Abstract. We Consider a Uni Ed Setting for Studying Lo Cal Valuated

Home , Limit (category theory)

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS

Fred Richman ElbertA.Walker

FloridaAtlantic University New Mexico State University

Boca Raton FL 33431 Las Cruces NM 88003

Abstract. We consider a uni ed setting for studying lo cal valuated

groups and coset-valuated groups, emphasizing the asso ciated ltrations rather

than the values of elements. Stable exact sequences, pro jectives and injectives

are identi ed in the encompassing category, and in the category corresp onding

to coset-valuated groups.

1. Introduction

Throughout, R will denote a discrete valuation domain with prime p, and module will

mean R-mo dule. In the motivating example, R is the ring of integers lo calized at a

prime p. In that case, a mo dule is simply an ab elian group for whichmultiplication

byanyinteger prime to p is an automorphism|a p-lo cal ab elian group. The inde-

comp osable, divisible, torsion mo dule Q=R, where Q is the quotient eld of R, will

. b e denoted by R

The notion of a valuated mo dule v-mo dule arises from considering a submo dule

A of a mo dule B , together with the height function on B restricted to A. The dual

notion of a coset-valuated mo dule c-mo dule comes up when considering the quotient

mo dule B=A with a valuation related to the height function on B .Traditionally, [2],

[4], one sets

v b + A = sup fhtb + a+1 : a 2 Ag:

For nite ab elian p-groups, the v-group A tells all ab out how the subgroup A sits

inside the group B in the sense that if the subgroups A and A are isomorphic as v-

groups, then there is an automorphism of B taking A to A [6]. For isotyp e subgroups

A of simply presented p-groups B , the c-group B=A tells all ab out how A sits inside

B [4].

In this pap er we consider these two notions in terms of ltered mo dules, fo cusing

on the submo dules

B =fb 2 B : vb g

rather than on the valuations themselves. This has the virtue, if you are so inclined,

that the structure is de ned in terms of submo dules, not elements, so can b e dealt

with in purely categorical terms. Indep endent of that, or p ossibly b ecause of that,

many of the ideas take a more natural form when the valuations are suppressed. In 1

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 2

particular, the relationship b etween v-mo dules and c-mo dules app ears more natural,

and we are not forced to consider the somewhat arti cial traditional de nition of the

coset valuation.

We consider a category of ltered mo dules that includes b oth v-mo dules and

c-mo dules. Every ob ject in this category is b oth a quotient of a v-mo dule and a

submo dule of a c-mo dule. The stable exact sequences, the elements of Ext, are

identi ed in this category and in the category of c-mo dules, as are the pro jectives

and injectives.

2. Height

A general setting for height is a forest with a unique zero, whichwe will call simply a

forest. This consists of a set X together with a function : X ! X such that has

a unique p erio dic p oint, which is a xed p oint, called 0. In the motivating example,

X is a p-lo cal ab elian group, and x = px. The elements of a forest are often called

no des.Amap between two forests is a function f such that f x=f x for all

no des x.

If x = y , then wesay that y is the parent of x and that x is a child of y .If

x = y , where n can b e 0, then wesay that x is an ancestor of y , and that y is

a descendant of x. A nonzero no de whose parent is 0 is called a ro ot,a childless

no de a leaf.

A subset S of a forest X is a subforest if S S .IfS is a subforest of a forest

X , then so is S , the set of all parents of no des in S .For each ordinal de ne S

inductively by

S . S =

S . In particular, S = S , and, if is a limit ordinal, then S =

If X = X , then X = X for each > . The length of X is the least

such that X = X . A forest is torsion if for each x there is n such that

x =0.Ifx is a no de in a forest, then the order,or exp onent,of x is the smallest

nonnegativeinteger n such that x =0.Ifnosuch n exists, then x is said to have

in nite order.

A mo dule b ecomes a forest up on setting x = px|we forget all its structure

except multiplication by p. Conversely,ifX is a forest, then we can construct a

mo dule S X by taking the free mo dule on X mo dulo the submo dule generated by

fy px : y = xg:

Note that 0 in X b ecomes 0 in S X b ecause 1 p is invertible. The functor S from

forests to mo dules is the left adjoint of the forgetful functor from the category of

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 3

mo dules to the category of forests. A mo dule isomorphic to some S X is said to b e

simply presented.

