 1, and where d = 1 + ?

- v, r' = -L- -

r am/ s7 = -^y

- 5. //ere

(g) =

Cpi, C w # primitive p2nd root of unity.




Proof Using the methods of [G], theorem 3.12, we establish this result by first showing that Az/(r/,.s/) C Hv(s,r)*, then checking that disc(Avt(rf,s'))

=

disc(Hv(s,r)*).

Case 1. pr < s, psf < r7. We note immediately that the extra condition psf <

r* implies that A^tf^') is the Greither order //,/(/,$') Clearly the generator

^-i G Hv(s,r)*, since there is a Hopf inclusion H(/) ?

Hv(s,r)*. To show xri

^^ G Hv(s,r)*, we establish that the pairing xs/

where y is any element of H(s). Here (, ): H x H* ? /? denotes the dual map,

with the g in ^--^ identified with the character y of order z?2. Put C = ^?, ? J?> Xr

D = ^tt-1. By the module-algebra property

= ^c^(Ajfcp2(/>Ky?c'))

= (* S> (<?', D) + (y, D) (C', a,,) + ?KCp2(fiy

? C))

where/ =

-jj-*--. (Cf. [G], p. 66 and [U2], p. 426.) One shows by

direct calculation that (y,g)(G,D) and (y,D)(CJ,ats) are in R. Moreover/ G

H(r')?H(r') since A?rCp2(Z>)

? ̂ (r7, j/)??|/(r/,/). Thus v>KCp2(f(y?0))

G fl,

implying AVf(r/,sf) =

Hv'(r',s') C Hv(s,r)*. One then shows disc^^^,s')) =

disc(Hv(s,r)*) using [G], lemma 1.3 and theorem 2.2.2.

Case 2. pr < s, pr* > rJ. We proceed exactly as in Case 1. However, to show

that/ G //(r') (8) //(r7) we first use Case 1 to write the Greither order Hv(\/(p -

arf?ar,-AKC (art)

1), r) and its dual Hv<(r', 0). Then xs>f =

-^??- is in //(r7) ? //(r7) C

HAr*,0) ? //^(r7,0), which says that/ G //(r7) ? //(r7) C //,,(*, r)* ? //?(*, r)*. We then have the inclusion Af/(r/,5/) C //,,(s, r)*. D

We can now denote an arbitrary R-Hopf order H in KCp2

as H = A?(s, r), for

the appropriate ^, 5, and r.

Recall that in ?2.1 we defined L(H), the largest Larson order contained in

the R-Hopf order //. We now compute the largest Larson order contained in

an arbitrary order in KCp2.

We have the following theorem which is an easy

consequence of [U2], theorem 1.4.0:




Theorem 3.1.1. Let Av{s, r) be an arbitrary R-Hopf order in KCpi.

Then

L(Av(s,r)) = H(s,r) if v Usie+re. Otherwise L(Av(s,r))

= H(s,s), where s =

s ~ ^t

+ v(l -

v) = E(Av(s, r))(gp\ (g) = Cp2.

4. Classification of R-Hopf orders in KCp3.

4.0. Valuative condition for n = 3. In this section we develop the "valua

tive condition for n = 3" which is a key step in classifying R-Hopf algebra orders

in KCp3.

We assume throughout this section that (g) =

Cp3 and ( is a primitive

p3rd root of unity.

Lemma 4.0.0. Let H be any order in KCp3.

Then H induces the s.e.s.'s A v(s, r) ?>

H -+ H(q) and Aw,(q', r7) -> //* -> //(s7), w/iere A^7, r7) w ffce rfwa/ 6>/f/ie /.

order AH,(r,^).

Prao/. We have that H induces the s.e.s. of R-orders J~X(H) -> // -> P(//) J p _i

from the s.e.s. over /T: J_Xy

?> JTCy

?> /CCp, where J '(//) is an arbitrary order

in KCp2

and P(//) is an arbitrary order in KCP (cf. Theorem 2.2.0). Hence, using

?3.0 and ?3.1 we can write the s.e.s. Av(s,r) ? //? //(<?), for some v, s, r and

g. We note that these parameters are uniquely determined by //. Moreover, from

gr2->\ the s.e.s. KCP ?>

KCp3 ?>

KCpi we obtain the s.e.s. H(x) ?> H ?

Aw(y,z), for some vv, jc, y and z (again by ?3.0, ?3.1). Since

H(x) = KCPC)H

= KCP fl (// H

KCpi) =

KCp fl A?($, r) = H(s),

by uniqueness, we conclude jc = _?. In addition, the image of the algebra generator * ~Qr

under the map g^ h-> 1 is ^-,

with (/i) =

Cp, implying that y - r.

Finally, an argument on the value of the integral of H shows that z = q (use Theorem 2.2.2). Now the s.e.s. H(s) ?>//?> Aw(r,q) dualizes as Awi(q',rJ)

?

//* - //(s7), by Theorems 2.2.1 and 3.1.0, thus the R-Hopf order H induces the

desired pair of s.e.s.'s.

Lemma 4.0.1. Let H induce the s.e.s.'s Av(s, r) ?> H ?> H(q) andAwt(q\ r7) ?

//* ?> //(57) as in Theorem 4.0.0. Then there exists injections j;Aw(r, q) ?

Av(s, r),

and k: Av>(rf, sf) ?> Aw>(<77, r7) of R-Hopf orders.

gP2v-+\ Proof. First note that the s.e.s. /TCP

? AXy

? _?Cp2

induces the s.e.s.

H(s) ?>//?> Aw(r,_7). Now let [/?]: // ?> // be the map defined by formal

multiplication by /?, that is

[p](h) =

M//(/i// ? /) Ox/* ? I?p-2)(I?p-2 ? A?) (/ ? A//)A//(/i)



VALUATIVE CONDITION AND R-HOPF ALGEBRA ORDERS IN KCp* 711

where /x#: H?H ?> H is multiplication in //. One notes that [p] is a nontrivial R

Hopf algebra homomorphism, which over K yields the map [/?]: KCp3 ?

JTCy,

given by gpl ^ 1 with lm([p])

= KCpi.

It follows that [/?](//) C tfC^,

and thus

[p](H) C #Cy

H // = A?($,r). In addition, we observe that H(s)+H =

fo?r([p]), and hence there is a Hopf algebra isomorphism [p](H) = Aw(r, q), implying a

Hopf algebra injection/ Aw(r, q) ?> Ar(s, r). By a similar argument we conclude

the existence of an injection k: At/(/,sf) ?? ^/(g', /). D

Lemma 4.0.2. Lef // te an arbitrary R-Hopf order in KCp3, inducing by The

orem 4.0.0 the s.e.s.'s Av(s, r) -+ H -+ H(q) andAw>(qf, /)

- //* - //(A 77ien

e/fter z/(^ ?

vv) > s' +pq or v(v ?

vv) > q+ps'.

Proof The order // gives rise to the s.e.s.'s

E\:H(s)^H -

Aw(r^):=/f -,- ,

E2: H(q)-> H -? A,/(r ,s ) :=/? -,- , Jiff Xg?

with (g) =

Cp3, and 77 the image of g under the cannonical map Cp3

? ty. The

R-module surjections

2 2

P: H gP^X Aw(r,q) and />: //* ^^ iMrV)

give rise to R-module homomorphisms (lifts)

/,: Aw(r,q) -> H and /2: Avl(r',s') - //*

satisfying /1 o P\ = id//, l2o P = idn*, since A^r,*?) and A^(r/,5/) are projective R-modules. We are interested in characterizing the lifts of the algebra generators

^^ and

^^f- under l\ and l2, respectively. For example, we assert that the lift

h(!Lyi}*L) is of the form

^^ where bx,y is a certain element of H dependent on

units x,yinR with f = vv. To this end let /K1^) = M, /2(!^-1)

= N, implying

by linearity /i(^) = xQM,

h(!L^-) =

*,/#. Observe that H = A?(s,r)[M],

//* = A^ (</,/ ')[#]> where AZ;(s,r)[M] and Atf(<U)W\ represent the Av(s,r),

Aw'(q\ r') spans of the powers M\ Nl, i = 0.. .p -

1, respectively. Noting that

Av(l/(p -

1), r) S ACp(l/(p -

1), r) and iMl/(p -

1), /) ? Ac,(l/(p -

1), /), it follows that there exists R-Hopf orders of the form

A=ACP(l/(p- l)9r)[xqAf] and B = ACp(l/(p

- l),rf)[xsfN]




giving rise to s.e.s's

/_(/ ',0)* a ACP(l/(p

- 1), r) -> A -?_/(0),

?3: //(i/(p -

i? -> a -> j? [2^ll,

!L__fd ,

//(r,0)* ? V('/(P

- l)y>

- * - ?(0),

?4: H(l/(p - D) - B - * f^ll, ^^1 .