As an example of a forest, whichwe will use later, consider the forest F con-

structed from ordinals and n, where n ! . A no de of the forest F is either

nite, strictly increasing, string a a :::a of ordinals less than , or the symbol t ,

1 2 m k

where k n is a nonnegativeinteger, and t is the empty string. The function is

de ned by

a a :::a = a :::a if m 1,

1 2 m 2 m

t = t .

minn;k +1

Clearly F is a forest of length + n.If1 n

;n ;n

unique ro ot t and zero t .Ifn = ! , then F has no ro ots or zeros.

n1 n ;n

Related forests are F and F . The nonzero no des of F are x , with n a

1;1 1;! 1;1 n

nonnegativeinteger, satisfying x = 0 and x = x . This is a torsion forest,

0 n+1 n

. In F the nonzero no des x are indexed by and S F is isomorphic to R

1;! n 1;1 p

the integers, and x = x throughout. The mo dule S F is isomorphic to the

n+1 n 1;!

quotient eld of R.

Anode x in a forest X is said to have height if x 2 X n X .Ifx 2 X ,

where is the length of X , then x is said to have height 1. In F , the no de

a a :::a k has height a ,ifm 1, and the no de k has height + k . The length of

1 2 m 1

F is +1+n.

3. O-modules

We are interested in mo dules, and forests, with descending ltrations indexed by the

ordinals. For any index class I , not just the ordinals, wemay consider an I -mo dule

to b e a mo dule G together with a family of submo dules G indexed by I . A map

f : A ! B of I -mo dules is a mo dule homomorphism such that f A B for

each in I .

The category of I -mo dules is preab elian:every map has a kernel and a cokernel.

The kernel of a map f : B ! C of I -mo dules is A = fb 2 B : f a=0g with

A = A \ B . It is easy to see that this is the categorical kernel, that is, if g is a

map from an I -mo dule into B such that fg = 0, then g factors uniquely through A.

A B ! C

- " g

The cokernel of a map f : A ! B of I -mo dules is C = B=f A with C equal to

the image of B in C . This is the categorical cokernel: if g is a map from B into

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 4

an I -mo dule such that gf = 0, then g factors uniquely through C .

A ! B ! C

g .

If the class I has some structure, like the class of ordinals, wewould normally

want the family of submo dules G to re ect that structure for example, to b e

a descending ltration in the case of ordinals. These conditions will b e relatively

harmless if whenever A and B are ob jects in the more restrictive category, and f :

A ! B is a map, then the kernel and cokernel of f in the larger category are in

the smaller one. Taking I to b e the ordinal numb ers, we put on three such harmless

restrictions.

An o-mo dule is a mo dule G with a family of submo dules G indexed by the

ordinals such that

If < , then G G ,

G0 = G,

pG G + 1.

In general we denote p G by p G . Call such a family of submo dules an

O - ltration. Set G1= G . Wesay that G is value reduced if G1=0.

There is an ordinal such that G=G1. The smallest such ordinal is called

the value length of G.

Note that wehave not referred to the additive structure of G, except that the

G are submo dules. The same de nition go es through if G is a forest and the G

are subforests. In this case we sp eak of an o-forest.Obviously an o-mo dule is an

o-forest. Moreover if F is an o-forest, then A = S F b ecomes an o-mo dule if we let

A b e the submo dule generated by F , and S is the left adjoint of the forgetful

functor from o-mo dules to o-forests. The left adjoint prop ertysays that any o-forest

map from an o-forest F to an o-mo dule C extends uniquely to an o-mo dule map

S F ! C .

If we put on two more conditions, wehavecharacterized the submo dules p G, the

height ltration.

G + 1 continuity, 1. G

2. G +1 pG divisibility.

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 5

Wesay that an ordinal is a limit ordinal if = sup f : < g, so a limit ordinal

is an ordinal with no immediate predecessor 0 is a limit ordinal. We mightaswell

restrict 1 to limit ordinals .For = 0 it says that G0 = G. An o-mo dule that

satis es 1 is called a valuated mo dule, one that satis es 2 is called a c-valuated

mo dule Hill and Megibb en [4]. We will use the shorter terminology v-mo dule and

c-mo dule. If an o-mo dule G is b oth a v-mo dule and a c-mo dule, then G =p G

and wesay that G is a mo dule with the natural height ltration, or an h-mo dule,

for emphasis.

Note again that the same de nitions can b e applied to o-forests, resulting in

de nitions of c-forests and v-forests.

The continuity and divisibility conditions are not harmless in the way that the

other three conditions are. The cokernel in the category of o-mo dules of a map

between two v-mo dules need not b e a v-mo dule, and the kernel in the category of

o-mo dules of a map b etween two c-mo dules need not b e a c-mo dule.

Finite Jordan-Holder length c-mo dules are h-mo dules, but nite length o-mo dules

need not b e v-mo dules. The idea of a c-mo dule was intro duced byFuchs in [1] as a

group with a coset valuation, whence the \c". He showed that every torsion c-group is

the quotient of a simply presented torsion group. Note that the data for a c-mo dule

A can b e provided by sp ecifying the submo dules A only for a limit ordinal,

sub ject to the condition that A + ! p A .

If B is an h-mo dule and A is a submo dule of B , then A is an o-mo dule kernel

exactly when A is an isotyp e submo dule of B , that is, if p A = A \ p B for each

ordinal . Alternatively, the o-submo dule A of B is an isotyp e submo dule exactly

when A is a c-mo dule. That is b ecause A is already a v-mo dule, b eing a submo dule

of the v-mo dule B ,so A is a c-mo dule if and only if it is an h-mo dule|but b ecause

A is an o-submo dule of the mo dule B , this is the same as saying A is isotyp e.

The next theorem characterizes when a submo dule of a c-mo dule is a c-mo dule.

First a lemma isolating a familiar maneuver used with isotyp e subgroups. Here the

m n m n

usual convention is that p C [p ]= p C [p ].

Lemma1. Let m and n b e nonnegativeintegers, and A B ! C a short exact

sequence of mo dules. Then the following conditions are equivalent.

m n n

1. p C [p ] f B [p ],

m+n n

2. A \ p B p A.

m+n m+n

Pro of. Supp ose 1 holds and let a 2 A \ p B . Then a = p b for b 2 B .

m m n n

f p b 2 p C [p ] f B [p ]

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 6

m 0 0 n n m 0 n

whence p b b 2 A for some b 2 B [p ], so a = p p b b 2 p A.

m n m 0 0 m+n 0

Now supp ose 2 holds and c 2 p C [p ]. Then c = p c for c 2 C ,so p c =0.

Cho ose b so that f b=c . Then

m+n m+n n

p b 2 A \ p B p A

m+n n n m m n

whence p b = p a for a 2 A.Sop p b a = 0 and c = f p b a 2 f B [p ].

We will use this lemma again later. For nowwe only need the case m =0.

Theorem 2. Let 0 ! A ! B ! C ! 0 b e an exact sequence of o-mo dules, where

B is a c-mo dule. Then the following are equivalent.

A is a c-mo dule,

n n

f B [p ] = C [p ] for each ordinal and nonnegativeinteger n,

f B [p] = C [p] for each ordinal .

Pro of. If A is a c-mo dule, then

n n

A \ p B =A \ B + n=A + n=p A

n n

so C [p ]=f B [p ] by the lemma. If f B [p] = C [p] for each ordinal

, then, by the lemma,

pA =A \ pB =A \ B +1= A +1

so A is a c-mo dule.

This is just the condition that B maps purely onto C for each .

A one-to-one map A ! B of o-mo dules is said to b e an emb edding if the inverse

image of B isA for each ordinal .Soevery kernel is an emb edding, and vice

versa. If A is a submo dule of a mo dule B , then the mo dule B=A is an o-mo dule

cokernel if and only if A is a nice submo dule of B .