L -V *0 J

But the above s.e.s's are all classified using the following easy consequence of

Theorem 4.2.7, appearing in ?4.2.

Lemma 4.0.3. Suppose the R-Hopf orders //(f,0)*, //(0), l/(p -

1) > t > 0

are involved in a s.e.s. H(t, 0)* ?> H ? H(0)for some order H in

KCp3. Then H

must be of the form H = H(t, 0)*[g~ **'], where bx,y is an element in H, Xq

p-\p-i

1=0 j=0

for some units jc, y in R and where the epi+j are the idempotents for the maximal

integral order in KCpi.

Proof. For the proof use Theorem 4.2.7 with s = t, r = 0, q = 0.

Now employing the lemma we see that the lift of ^^-

in s.e.s. ?3 must be of the

form g~jcy',V|

for some units x\, y\ in /., hence M = 5z___j___l Similarly we conclude

that N = *~bxxfl

for some jc2, y2 ?/(/?). The dual mapping (, ): H x //* -> /.,

with g in _V identified with the character \ of order/?3 yields (g~/''V|, g-z-~f1-) E

/?, implying

KC -

3W -

?2*2 ~

(^,,v,, &r2,y2)) > 57 + ^ > 0.

But this says that the pairing (bXuy^bXlyy2) must be a unit. Direct calculation

shows that (^,^,^2) ls a un^ onty *f at êast one ?f 3M. ?2 is 1. Moreover if

either y\ or yi = 1 then (^.ypôs) = 1- We assume, without loss of generality

that y\ = 1. Then bXu\

= ^,, hence vv = 1. One also checks that fl(-W"?il G //,

where (>>2*2)" = 1 + C

~ J2^2 is a generalization of the dual unit 1/ associated to



VALUATIVE CONDITION AND R-HOPF ALGEBRA ORDERS IN KCpy 713

orders in KCpi. (Recall 1/ = 1 + C?

? v.) Thus we may consider the lift M to be

of the form *Z3ffia2_!. Let z = (r^)". Observe that the generators *=&, iz^z Xq Xq xsi

satisfy monic degree p polynomials with coefficients in Av(s,r) and Aw>(q\ r7),

respectively since //, //* are finitely generated. It follows that the dual pairings

(__zf__, _!____) and <?_i, f_^___.) ^ i? R. Here gP in ^ and

^&* is \ JC-/ Jfw

' ^ Xq Xpst

i ? Xpq Xpsi

identified with p which generates the group of characters of order p2. From this

it follows that

H(p - zp - if +1) = v{v - (f24)') >s' +p ps' + q.

Here (y^^Y = 1 + Cp

? yi*!- Thus if ps' + q > s' +pq, then v(v

? w)>pq + _/,

and if s' +pq > ps' + q, then v{v -

vv) > q+ps'. This completes the proof of

Lemma 4.0.2.

Before we state and prove our valuative condition for n = 3, we need to put another lemma on record.

Lemma 4.0.4. Let H be an arbitrary order in ATCy inducing the s.e.s.'s Av(s, r)

?

H ? //(#) andH(s) ?>//?> Aw(r, #) w/f/. z/(l

- z/) > ^(1

? vv) andpr < s,pq < r.

Thenpq < E(Av(s,r))(gp), with (g) =

Cp3.

Proof. We desire to show that pq < s ? -^

+ v{\ ?

v). First note that

1 , q 1 s- + r +

- < s-+ i/(l -

v) p-\ p p-\

since v(\ ?

v) > v(\ ?

vv) > r7 + 2 hence

11 q 1 5- +-

- r + - < s- + i/(l

- */)

p-1 p-1 p p- 1

s -

r+- < s- + v(\ -

v). P P~ 1

Now observe that 0 < (p ?

l)s ?pr + q, since s -pr > 0, so that s < ps-pr + q, hence * < s - r + * which implies

* < s ? ~zy + ^(1

- f). At the same time

pr < s and pq < r imply pq < ^.

Theorem 4.0.5. Lef // fee aw arbitrary R-Hopf order in KCp3, inducing by

Theorem 4.0.0 the s.e.s.'s Av(s9r) - // -> //(#) andA^tfy) -+//*-> H(s').

Then either pq < E(Av(s,r))(gp) or ps' < E(Aw>(q',r'))(gp). In other words, the

Valuative Condition for n = 3 holds.




Proof Case \. ps' + q> s' +pq. In this case, by Lemma 4.0.2, we have v(v

? vv) >

s' +pq. Now if v(\ -

v) ^ v(\ -

vv), then v(\ ?

v) > s' + pq, hence

pq < ^(1 -

v) ?

s', or

pq < v(\ -

v)- + s, or P- 1

pq < E(Av(s,r))(gp).

So we assume that v(\ ?

v) = v(l

? vv). Then if v(\

? v) = v(\

? vv) >

ptl\y

v(\ ?

tf) = v(l

? vv')

= pF~Y)

which says that pr < s and pq < r. Now

by Lemma 4.0.4, pq < E(Av(s,r))(gp). If v(\ -

v) = v(\

- vv) <

^--, then

i/(l ?

i/) = i/(l

? vv') <

p(p!_i) which says that pr < s, pq < r, ps' < r' and

pr' < q'. In view of [U2, figure 2.7], these conditions on the pairs (r, s) and (q, r)

imply that s = ^,

r = ^ryj,

q = 0, thus pq < E(Av(s,r))(gp). Finally, we

consider the case v(l ?

v) = v(\ ?

vv) = ( X_X), implying prf < q', ps' < r*. First

suppose v(l -

wf) > v(\ -

i/), then ps' < E(Aw'(qf,rf))(gp), by Lemma 4.0.4.

Otherwise, if v(l ?

vv') < v(l ?

i/), then v(v ?

vv) = ^(w7

? z/)

= i/(l ?

vv') hence

v((p ?

vv) > s' + /7^, yielding

(i) KC^-^^p^+pV

We claim the R-algebra A = fltf-1-1, ^-^] is an R-Hopf algebra order in KCD2. b

x(Ps')f xpq r o p

First note that (1) implies 8 ~Gw

satisfies a monic degree p polynomial with Xpq

coefficients in the Larson order H((ps')') = R[%

~ ]. So we only need to show

x(ps')'

that A is a coalgebra, i.e., we claim Akc 2(A) ^ A<g>A. But this follows exactly as in Theorem 3.1.0 using the module-algebra property of the R-Hopf order

Aw((psfy,0) and its dual Aw'(l/(p ?

l),psf). Thus we write A = Aw((ps')',pq)

with its dual Aw>((pq)',ps'). By the structure theorems for Hopf orders in KCp2

we must have either

v(\ ?

vv) > ps' + q and v(\ ?

vv) > s' +pq

or

i/(l -

vv') > pq + s' and v(\ ?

vv') > q+psf.

Thus either v(l-w) > s'+pq, implying v(\ ?

v) > s'-\-pq and pq < E(Av(s, r))(gp),

or v(\ ?

vv') > q+psf, which says psf < E(Aw'(q'y))(gp).



VALUATIVE CONDITION AND R-HOPF ALGEBRA ORDERS IN KCp> 715

Case 2. s' + pq > ps' + q. In this case v(v ?

vv) = v(w'

? if)>q + ps' by

Lemma 4.0.2. Thus by an argument symmetric to the above we conclude either

pq < E(Av(s,r))(gp) or ps' < E(Awf(q',rf))(gp). This completes the proof of

Theorem 4.0.5.

4.1. Generalized Greither orders in KCpn. Now that we have established a

valuative condition for orders in KCp3,

we next construct a class of R-Hopf orders

in KCpn which generalizes [G], part I. In that paper, the author gives a series of

theorems which culminate in the development of Greither orders as the middle

terms of the Baer product Ev E, where Ev is a generically trivial 1-extension of

Tate/Oort Hopf orders and E is a "distinguished" extension generically equivalent to KCP

?> KCp2

? KCP. We present a straightforward generalization of Greither's

sequence of theorems resulting in the construction of a new class of R-Hopf orders

in KCpn, which we call generalized Greither orders. For the convenience of the

reader we state Greither's theorems and then give our generalizations. Let s, r be rational in the range of v so that pr < s. Let /?[^--, T~l] denote Xf

the algebra over R on the generators ^=-- and T~l. Let H(s) be the Larson order

in KCP determined by s. H(s) and /?[^-^, T~l] are commutative, cocommutative r Xf

R-Hopf algebras which give rise to R-group schemes which we denote G(s) and

T(r), respectively (generically \ip and Gm). The group of 1-extensions of G(s) by T(r) as sheaves, denoted Ext1 (G(s), T(r)),

is seen to be isomorphic to the group of 1-extensions of G(s) by T(r) as presheaves

(cf. [G], theorem 2.1). Therefore Ext1 (G(s), T(r)) is constructed exactly as we

would construct the 1-extensions of one abstract abelian group by another (cf.