It will b e convenienttointro duce the symbol , for each nonzero limit ordinal

, to serve as a predecessor to with the prop erties that < and < for any

ordinal < . Moreover|and this is the purp ose of the notation|we set

G = G <

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 7

so the continuity condition is simply G =G for any nonzero limit ordinal .

It will b e convenient to let G0 =G, which is reasonable for an emptyintersection,

even though we don't want or need a symb ol to precede 0. Finally,we set 1=

for a nonzero limit ordinal. So, for any ordinal >0wehave

G 1 = G :

A sequence 0 ! A ! B ! C ! 0 of o-mo dules is exact if 0 ! A ! B !

C ! 0 is exact for all ordinals .Every short exact sequence o-mo dules is stable:

pushouts of kernels are kernels, and pullbacks of cokernels are cokernels. This is

exactly what is needed for the short exact sequence to represent an element of Ext

[5].

Theorem 3. In the category of I -mo dules, pushouts of kernels are kernels, and

pullbacks of cokernels are cokernels.

Pro of. Consider the pushout diagram

A B

0 0

A ! B

where A = A \ B for each in I .We construct B as

A B

B =

fg a; a:a 2 Ag

0 0 0 0 0

so the map A ! B is one-to-one. Wewanttoshow that A =A \ B . Because

0 0 0 0 0 0 0

A ! B is a map, A A \ B . Now B is the image of A B . If

0 0 0

x; 0 represents an elementof B , then x; 0 = a ;b+ g a; a with a ;b 2

0 0 0

A B . Then b = a,so a 2 A , and x = a + g a, so x 2 A .

For the second part, consider the pullback diagram

0 0

B ! C

where C = f B for each in I .We construct B as

0 0 0 0

B = fb; c 2 B C : f b=g c g:

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 8

0 0 0 0

Wewant to show that each element c of C is in the image of B . As g c 2

0 0 0

C , we can cho ose b 2 B such that f b=g c . Then b; c 2 B and

0 0

f b; c =c .

The set of all O - ltrations on a mo dule A is closed under intersection. So it is

also closed under join: if A is a family of O - ltrations on A, then the smallest

A . O - ltration containing them all is given by setting A =

The largest O - ltration on a mo dule A is obtained by setting A =A for each

ordinal . The smallest is given by setting An= p A for n< !, and A! =0.

Each c-mo dule structure on A lies b elow the h-mo dule structure on A, and eachv-

mo dule structure on A lies ab ove it. Note that the largest ltration gives a v-mo dule

structure, which is generally not a c-mo dule structure, and the smallest ltration

gives a c-mo dule structure, which is not generally a v-mo dule structure. The set of

c-mo dule structures is closed under suprema, the set of v-mo dule structures under

in ma.

Given an o-mo dule A, there exists a v-mo dule A , the re ection of A, and a map

' : A ! A of o-mo dules such that if B is a v-mo dule, then any map A ! B factors

uniquely through '. This is a left adjoint of the inclusion functor F from v-mo dules

to o-mo dules:

HomA; F B = HomA ;B:

Thus the category of v-mo dules is a full re ective sub category of the category of

o-mo dules, like the category of torsion-free ab elian groups in the category of ab elian

groups. This construction is describ ed in [6, Theorem 3]: you simply take the inter-

section of all continuous O - ltrations on A that contain the given one.

00 00

Dually, there exists a c-mo dule A , the core ection of A, and a map ' : A ! A

of o-mo dules such that if B is a c-mo dule, then any map B ! A factors uniquely

through ':

HomFB;A = HomB; A ,

so this is a right adjoint of the inclusion functor F from c-mo dules to o-mo dules. The

ltration on A is the sum of all the O - ltrations on A that give c-mo dules and are

contained in the given ltration. So A is just A with the ltration given inductively

00 00

pA . A = A \

Clearly v-mo dules are closed under submo dules, and c-mo dules are closed under

quotients. Both are closed under extensions.

Theorem 4. Let 0 ! A ! B ! C ! 0 b e an exact sequence of o-mo dules. If A

and C are v-mo dules c-mo dules, then B is a v-mo dule c-mo dule.

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 9

B . Pro of. Supp ose A and C are v-mo dules, is a limit ordinal, and b 2

0 0 0

Then f b 2 C =C , so f b= f b where b 2 B . So b b 2

A =A . Therefore b 2 B .

Now supp ose A and C are c-mo dules and b 2 B + 1. Then f b 2 C +1

0 0 0

pC , so f b=pf b with b 2 B . Thus b pb 2 A +1= pA , whence

b 2 pB .

Each o-mo dule A is the quotient of a v-mo dule in a canonical way. There is

a canonical v-forest F asso ciated with A. For each ordinal not exceeding the

length of A, and each element t in A , let x have order the same as t, and

n n n

v x = + n if p t 6= 0 and <, and v x = 1. Then wehave a natural

t; t;

pure quotient map

S F ! A

that takes x to t. Note that S F is torsion if A is. The following theorem

t; A

generalizes [6, Theorem 1] which shows how to obtain a nice emb edding of a v-mo dule

in a mo dule.

Theorem 5. Every o-mo dule can b e emb edded in a c-mo dule with a simply pre-

sented torsion mo dule quotient.

Pro of. Let A b e an o-mo dule. Clearly we can embed F in a forest F made

up of the forests F . Consider the push-out

0 ! S F ! S F ! T ! 0

0 ! A ! ! T ! 0

Here K is the kernel of the quotient map S F ! A. The o-mo dule is a c-mo dule

b ecause it's a quotient of the h-mo dule S F the cokernel of the map K ! S F |

the push-out of a quotient map is a quotient map.

In particular, every v-mo dule can b e emb edded in a mo dule with a simply pre-

sented torsion mo dule quotient, b ecause if A is a v-mo dule, then is b oth a v-mo dule

and a c-mo dule, hence a mo dule. This gives [6, Theorem 1] b ecause the sequence is

exact in the category of o-mo dules, hence is nice.

Theorem 6. Every torsion o-mo dule C is a quotient of a torsion v-mo dule B by

a mo dule A so that B maps purely onto C for each ordinal .

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 10

Pro of. Let C b e an o-mo dule and S F ! C the canonical quotient map

from the v-mo dule V = S F . Note that V maps purely onto C . Emb ed the

kernel K of this map in a mo dule A with A=K a simply-presented torsion mo dule [6,

Theorem 1]. Consider the pushout

0 ! K ! V ! C ! 0

0 ! A ! B ! C ! 0

The inclusion K A is nice, so B is a v-mo dule b ecause V and A are. Moreover

B maps purely onto C b ecause V do es.

If C is a c-mo dule, then B is a mo dule, so every c-mo dule is isomorphic to the

quotient of a mo dule. As A is a mo dule, and an o-mo dule kernel, it is isotyp e in

B . We see that the o-mo dules are characterized as quotients of v-mo dules and as

submo dules of c-mo dules. They are exactly what you get if you start with mo dules,

and rep eatedly take submo dules and quotient mo dules with the induced O - ltrations.

We can get a little ner information ab out writing a c-mo dule C as a quotientof

a mo dule B .For example, Hill and Megibb en show that we can take B to b e simply

presented torsion if C is a p-group [4, Theorem 2.8]. That is a consequence of the

following characterization of c-mo dules among o-mo dules.

Theorem 7. Let C b e a c-mo dule and F a v-forest. Then any o-forest map ' from

a subforest F of F to C can b e extended to an o-forest map from F to C .

n 0

Pro of. If x=2 F for each p ositiveinteger n, then set 'x = 0. Otherwise

n 0

induct on the smallest n such that x 2 F .Sowemay assume that ' is de ned

on x but not on x.Ifvx = , then x 2 F +1so 'x 2 C + 1. As C is a

c-mo dule, there exists c 2 C such that pc = 'x. Set 'x= c. Much the same

argumentworks if vx = 1.

A KT-mo dule balanced pro jective mo dule is a mo dule of the form S F where

F is a forest with the prop erty that for eachnode x there exists a p ositiveinteger n

n n+i n

such that either ht p x = 1 or ht p x =ht p x + i for each p ositiveinteger i.A

torsion mo dule is a KT-mo dule if and only if it is simply presented.