[B], pp. 91-94). Hence Ext1 (G(s),T(r)) *

H2(G(s),T(r)), where H2(G(s),T(r)) is the 2nd cohomology of the complex

Mor(G(s)k~\T(r)) -+ Mor(G(s)k, T(r)) -> Mor(G(s)k+l ,T(r)).

Thus H2(G(s), T(r)) = C2(G(s), T(r))/B2(G(s), T(r)) where

C2(G(s), T(r)) = {6e Mor(G(s) x G(s), T(r)) \66 = 0} and

B2(G(s), T(r)) = {6i\te Mor(G(s), T(r)).

Now by the Yoneda Lemma (cf. [W], p. 6), each element 0 G C2(G(s), T(r))

corresponds to an R-algebra map

R ??, T~x -+ H(s) ? H(s) C RCp ? RCp

3? Rp ? RP, L xr J

thus 9 corresponds to a unit u0 in RP?RP so that ^-

G H(s)?H(s) CRp?Rp.

Likewise, each element 6l g B2(G(s), T(r)) can be viewed as the boundary of

some unit uL G Rp so that ^=-- G H(s). Greither then establishes that *-__i e //(5)




if and only if

M_C (

*

)(~ 1)/m') >^' + r> for it = l,...,p- 1,

where uL = (wo, , uP-\). Moreover, ^^-

e H(s) ? H(s) if and only if

*(EE(;) (lj)(~l)i+Juin >(* + 0*' + r, for/:,/=l,...,p-l,

where w# is an element of MatpxpU(R) with ij the entry iify (cf. [G], lem

mas 3.2, 3.3.). It is these conditions which allow Greither to describe the quotient

H2(G(s), T(r)), and thus Ext1 (G(s), T(r)) as a quotient of units groups in R as

follows ([G], theorem 3.4):

Theorem 4.1.0. (Greither) Ext1 (G(s), T(r)) *

//2(G(s), T(r)) *

Ups,e+re/Up,e+re.

Moreover, the group of generically trivial extensions, i.e., those elements

of Ext1 (G(s), T(r)) which tensor to Gm ?> Gm x \ip ??

up over K, denoted

ExtJLfl (G(s), T(r)) can be calculated as follows ([G], theorem 3.5a):

Theorem 4.1.1. (Greither) Ext]K/R(G(s),T(r))

= Us,e+{re/p)/Us>e+re.

This, however, is not quite what we want; we desire Extl^(G(s),G(r)), where G(r) is the R-group corresponding to the Larson order H(r). Greither

proceeds to calculate ExtjL^ (G(s), G(r)) indirectly, by first letting T(pr) be the

R-group scheme corresponding to the R-Hopf algebra R[^-,T~l] and proving

the following (cf. [G] theorem 3.6b):

Theorem 4.1.2. (Greither) Ifpr < s, then

ExtJ^//? (G(s), G(r)) * Ker (ExtJ^//? (G(s)9 T(r)) ̂

ExtJ^^ (G(s), T(pr))) ,

where tt denotes the pth power map.

We then can use Theorem 4.1.1 to obtain

Theorem 4.1.3. (Greither) Extj^//?

(G(s), G(r)) = UsieHn,/p)CiU{sie/p)+y /Us'e+n!.

Note that a class [v] G Usfe+{re/p)

fl U{sie/p)+re/Us>e+re

will give rise to a s.e.s.

of the form __?,,: H(s) ?> H ?> //(r) which over A' is isomorphic to /fCp ?

A'Cp x ATCp

? KCP. The middle term // of this s.e.s. is an R-Hopf order in the




group ring K(CP x

Cp). If we take the Baer product of Ev with the distinguished

extension E of Larson orders E: H(s) ?

H(s, r) ?> H(r), we will produce a

s.e.s. Ev - E, the middle term of which is the Greither order Hv(s, r). Moreover,

any R-Hopf order H inducing the s.e.s. H(s) ??//? H(r) where pr < s must

correspond to some product Ev E, and thus must be a Greither order (cf. [G],

corollary 3.6b). Now let G((c))* be the R-group whose representing algebra is the dual of

the one-parameter Larson order H((c)) in KC^-x

and let T(q) denote the R

group represented by R[^-9T~l]9 with pq < E(H((c))*)(gp). Here (g)

= C>.

The critical step in generalizing the above construction is to realize that H(s) =

H(s')* and //((c))* are both duals of one-parameter Larson orders in KCP and

KCp,i-\, respectively. The parameter c generalizes the parameter s'; the powerpn~x

generalizes p. Moreover, we see that the valuative condition/*? < E(H((c))*)(gp)

generalizes the condition pr < E(H(s')*)(hp) = s (here (g) = Cpn and (h)

= ty).

Generalizing Greither is a matter of checking the above theorems with //((c))*

replacing H(s) and T(q) replacing T(r) under the condition pq < E(H((c))*)(gp). Hence Ext1 (G((c)T,T(q))

* H2(G((c)f,T(q)), where H2(G((c)f,T(q)) is

the 2nd cohomology of the complex

Mor(G((c)fk-\ T(q)) -* Mor(G((c)fk, T(q)) - Mor(G((c)fk+x ,T(q)).

Thus H2(G((c)T,T(q)) S C2(G((c)T,T(q))/B2(G((c)y,T(q)) where

C2(G((c)f, T(q)) = {0 G Mor(G((c)f x G((c)f, T(q)) \60 = O} and

B2(G((c)T, T(q)) = {6i\te Mor(G((c))*, T(q)\

Again by the Yoneda Lemma we can associate to each element 0 G C2(G((c)Y,

T(q)) a unit ue in Rf~x ?RP"~] so that ^pl G //((c))* ? //((c))* C /JP""' 0^',_l

We further identify Z?^-1 <g)Rp"~* with Matpn_Xxpn_x(U(R)) via the map

(a0, , V-1) ? (*o, , V-1) -*

(fl/*/)(/

Thus ue corresponds to a /?" - 1 x /

- 1 matrix with i/th entry ?/,.

Likewise, each element ^^ G 52(G((c))*, T(q)) can be viewed as the boundary of some unit uL G RpH~ so that ^-^ G //((c))*. Using the methods of [G], lemmas

3.2, 3.3, we easily establish that ^f1

G //((c))* if and only if

M-M *

M"1)'"') >*c + 9? forfc=l,...,pn-1-l,




where uL -

(w0, , upn-\_x). Moreover, ^*pi

G //((c))* ? //((c))* if and only if

^(EE( ki)(j )(-l)/+X) >(* + 0* + * forkJ=h...,p"-l-h

where w# = (?,y).

It is these conditions which allow us to describe the quotient //2(G((c))*, T(q)\ and thus Ext1 (G((c))*, T(q)), as a quotient of units groups in R as follows.

Theorem 4.1.4. Ext1 (G((c))*, T(q)) 3* //2(G((c))*, 7X0) *

Upn->ce+qe/u?qe.

Proof. This follows exactly as in the proof of [G] theorem 3.4.

Moreover, the group of generically trivial extensions, i.e., those elements of

Ext1 (G((c))*,T(q)) which tensor to Gm ?

Gm x /y;-i

?> /yj-i

over K, denoted

ExtL^ (G((c))*, T(<7)) can be calculated as follows:

Corollary 4.1.5. ExtlK/R(G((c))\T(q))

* Uce+{qe/^-^/Uceê.

Proof. This follows as in the proof of [G] corollary 3.5, with q < E(H((c))*)(gp) playing the role of Greither's condition "i1 +j < e7". For example, with n = 3, we

note vv G Upice+qe corresponds to a generically trivial extension if and only if w

is a p2nd power in /.. Let if = vv. Now with (g) =

Cp3, ? a primitive p2nd root

of unity

q < E(//(pc,c)*)(gO = E(Ac(c7,(pc)W)

= C - -L- + i/(i -

o = ,

* n

- c, p -1 p(p -1)

hence, c + q < ( }_}).

It follows that

(1) p2c + q<(p2- l)c+-- < l+pc + (0/p). P(P

- 1)

Now for i; G UceHq/p2),

(1 _

^ =

(l -

if2) + (terms of value > 1 +pc + (#/p)).

Thus since v((\ -

vf ) > p2c + <?, we must have v(\ ? tf ) > p2c + # in

view of (1). We conclude that c + q < ( l_{) implies v G

Uce^q/pi^ <& if G

Upice+qe. Hence the generically trivial extensions are characterized by the quotient

Uce+(qe/p2) I Uce+qe a




We can now calculate ExtLR(G((c))*,G(q)) indirectly, by first letting T(pq) be the R-group scheme corresponding to the R-Hopf algebra R[^~,T~X]

and

realizing the following.

Lemma 4.1.6. Ifpq < E(H((c))*)(gp) then

Extj^//? (G((c))*, G(q)) * Ker

(ExlxK/R (G((c)f, T(q)) ̂ ExtxK/R (G((c)f,

T(pq))) ,

where n denotes the pth power map.

Proof This follows exactly as in the proof of [G] corollary 3.5a.

We then can use Lemma 4.1.5 to obtain

Theorem 4.1.7. Ifpq < E(H((c))*)(gp) then

Exttf//?(G((c))*,G(<?) -

Uce+qe/p"-* n Uce /p+qe / Uce+qe

-

Proof This follows exactly as in the proof of [G] corollary 3.5a.

Note that an element vv G LfCe^{qel^-x)n U(ce/P)+qe

will give rise to a s.e.s. of

the form Ew: H((c))* ?> H ?> H(q) which over K is isomorphic to

KC^-\ ?

KCpn-\ x

KCP ?

KCP. The middle term H of this s.e.s. is an R-Hopf order

in the group ring K(Cpn-\ x

Cp). If we take the Baer product of Ew with the

distinguished extension E of R-Hopf orders

E: H((c)f - H((c)f ??L ^ H(q), (g) = Cpn xq J

(which must exist because ofthe condition pg < E(H((c))*)(gp)), we will produce a s.e.s. EWE, the middle term of which is our generalized Greither order in KCpn.

Moreover, any R-Hopf order H in KCpn inducing the s.e.s. //((c))* ?> H ?> //(#)

where /?<? < H(//((c))*)(^) must correspond to some product Ew E.

It remains to discuss the structure of our generalized Greither orders in KCpn. By Theorem 4.1.7, we see that each generalized order corresponds to the middle

term of a Baer product Ew E, where vv G Uce+(qe/pn-i)

fl U(ce/p)+qe.

It follows

that the generalized Greither order is an R-Hopf order in KCpn of the form

#((c))*[_^], where ̂ =

Cpn The quantity bw is an eiement in H((c))* of the form

bw =

ô + wcj + + wp ~xepn-\_l,




where the e, are the idempotents for the maximal integral order in AXy-i (cf.

[G], proof of corollary 3.5).

4.2. Calculation of Ext1 (SpH(s, r)*, G(q)) when pq < E(H(s, r)*)(gp). In this section we compute the 1-extensions Ext1(SpH(s, r)*,G(#)), pq <

E(H(s,r)*)(gp), (g) =

Cp3, which over K appear as pp

?> /jLp3

?> iipi. Here,

we do not assume that pr = s, as was the case in ?4.1. Let ? denote a primitive

p2nd root of unity. We first note that pq < E(H(s,r)*)(gp) implies the existence

of a distinguished s.e.s. of R-Hopf orders:

E: //(s,r)*^//(s,r)* ??!- ^ H(q), L ** J

where H(s,r)*[&^-] denotes the H(s, r)*-span of the elements (&^-)a for a =

0,... ,p- 1. As in ?4.1, we need only to find ExtJL^(SpH(s,r)*,G(q))9 the group

of generically trivial extensions, i.e., those of the form [ip ?>

\xp x /y

?> jiy

over

A'. The Baer product of the generically trivial extensions with the distinguished extension will then give us the desired collection of extensions.

As before we must calculate Ex\}K,R (SpH(s, r)*, G(q)) indirectly by first deter

mining Extl/fl(Sp//(s,r)*,T(q)) and

Ext]K,R (SpH (s,r)*,T(pq)), and once again

realizing that

ExtlK/R(SpH(s9r)\G(q)) * Ker

(Ext^/i? (SpH(s, r)*, T(q)) A

ExtJ^//? (Sp//(s, r)*, T(pq)))

where 7r is the pth power map (cf. [G], corollary 3.6b). We can compute ExtlK,R (SpH (s,r)*,7Xg)) using the technique outlined in

?4.1, i.e., we first calculate

Ext1 (SpH(s, r)*, T(q)) 9. H2(Sp(H(sy r)*, T(q)\

where H2(SpH(s, r)*, T(q)) is the 2nd cohomology of the complex

Mor(SpH(s, r)*k~l, T(q)) ->

Mor(SpH(s, r)**, 7(^)) -> Mor(SpH(s, r)*k+l, 7(^)).

We have that

H2(SpH(s, r)*, 7X</)) ̂ C2(SpH(s, r)*, T(q))/B2(SpH(s9 r)*, T(^))

where

C2(SpH(s, r)*, 7X<?)) = {0 G Mor(SpH(s, r)* x Sp//(s, r)*, T(^)) | 66* = 0}



VALUATIVE CONDITION AND R-HOPF ALGEBRA ORDERS IN KCp* 721

and

B2(SpH(s, r)*, T(q)) = {6l\l? Mor(SpH(s, r)*, T(q)).

The critical step in the calculation of Ext1 (SpH(s, r)*,T(q)) is to charac terize the collection Mor(SpH(s, r)*,T(q)). As in ?4.1, we identify elements of

Mor(SpH(s,r)*,T(q)) with vectors u = (uq, ... ,up2_x) G RP . We can rewrite

the subscripts of the vector u = (uq,u\,. .

.,up2_{), yielding u = (uPj+j), with

i,j = 0,... ,p

? 1. With this in mind, we have the following:

2 Theorem 4.2.0. The unit u =

(U(Pj+j)) = (u$,... ,up2_\) G RP corresponds to

an element in Mor(SpH(s, r)*, T(q)) if and only if

AEE ( ) ( j ) ( -

D^W^)) > ks + lr + q,

for k, I = 0,... ,p ?

1, except k = / = 0.

Proof By Yoneda's Lemma, the collection of morphisms i G Mor(SpH(s, r)*,

T(q)) are in a 1-1 correspondence with R-algebra maps from R[^-,T~X] to Xq

H(s,r)* and hence are characterized by units uL = (uo,...,up2_\) G RP with

^-"-i eH(s,r)*. Since

xq

\ x '

) k,l=0,...,p-\

is an R-basis for H(s,r) (cf. ?2.1), we conclude

Xq Xs Xr

\ xq \ xs J \ xr J I

for k, I = 0,... ,p

? 1. Hence

^ H(s,rf ?? H EE

( / ) ( j )

< "

D^V^)) > ks + lr + q.

for &, / = 0,... ,p ?

1, except k = / = 0. a




Theorem 4.2.1. For v, vv in U(R), the unit u = (ifi+jvJ\ 0<ij<p-l, in Rpl

corresponds to an element in Mor(SpH(s, r)*, T(q)) if and only ifv(\ ?

vw) > r + q

andv(\ -

if) > s + q.

Proof Observe that with u$ =

1, u\ = vw, m2

= ^2vv2,

... ,ua =

if^w1', and

so forth,

" (EI] (ki)(lj )

(" Vm*Pi+J))

=^(1 ~ ̂ ) + Ml " ̂ ),

forik,/ =

0,...,p- 1.

Thus by Theorem 4.2.0, v(\ ?

vw) > r+q and v(\ ?

if) > s + q are sufficient

conditions for the element (ifl+Jw^) to be a member of Mor(H(s, r)*, T(q)).

For the converse, suppose (if'^w^) is a member of Mor(H(s, r)*, T(q)). Then

by Theorem 4.2.0, v(\ ?

vw) > r + q and is(l ?

if) > s + q. D

We now proceed to characterize our collection of morphisms

6 G Mor(SpH(s, r)* x SpH(s, r)*, T(q)).

2 2

This collection is in a 1-1 correspondence with units uq G RP ? RP satisfying

^-^- G //(s, r)* ? H(s, r)* C fl"2 ? Rpl. Xq

As in ?4.1, we associate Rr ?RP with Matpixpi(R)

via the map

(a0,..., api_ j) ? (bo, , V-1)

~* (aibj)ij

Hence by Theorem 4.2.0,

^-Zl e H(s, r)* ? H(s, r)* xq

> (k + a)s + (I + r)r + q, for k, /, a, r = 0,... ,p -

1,

except k = l = a = r = 0.