Corollary 8. Every torsion c-mo dule C is the quotient of a torsion KT-mo dule

B by an isotyp e submo dule.

Pro of. Emb ed the v-forest F in a forest F made up of the forests F . The

C ;n

canonical o-forest map F ! C extends to an o-forest map F ! C by the theorem.

The induced map from B = S F to C is the desired quotient map. Tosay that the

kernel A is an isotyp e submo dule is to say that A is a c-mo dule. This follows from

Theorem 2 b ecause B [p] maps onto C [p], for each .

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 11

4. Valuations

An alternativeway to view an o-mo dule is by means of a valuation v .IfG is an

o-mo dule, and x 2 G n G + 1, then we set vx = .Ifx 2 G for all ordinals

, then we set vx = 1.If is a limit ordinal, and x 2 G n G , then we set

vx = .Thus vx is either an ordinal, the symbol 1,or where is a nonzero

limit ordinal. Note that v-mo dules are characterized by the prop erty that vx is always

either an ordinal or 1.

The valuation v on an o-mo dule has the following prop erties.

vx is either an ordinal, 1,or for a nonzero limit ordinal ,

vpx > vx if vx is an ordinal,

v x + y minvx; vy,

vux = vx if p do es not divide u,

v 0= 1.

For G an o-mo dule and an ordinal, wehave G =fx 2 G : vx g, so the

valuation and the ltration are equivalentways of viewing an o-mo dule.

It is instructive to compare the coset valuation as de ned in [1] and [4] with how

we view it here. The setting is an epimorphism ' : B ! C of mo dules. They set

vc +1 if c 2 'p B , for any ordinal, and

vc if vc + 1 for some < , for a nonzero limit ordinal.

So knowledge of the submo dules 'p B = fc 2 C : vc +1g suces to sp ec-

ify the coset valuation. From the ltration p oint of view, we simply lter C with

the submo dules 'p B , indexing 'p B by |end of story. But if we demand a

description in terms of an ordinal-valued function v , then we are forced to makean

arti cial shift by 1 in order to makeroomforavalue for the elements of 'p B

that are not in 'p B , if is a limit ordinal. This has the annoying consequence

that if ' is the identity map, then the coset valuation on C is not the same as the

valuation on B . It could b e argued that this is a small price to paytoavoid intro-

ducing elements , but there is no need intro duce such elements from the ltration

p oint of view, and it is certainly more natural to index 'p B by than by +1.

Another illustration of the advantage of the ltration p oint of view is provided by

Hill's notion [3] of compatible submodules H and K . The valuation de nition is

For all h 2 H and k 2 K , there exists x 2 H \ K such that v h + x v h + k .

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 12

The ltration de nition is

H + K =H +K for all ordinals .

Not only is the ltration de nition simpler and easier to rememb er, but it makes it

obvious that compatibility is a symmetric relation.

A submo dule A of an o-mo dule B is nice if every coset b + A has an elementof

maximum value. Equivalently,ifb + A \ B is nonempty for all ordinals in

B is nonempty. That is, the submo dule A has the some set S , then b + A \

b est approximation prop erty:ifb is any elementof B , then there exists a in A

such that any ball b + B centered at b that intersects A contains a. An alternative

way of expressing this is to say that, for any set S of ordinals,

\ \

B +A= B +A:

2S 2S

This de nition has the advantage that it is expressed purely in terms of submo dules,

not elements. If A is a submo dule of a v-mo dule B , then the quotient B=A is a

v-mo dule if and only if A is nice.

A nonzero element of the divisible part of a mo dule really has height 1 , and

only 0 has height 1, but the use of 1 for the height of elements in the divisible part

is well established. Note that with this usage 1 b ehaves much more like than

like an ordinal.

5. Projective and injective o-modules

The pro jectives in the category of o-mo dules are easily characterized: they are the

valuated mo dules that are free on a valuated set.