In fact, the boundary map

<S: Mor(SpH(s, r)*, 7(^)) -+ Mor(SpH(s, r)* x Sp//(s, r)*, T(^))




is the restriction of the map 8: RP ?* Matp2xpi(U(R)),

defined

6(u) = (uvUuIuv+u)vu Q Matp2xp2(U(R)),

where v + uj is taken mod p2 and u = (u$,... ,up2_x). We now can state the

following lemma:

Lemma 4.2.2. A representative 0 in H2(SpH(s,r)*,T(q)) corresponds mod

B2(SpH(s,r)*,T(q)) to a matrix in Matp2xpi(U(R)) of the form Ny Mx, where

Ny is the p2 x p2 matrix with entries in U(R) of the form

<? p blocks

?

/ my my - - -

my \

my my :

Ny- p blocks '

i m y my my my

\ my my my my J

where each block my is the p x p matrix

/ i i i \ 11 \ y 1 1 ... 1 y y

mv = .

: : y I 1 y y

\\ y y y )

and Mx is the p2 x p2 matrix

/ 1 1 1 \

II 1 x 1 1 1 x x

: : x

1 1 x x

\ 1 x x x J

for some x, y G U(R). Here denotes entry-wise multiplication.

Proof First note that a representative 0 in H2(SpH(s,r)*,T(q)) corresponds to an element of H2(Z/p2Z, U(R)). From our knowledge of the cohomology of the abstract group Z/p2Z, we see that uq can be written 6(a) M^ where




2 a = (ao,... ,ap2_})

G RP and Mf, is an element of Matp2xp2(U(R))

of the form

/ 1 1 .. 1 \ 11 lb

111-. 1 b b Mb =

: : b 1 1 b b

\ 1 b b .. b J

We now claim that ug = (uVUJ) is congruent mod B2(SpH(st r)*, T(q)) to an element

M, M G Matp2xp2(U(R))

whose 1st row contains all ones, whose 2nd row has

entries

u\u = lif0<a;</7

? 2

and whose (p + l)st row is so that

upu; = 1, for 0 < uj < p2

? (p

- 1).

First note that as in [G] theorem 3.4, we can assume the uppermost row and

leftmost column of uq contain all ones. Now let u\K, k < p ?

2, be the first

entry in the 2nd row of uq which we desire to be 1, as above, but is not 1.

Then we can find an element c = (co,... ,cp2_x) of Mor(SpH(s,r)*,T(q)) so

that the lrcth entry of 8(c) (u$) is 1. We do this by using the technique in

[G] proof of theorem 3.4, which we review here. We know that since u$ G

Mor(SpH(s,rT x SpH(s,rf ,T(q)),

by the criterion for membership in Mor(SpH(s, r)* x SpH(s, r)*, T(q)). Thus v(u\K

- 1) > (1 + n)r + q, and hence the vector

c = (C(Pi+j))

= (1,1,..., 1, u\K), (k + 1 ones, k + 2 entries),

satisfies

v (-C (

l- )

<~ iycj)

^lr + ̂ for k = 0,1= 1./c+ 1.

Thus c satisfies the first k+ 1 conditions for membership in Mor(SpH(s, r)*, T(q)).




Moreover, we can choose cK+\,.. .,cpi_x so that

"(EE( J)(j)c -I*-***,)-* * *

for &, / = 0, ..., k + 2, ..., p ?

1, except k = / = 0, using the induction technique of [G] theorem 3.4. Hence c will be in Mor(SpH(s,r)*,T(q)) by Lemma 4.2.0, and the l^th entry of 6(c) (uq) will be 1. We can proceed in this manner until

the matrix uq is congruent to a matrix M with u\^ = lif0<u; pr + q is not a sufficient condition for

c = (c(l,/+/))

= (1,1,..., 1, iii(p-i)), (p ones,p + 1 entries),

to be extended to an element of Mor(SpH(s, r)*, T(q)). Now let up\, 1 < A < p(p

? 1), A =

pm + n, be the first entry in the (p + l)st row of uq which we desire to be 1, as above, but is not 1. We know that since

ue G Mor(SpH(s,r)* x SpH(syr)\T(q)\

? (E ( I ) ( I )

( " D(1+a+/?V^))

> (1 +a)s + rr + q9

for a, r = 0, ..., m, n, except a = r = 0, by the criterion for membership in

Mor(SpH(s, r)* x SpH(s, r)*, 7Xg)). Thus i/(?pA -

1) > 0 +m)s + nr+q, and hence

the vector

c = (c(Pi+j))

= (1,1,..., 1,upx),(A +p ones, A +p + 1 entries),

satisfies

/:. = 0, ..., m + 1, / =

0, ..., n, except k = I = 0. Moreover, we can choose

ca+,7+1,.. .,cpi_{ so that

"(_:t(?)(;)(-'^-*.+^ for &, / = 0, ..., p

? 1, except & = / = 0, using the induction technique of [G]

theorem 3.4. Hence c will be in Mor(SpH(s,r)*,T(q)) by Lemma 4.2.0, and the

pAth entry of 6(cg) (uq) will be 1.




We can proceed in this manner until the matrix uq is congruent to a matrix

M which has ones in the desired places. It follows that there exists d = (d0, ...,dp2_x)

G Mor(SpH(s, r)*, T(q)) with

8(d) %uq = M. Hence

M = 8(d) uq = 8(d) 8(a) Mb, or

M = <5(d<z) Mb =

<5(m) Mfc, or

M = <5(m) Mb,

with m = (mi), m, = d/fl,. Now using the definition of 8, and equating correspond

ing entries in M and 8(m) Mb, (using the same technique outlined by Greither

in [G] theorem 3.4 second step), we conclude that m must be of the form

m = (\,a,...,ap-x,ap/(3,ap+x/(3,...,a2p-x/^,(ap/f3)2,...,ap^

for some a, (3 in U(R). For example, we have that the entry u$P-$ in M is say, (3, while the corresponding entry u\{P-\) in Mb is 1, hence once we establish that the

first p entries of m are 1, a, ..., ap~x, a G U(R) (exactly as in [G] theorem 3.4), we see that (a)(ap~x)/mp

= (3 by the definition of 8. Thus mp

= ap//3. Now

since the entry up\ = 1 in both M and Mb, we realize that ((a)(aP / (3)) / mp+\

= 1.

Hence ra^+i = ap+x /(3. It is clear that we can proceed in this manner until m has

the claimed entries.

Now let a = vw be a factorization of a in /?. Then with the substitutions

a = vw, and /? = wp, m can now be written

m = (vpi+jwi),0< i,j <p- 1.

Moreover, by the definition of 8, 8(m) is of the form Nwp Mi)Pi. It follows that

uq is congruent mod B2(SpH(s, r)*, ZXg)) to a /?2 x /?2 matrix of the form Ny9Mx, for some x, y G ?/(/?).

Now, in light of Lemma 4.2.2, we may assume without loss of generality that every representative 0 in H2(SpH(s,r)*,T(q)) corresponds to a matrix uq in

Matpixp2(U(R)) of the form A^ Mx, for some x, y G U(R).

Additionally we have

Lemma 4.2.3. A matrix u = (uVUJ) in

Matp2xp2(U(R)) of the form Ny Mx,for some x, y G U(R) corresponds to an element 0 G Mor(SpH(s, r)* x SpH(s, r)*, T(q))

if and only ifv(\ -

x) > ps + q and v(l -

y) > pr + q.




Proof Let u = (uVUJ) be of the form Ny MX9 where Ny is the p2 x p2 matrix

with entries in U(R) of the form

<? p blocks ?

/ my my my \

my my :

Ny. = p blocks

mv wv /wv my

\ mv rriy my mv /

where each block mv is the /? x p matrix

/ 1 1 ... 1 \

ill 1 y I 1 1 1 y y \ mv

=

I 1 j y | V i j y -

y /

and Mjc is the p2 x p2 matrix

/ 1 1 .. 1 \ II 1 x I 1 1 ... 1 X X \

Mx=\ , .

1 1 X " - X \

\ 1 X x X j

for some x, y G [/(/.). Suppose also that z/(l -

jc) > ps + q and i/(l -

j) > pr + q. Then we claim that (uVUJ) corresponds to an element

6 G Mor(SpH(s, r)* x SpH(s, r)*, T(^)),

i.e., we claim that

1 ?

\UVUJ) * * - G //(s, r) ? //(_;, r) .

xq




To this end recall

- G H(s, r) ?H(s,r) Xq

- "(S5S?0 GO ft) ft)*-"^-^---)

> (k + o)s + (/ + r)r + q, for k,l,o,r = 0,... ,p -

I,

except k = l = a = r = 0.

Now because (uuu) =

Nym Mx, direct calculation yields

= oo if 0 <(pk + l) + (per + r) l+v(l -y) ifp + 1 < (pk + 1) + (pa + t) <p2 > i/(l

- jc) if (pfc + /) + (po + r)

= p2.

Moreover, for p2 + 1 < (pk + /) + (per + r) < 2p2 ?

1, we calculate

> 1 + 1/(1 -

jc) if (k + a) < p, (/ + r) > p oo if (fc + cr) = p, (/ + r) z/(l -y) + v(l -x) if (Jt + <7) =

(/ + r)=p > l + i/(l -y) + i/(l -jc) if (fc + o) =p,(l + r) > p > 2 + v(\

- jc) if (A; + cr) > p, (/ + r) > p.

Hence we can easily check that v(\ ?

x) > ps + q and v(\ -

y) > pr + q imply that !~(Mt/u,) satisfies all the conditions for membership in H(s, r)* ? H(s, r)*. Xq

For the converse suppose x~^ g H(s, rf?H(s, rf. Then 1~("^) satisfies

the conditions for membership in H(s, r)* (8) //(s, r)*, and from the above tables

of values we conclude v(\ ?

x) > ps + q and v(\ ?

y) > pr + q.




At this point, we need a generalization of the group of units Un. For any unit

7 of /?, let U(i)n denote the group of units in R of given by

f/(7)n = {p U(R) | p 7m + Rpn/eh

for some positive integer m. Note that ?/(l)n = Un. We are now in a position to

prove the following:

Theorem 4.2.4. Ext1 (SpH(s,r)*,T(q)) ?

(Upse+qe x

Upre+qe)/((Use+qe ^ C/(w )r^+^) X U(V )re+qe)'>

where a pair of units (*, j) G Upse+qe x

Upre+qe corresponds to an isomorphism class

0,0e H2(SpH(s, r)*, 7X#)) ^ Ext1 (5p//(^, r)*, T(q)\ with u0 = Ny Mx. Here vv

is a pth root ofy and v is a p2nd root ofx.

Proof. By Lemma 4.2.2 and Lemma 4.2.3 every element of C2(SpH(s,r)*,

T(q)) corresponds to some p2 xp2 matrix Ny9Mx with y G Upre+qe and x G Upse+qe. Moreover, an element 0 =

Ny Mx of H2(SpH(s, r)*, T(^)) will be trivial if and

only if (9 G B2(SpH(s, r)*, T(^)), i.e., if and only if NymMx is of the form 6(uL) for

some t G Mor(SpH(s, r)*, T(^)). This implies that w, is of the form (if^w1) G /?p2, where v and vv are p2nd and pth roots: x = if , y = wp. Now by Lemma 4.2.1, we must have v(\

? vw) > r + q and i/(l

- if) > s + q. Observe that

i/(l -

vw) > r + q <=$> v(v~x ?

vv) > r + q <=> z/(w-1 ?

v) > r + q.

Thus trivial elements of H2(SpH(s, r)*, 7X#)) are elements of the form _Vy M* where jc is a pth power in Use+qe and p2nd power in U(w~l)re+qe and where y is

a pth power in U(v~l)re+qe. Hence

Ext1 (SpH(s, r)*, T(q)) *

H2(SpH(s, r)*, T(q)) *

(Upse+qe X

Upre+qe) / ((UPse+qe H

t/(w_1)ê) X

îT1)^*).

Our goal, however, is to compute the generically trivial 1-extensions of T(q) and T(pq) by //(s,r)*. We have the following corollaries.

Corollary 4.2.5. Extj^//? (SpH(s, r)*, T(q))

*

(U(s/p)e+(q/p2)e X

Ure+(q/p)e)/

((U(s/p)eHq/p)e H

(/(w_1)re+^) X

f/(^_1)^+^)




Proof This follows as in the proof of [G] corollary 3.5. We remark that our

condition pq < E(H(s,r)*)(gp) implies q < E(H(s,r)*)(gp) which plays the role of Greither's condition "j'+y < eh\ Observe (jc,y) G Upse+qe

x Upre+qe corresponds

to a generically trivial extension if and only if jc is a p2nd power in R, and y is a

pth power in R. Let if = jc, wp = y. Now with C denoting a primitive p2nd root

of 1,

q < E(H(s,rT)(gp) = E(Ac(rf,s,))(gp)

= r'- -i- + 1/(1-0= , l

n " r,

p - 1 p(p

- 1)

hence, r + q < p<p_Xy

It follows that

pr + q = (p- l)r + r + q<(p- l)r+---.

P(P ~

1)

Now since H(s, r) is a Larson order, r < * t) implying (p

? l)r < i Thus

(P ~

l)r+-rr < - +-rr =-r < 1 +r + (^/p).

p(p -

1) p p(p -

1) p - 1

We conclude

(1) pr + q< l+r + (#/p).

Moreover, <? < E(H(s, r)*)(gp) < s', hence s + q < -ly.

It follows that

ps + q < (p ?

l)s + s + q < (p ?

1)5 +--. p- 1

Again, since H(s, r) is a Larson order, s < -^\.

Thus

(p -

2> < ̂ ?, or p- 1

1 p-2 1 (p

- 2)5 + s +- <- +- + 5, or

p-1 p-1 p-1

(p -

1)5 +- < 1 + 5 + (q/p). p- 1




We conclude

(2) ps + q < 1 + s + (q/p\

hence

(3) s + q/p< \+s/p + (q/p2).

Now let (v9w) G U{s/p)eHq/pi)e

x UreHq/p)e.

Then

(1 -

vf = (1 -

if) + (terms of value > 1 + sjp + (q/p2)\

and in view of (3), v((\ -

if)) > s + q/p since i/((l -

vf) >s + q/p. It follows

that

(1 -

vf2 = (1

- if2) + (terms of value > 1 + s + (q/p)).

Moreover,

(1 -

wf = (1 -

wp) + (terms of value > 1 + r + (q/p)).

2 Thus since i/((l

? vf ) > ps + q and v((\

- wf) > pr + q, we must have

v(\ ? if ) > ps + q in view of (2), and v(\

? wp) > pr + q in view of (1). We

conclude that q < E(H(s,r)*)(gp) implies

2 (^,VV) G

U(s/p)e+iq/pi)e X

UreHq/p)e ?=> (if ,w0 G Upse+qe

* Upre+qe>

Hence the generically trivial extensions are characterized by the claimed quotient.

Corollary 4.2.6. Extj^//? (SpH(s, r)*, T(pq))

=

(U(s/p)e+(q/p)e x

Ure+qe)/

((U(s/p)e+qe H

U(w-l)re+pge) X

U(V~l)re+pqe).

Proof Since pq < E(H(s, r)*)(gp) we can simply replace q with pq in Corol

lary 4.2.5.

Recalling the isomorphism

ExlxK/R(SpH(s,r)\G(q)) <* Ker

(ExixK/R (SpH(s, r)*, 7(^)) ^ ExtJ^//? (5p//(5, r)*,

T(p9)))




where n is the pth power map, and employing the above corollaries, yields

ExtlK/R(SpH(s,r)*,G(q)) * (U x V)/(W x Y),

where

U = (U(s/p)e+(q/p)2e n (U(s/p2)e+(q/p)e n f/(W~ )(r/p)?+^))

V = (Ure+iq/p)e

D U(V~l)(r/p)e+qe)

W = (^/pjeH^nl/^"1)^)

Using the methods of Greither [G] corollary 3.5, it follows that a pair of units

(v, vv) in the above Cartesian product corresponds to a generically trivial s.e.s. of

R-Hopf orders of the form

H(s,rf-+H(s,r)* ?^ ^H(q),(h) = Cp,H(q)CC(h)

xq

2 where bW4, is an element in H(s, r)* C RP with

p-ip-i

Over A', the above s.e.s. would appears as KCp2

? Jf(Cp2

x C(h)) ?>

K(h). Moreover, we can now state the main theorem of this section:

Theorem 4.2.7. The elements c/Ext1 (SpH(s, r)*, G(q)), pq < E(H(s, rf(gp\

(g) =

Cp3 which over K appear as fip

?> p,p3

?? fip2,

are in a 1-1 correspondence with classes [(v, vv)] in

(U x V)/(W x Y),

where

U - (U(s/p)eHq/p2)e n (U(s / p2)e+(q / p)e n U(w~ \r/p)e+qe))

V = (Ure+{q/p)e

fl U(V~ Xr/p)e+qe)

w = (u(s/p)eHq/p)e

n U(w~x)re+qe)

V = (/(^1W<7,.



VALUATIVE CONDITION AND R-HOPF ALGEBRA ORDERS IN KC^ 733

These 1-extensions of group schemes correspond to s.e.s. of R-Hopf orders:

TT/ \* TT/ \* 8 ~

"W,V ff ~*X

Tr, v

H(s,r) -+H(s,r)- -+ H(q),

L xi J

(s) =

Cpi, bWyV as above. Over K, these extensions appear as KCpi

?> KCp3

?>

KCP.

Proof. This is simply a matter of taking the Baer product of the generically trivial extensions above with the distinguished extension:

H(s,r)* -+H(s9r)* ^LlA f^ H(q). xq

4.3. Calculation of Ext1 (SpA^s, r), G(q)) when pq < E(A^(_*, r))(gp). In this section we extend the methods of the previous two sections to compute the

1-extensions Ext1 (SpA(f)(s,r),G(q)\ pq < E(A<t>(s,r))(gp), (g) = Cy, which over

K appear as fip ?>

/y ?>

/y. Here A^(5, r) is an arbitrary R-Hopf order in KCpi,

with pq < E(A(j)(s,r))(gp). ? denotes a primitive p2nd root of unity. As in ?4.1

and ?4.2, the key is to calculate ExtJL^ (SpA^s, r), G(q)) via to the isomorphism

Ext^ (SpA^s, r), G(q)) * Ker

(Extj^^ (SpA^s, r), T(q))

-^ Extj^//? (SpA^s, r), T(pq))),

where n is the pth power map. And as we have seen, the key to calculating

Extj^ (SpA^s, r), T(q)) and ExtL^ (SpA^s, r), T(pq)) is to give an explicit char acterization of MortfpA^s, r), T(q)).

If cf) -

C, then A^Cs, r) = //(r7,^7)*, thus we can characterize Mo^SpA^s.r),

T(q))9 and hence compute Ext1 (SpA^s, r), G(#)) as in ?4.1 or ?4.2. So we assume

throughout this section that (f) ̂ (. Recall from ?3.1 our definition of the "dual"

quantity qb' = 1 + C

? <t>

2 Lemma 4.3.0. For (f) ̂ (, the unit u = (uo,..., upi_x)

G RP corresponds to an

element in Mor(SpA^(s, r), T(q)) only if for eachf 0 < j 1.

Proo/. By Yoneda's Lemma, u G MortfpA^s, r), 7(^)) ^ ^

G A^Cs, r).

Moreover, ^ G A^s, r) if and only if

(^,A^,(^,r)*) C /_, where the map




(,)??/? denotes the duality map H x //*? /?. In addition, by Theorem 3.1.0,

A^rf =

A,,(rV) = * -?^, -^-- I ,

I Xfi Xsi J

hence (^.A^.r)*)

C R if and only if

/_i/__\V!__y\ .

So we conclude that when k = 0, / = 1

//?__!J__f*\U ft or V\ *, *y //

i/(?i -

(?,/,',?)) > s' + <?.

Observe that

(a^,?) =

(4>Jepi+j,uoeo + + upi_xepi_x)

= p,(\ ?

<pjePi+j)(AKCp2(uoeo + +

up2_lep2_i))

p-i , /p-1 \

m=0^ \n=0 )

Hence

" ("l

- E ? (E

C"mn^'" )

ump\>s' + q>0.

Now since ?i is a unit

p-i , /p-i \

m=0^ \n=0 /

must be a unit. This implies that

ump =

woC^mp for 0 < m 1. It follows that

(u -

l,^7 -

a#) = u\

- u0(t> *,

where u? > 1. 2 2

Now since A^s, r) is stable under the cyclic shift automorphism RP ? /?p

given by

(m0,wi,...,V-i) "^

(Ki,...,iy_i,Ko) =

i/.

we have

, (/!_-l, _-?-))_

a ?

^(w2 ~

(a<t>',u')) >s' + q,

which says that

P-\ (p-\ \

is a unit. Thus as above,

Ump+\ =

wiC^mp for 0 < m 1. Note that under the cyclic shift automorphism u\

? ?o and wiC^p

? koC^ thus by linearity we may assume ip

= (p. We

conclude that fory = 0, 1,

ump+j =

"X^mp for 0 < m 1. Clearly, this can be repeated to obtain the desired

result for the components ump+j, j > 1.

Theorem 4.3.1. 77ie element u G RP of the form

M = (?o.?i.?p-i,iioCw.iip-iCJv.iioCp<2,p). iP-iCp(p"lv)




given in Lemma 4.3.0 with (p > 1, corresponds to an element in Mor(SpA,p(s, r), T(q)),

if and only if

v (E ( \ )

< - D'V**"0".-)

>ks' + q,foTk=l,...,p-l.

Proof. We have already seen that

*=? 6 A^,r) ?? (^.A^.r)*)

C fl <^

W (^.(-^-)*(^-),)6il.fcr*./.0.....p-l.

Suppose the element w = (wo,..., up2__i)

G /?p of the form given in Lemma 4.3.0

corresponds to an element in Mor(SpA(f>(s,r),T(q)). The condition (1) implies

that

(2) ^,(^6i?i?k,l.p-1.

We claim condition (2) is equivalent to the condition

v [it (

* )

(- 1)V**~?k? )

> ̂ + 4* for it = 1,... ,p - 1.

Indeed, for k = 1,

i/ f /^,_^\)

> 0 ̂ i/(Ml -

u0<t>'?) > s' + q, \\ xq xsf I)

where (p is an integer 1, by Lemma 4.3.0. When k = 2, we can compute

directly using Lemma 4.3.0. We find that

v (/^p-, (^7^)2))

> ? * ?(*2 - W* + u0<j>'2?) > 2s' + q.

Proceeding by direct computation we verify,

for k = 1,... ,p - 1. Thus if u = (wo, ,

V-i) G ̂ of the form given in




Lemma 4.3.0 corresponds to an element in MortfpA^s, r), T(q)), then

^(E(ki)(-î^k-i)ui\>ks' + q,

for k= 1,.. .,p ?

1.

For the converse, suppose the element u = (wo>..., Upi^x) G RP of the form

given in Lemma 4.3.0 is so that

for it = 1,... ,p - 1. Then ump+j

= UjC*mp implies

\^\ xq ' \ Xr> ) V xs> J J J

= (1 - (CD* fe ( J ) < -

D'^'"0".) - *'' - Is' - q.

Hence since

m-(.?Y)k)>?v p-\

and

-^r - y > 0,

P- 1

we see that the condition

for k = 1,... ,p - 1 is sufficient for w to be in MortfpA^s, r), _T(#)).

Corollary 4.3.2. Elements of Mor^pA^s, r), T(q)) correspond to p?tuples

of the form

{Cj) =

(?o4>,w, ?itf>'('-lv,..., ?p-i^n ?> > 1




satisfying

"[?<[) j*-1)1'*) ^ks' + <l> for/:=l,...,p-l.

Proof. The condition of the above corollary is easily seen to be equivalent to

the condition of Theorem 4.3.1.

Corollary 4.3.3. Thep-tuple

(Ci) =

(0'w v^p-x^,..., i?-l<j>'*), l,ve U(R)

corresponds to an element ofMortfpA^s, r), T(q)) if and only ifv((f),{p ?

v)> s'+q.

We now proceed to characterize our collection of morphisms

0 G MortfpA^s, r) x SpA^s, r), T(q)).

Consider the collection of all tensors of p-tuples

M' , im^'-1* ..., up-itf") ? (i*^, vi4>*p-1?, . . . , Vp-X^)

for elements k,-, v\ G U(R) and integers (p, ip > 1. We identify these tensors with

p x p matrices (a,y) via

(uo<f>'W9..., Up.X$*) ? (n>^\ , Vp-xtf*) <*

It follows that (fl,y) corresponds to an element oiMor(SpA^(s, r)xSpA^(s, r), T(q)) if and only if

"(ss(')0)(-i,w-)2(tt,y+* for it, / = 0,... ,p

? 1 (except k = I = 0). The coboundary map

6: MoriSpA^s, r), T(q)) -? A/or(SpA^(s, r) x SpA^(j, r), rfa))



VALUATIVE CONDITION AND R-HOPF ALGEBRA ORDERS IN KCpz 739

is defined

8((ui^p-^)) = (k^-^%^

for i, j = 0,... ,p

? 1. (Note i +j is taken mod p.)

Lemma 4.3.4. A representative 0 in H2(SpA(f>(s9r),T(q)) corresponds mod

B2(SpA<fi(s, r), T(q)) to a matrix A = (ay) in Matpxp(U(R)) of the form

I <p'pw <p,pw .

<p'pw \

^W ^ ÎPW y <P'p (P'p

... (\)'pw y y

Ay "

: :

<P'pzu <p'p y ...

y

\<t>fpw y y y 1

for some y G U(R) and some integer vo > 1.

Proof. In this case, we can once again use method of [G] theorem 3.4 to

show that the matrix (ay) of 0 is congruent mod B2(SpA<p(s, r), T(q)) to a p x p matrix of the claimed form.

Additionally, we have

Lemma 4.3.5. A matrix(atj) inMatpxp(U(R)) of the form A^1 (as inLemma4.3.4),

corresponds to an element 0 G Mo^SpA^s.r) x

SpA^(s,r),T(q)) if and only if

v^'p -y)>ps' + q,-G7> 1.

Proof. A direct calculation employing the criterion for membership in

MortfpA^s^r) x SpA^s, r), T(q)) yields

= oo ifO <k + l'Pw-y) if k + l = p >\ + v{<j)'Pw -y) ifp+1 <? + /< 2/7-1.

Now we can readily check that v{4>lpw ?

y) > ps1 + q is a necessary and

sufficient condition for A to correspond to an element of MoiiSpAîs, r) x

SpA^s,r),T{q)).




Theorem 4.3.6. Ext1 (SpA^s, r), T(q)) S U(c/)fp)pSfe+qe/(U((t>'fsfe+qe)

where a

unit y G U(</>'p)pS'e+qe> corresponds to an isomorphism class 0, 0 G //2(5pA^(5, r),

T(q)) = Ext1 (SpA^s, r), T(q)) whose matrix is A , for some w > 1.

Proof. By Lemma 4.3.4 and Lemma 4.3.5, every element of C2(SpA(j>(s,r),

T(q)) corresponds to some p x p matrix A^

with v(4>'p ?

y) > ps' + q for

some integer w > 1. Recalling the definition of U(^)n from ?4.2, we see that

y G U(<f)P)pS>e+qe. Moreover, an element 0 o A of H2(SpA<i)(s,r),T(q)) will be

trivial if and only if 0 G B2(SpA<j)(s, r), T(q)), i.e., if and only if 0 is the boundary of some i G Mor(A^(s, r), T(q)). Hence uL ?=> t must be of the form

(d) =

(0'w, V(j>,{p-l)*,..., if~l (/>'*), l,ve U(R), if=y.

Moreover, by Corollary 4.3.3, we have v((f>fip ?

v) > s' + q. Thus

Ext1 (SpA^s, r), T(q)) S H^SpA^s, r)T(q)) S l/^V^'W

D

Our goal, however, is to compute the generically trivial 1-extensions of T(q) and T(pq) by A^>(5, r). We have the following corollaries.

Corollary 4.3.7. ExtlK/R(SpA(f>(s,r),T(q))

= U(<j)')s/eHq/p)e/U((t)')s>e+qe

Proof. Here we employ the relation

q < E(A^(s, r))(gP)

= s -

?!-y + KI

- <t>)< s,

hence

since 5 < -ry. Now we can proceed as in Corollary 4.2.5 to show that

Corollary 4.3.8. Extj^j, (SpA^s, r), r(p?)) =

U{<t>')s,e+qelU{<j)')sle+pqe.

Proof. Just replace <? with p# in Corollary 4.3.7.




Recalling the isomorphism

ExtlK/R(SpA(s,rXG(q)) ? Ker

(.&?/*(SpA^s,r),T(q))

i Ext^(5pA^(5,r),r(pc7)))

where 7r is the pth power map, and employing the above corollaries, yields

Theorem 4.3.9. Ext^//? (SpA^s, r), G(q)) =

U(<p')sie+{q/p)e n

U((f)%f/p) +qe)/U(<i)')s'e+qe.

Proof. This follows exactly as in [G] corollary 3.5.

Finally, we can state the main theorem of this section.

Theorem 4.3.10. The elements of 'Ext1 (SpA^s, r), G(q)\ pq < E(A<t>(s, r)(^\ which over K appear as \xp

?> /y

?> /y,

are m a 1-1 correspondence with classes

[v] in

U(<p')s>e+(q/p)e H

U((p%f/p)e+qe)/U((pf)ste+qe.

These 1-extensions of group schemes correspond to s.e.s. of R-Hopf orders:

A^^-tA^r) 2-2- *- H(q),

I X(i

(g) =

Cp3, where

b* = e0 + vei + -- + tf~lep-X

+ Cp%

+

+ iP-lCp*e2p-i

+ {p(2*)e2p

+ + C^"1^^"1 V-i

/or some </? > 1. Over A', fAese extensions appear as KCpi

? AXy

? ^TCp.

Proof. This follows exactly as in [G] corollary 3.6.

4.4. Determining all R-Hopf orders in KCp3.

In this final, short section

we establish a complete classification of R-Hopf orders in KCp3.




Theorem 4.4.0. Let H be an arbitrary R-Hopf order in KCp3.

Then either H or

H* is of the form

A^(s,r) 8? ,(g) = Cp3 Xq J

for some R-Hopf order A^(s,r) in KCpi,

some appropriate quantity b, and some

value q.

Proof. By Theorem 4.0.0 the order H will induce the s.e.s.'s

A^s, r)-^H-> H(q) and A^(q', r') - //* -+ H(s').

Now by the Valuative Condition for n = 3, either pq < E(A<f>(s, r))(gp), or ps' <

E(A^(q\rJ))(gp). Thus if pq < E(A(f>(s,r))(gp)

\g~b~ HÂ^r)

5

L X(i

for some appropriate <j>, s, r, b, q, by Theorem 4.1.7, Theorem 4.2.7, or Theo

rem 4.3.10 (use Theorem 4.1.7 or Theorem 4.2.7 if </> =

? and Theorem 4.3.10

if (j> ̂ C with C a primitive p2nd root of unity). Moreover, if ps' <

E(A^(ql, r/))(gp) then by the appropriate structure theorem

we conclude //* is of the desired form.

DEPARTMENT OF MATHEMATICS, AUBURN UNIVERSITY AT MONTGOMERY, MONTGOMERY,

AL 36117 Electronic mail: [email protected]

REFERENCES

[A] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing

Co., Reading, Massachusetts, 1969.

[B] K. Brown, Cohomology of Groups, Springer-Verlag, New York, 1982.

[C] L. N. Childs, Taming wild extensions with Hopf algebras, Trans. Amer. Math. Soc. 304 (1987), 111-140.

[CF] J. W. S. Cassels and A. Frohlich, (eds.), Algebraic Number Theory, Academic Press, New York, 1967.

[DG] M. DeMazure and P. Gabriel, Groupes Algebrique, Tome 1, North-Holland Publishing Co., Amsterdam,

1970.

[G] C. Greither, Extensions of finite group schemes, and Hopf Galois theory over a complete discrete

valuation ring, Math. Z. 210 (1992), 37-67.

[HS] P. J. Hilton and U. Stammbach, A Course in Homological Algebra, Springer-Verlag, New York, 1970.

[J] J. C. Jantzen, Representations of Algebraic Groups, Academic Press, Boston, Massachusetts, 1987.

[Ll] R. G. Larson, Group rings over Dedekind domains, J. Algebra 5 (1967), 358-361.



VALUATIVE CONDITION AND R-HOPF ALGEBRA ORDERS IN KCp 743

[L2] _, Characters of Hopf algebras, J. Algebra 17 (1971), 352-368.

[L3] _, Orders in Hopf algebras, J. Algebra 22 (1972), 201-210.

[L4] _, Hopf algebra orders determined by group valuations, J. Algebra 38 (1976), 414-452.

[L5] _, Hopf Algebras via symbolic algebra, Computers in Algebra (Tangora, ed.), Dekker, New

York, 1988, pp. 91-97.

[LS] R. G. Larson and M. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J.

Math. 89 (1967), 75-93.

[M] J. S. Milne, Etale Cohomology, Princeton University Press, Princeton, New Jersey, 1980.

[S] P. Samuel, Algebraic Theory of Numbers, Houghton Mifflin, Boston, Massachusetts, 1970.

[Sc] H-J. Schneider, Cartan matrix of liftable finite group schemes, Comm. Algebra 5(8) (1977), 795-819.

[Sh] I. R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag, New York, 1974.

[Sw] M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969.

[TO] J. Tate and F. Oort, Group schemes of prime order. Ann. Sci. tcole Norm. Sup. (4) 3 (1970), 1-21.

[Ul] R. G. Underwood, Hopf algebra orders over a complete discrete valuation ring, their duals, and exten

sions of R-groups, Ph.D. thesis, State University of New York at Albany, 1992.

[U2] _, R-Hopf algebra orders in KCp2,

J. Algebra 169 (1994), 418-440.

[W] W. C. Waterhouse, Introduction to Affine Group Schemes, Springer-Verlag, New York, 1963.

Documents

The Valuative Condition and R-Hopf Algebra Orders in$\mathbf{\mathit{KC}}_{p^{3}}$