Theorem 9. The pro jectives in the category of o-mo dules are direct sums of torsion-

free cyclic valuated mo dules such that vpx = vx +1 for each nonzero element x,or

vx = 1 for all x.Every o-mo dule is the quotient of a pro jective.

Pro of. These o-mo dules are obviously pro jective. Conversely, it suces to

show that every o-mo dule is the quotient of such a direct sum b ecause, by the Azu-

maya theorem in an additive category [8, Theorem 5], such direct sums are closed

under summands as the endomorphism ring of a cyclic o-mo dule is lo cal. Let C

denote a torsion-free valuated mo dule with generator c such that vp c = + n,if is

an ordinal, and vp c = 1 if = 1.IfA is an o-mo dule of value length , construct

the direct sum

M M

C F =

x2A

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 13

where C is a copyofC . Map F onto A by taking the generator of C to x.

;x ;x

We turn to the injective o-mo dules.

Lemma 10. Let I b e an injective o-mo dule. Then I is a c-mo dule and I is a

divisible summand of I for each nonzero limit ordinal .

Pro of. If x 2 I + 1, then the cyclic submo dule generated by x can b e

emb edded in a cyclic o-mo dule C generated by y such that py = x and y 2 C .

As I is injective, we get a map from C to A that xes x,so x 2 pA . So I is a

c-mo dule.

Now let D b e a divisible hull of I , ltered by setting D = D and D =

I for . This gives an emb edding of I inD whichmust extend to a map

D ! I that is the identityon I . As D = D , this map go es into I .

Therefore I is divisible, hence splits out of I as a mo dule. But since for all ,

either I I orI I , this is a splitting of o-mo dules.

Note that as I 1= I for any limit ordinal b eyond the value length of I ,

the submo dule I 1 is a divisible summand of I . Note also that p I = I ! .

Lemma 11. Let D b e a c-mo dule and a limit ordinal, p ossibly zero. If p D =0

and D =D , then D is an injective o-mo dule if and only if the underlying mo dule

of D is algebraically compact.

Pro of. Note that p D =D + ! . Supp ose D is algebraically

compact. Given o-mo dules A B and a map f : A ! D ,wehave to extend f to B .

As f A + ! =0,wemay assume that A + ! =0=B + ! . First

extend f from A = A \ B to B . We can do this b ecause A and B

are v-mo dules, and D is algebraically compact, hence injective in the category

of v-mo dules. This gives an extension of f to A + B . If =0,we are done.

If not, then the c-mo dule D is divisible as D = D , so we can extend f to a

homomorphism from B to D . The result is a map of o-mo dules b ecause D = D .

Conversely, supp ose D is an injective o-mo dule. Let A b e a pure submo dule of a

mo dule B , and f : A ! D a homomorphism. If we show that we can extend f to

a homomorphism from B to D , we will have shown that D is pure injective,

! !

hence algebraically compact. Wemay assume that p B =0 as p D = 0. Put an

o-mo dule structure on B by setting B + n= p B . Then f is an o-mo dule map,

hence extends to an o-mo dule map from B to D , whichmust take B into D as

B = B .

We will b e esp ecially interested in the rank-one torsion divisible o-mo dule I ,

i n

where is a nonzero limit ordinal, I =I , and I + i= p I [p ].

+n +n +n +n

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 14

Also the rank-one divisible o-mo dule I where I =I for each , and I =

1 1 1 n

R=p R.

Theorem 12. The torsion cyclic c-mo dules are the mo dules I . The value reduced

quasicyclic c-mo dules are the c-mo dules I for a nonzero limit ordinal. The

torsion cyclic and quasicyclic c-mo dules are injective o-mo dules.

Pro of. If G is a c-mo dule, and p G = 0, then G! = 0, so G is a v-mo dule,

hence a mo dule. So the torsion cyclic c-mo dules are the mo dules I . If G is a

quasicyclic c-mo dule, let x b e a generator of G[p]. If vx = , then G = I . If

vx = + n, then G = I . The mo dules I and I are injectiveby Lemma 11.

+n+1 +n n

We can now show that the category of o-mo dules has enough injectives.

Theorem 13. Eachvalue reduced o-mo dule can b e emb edded as a submo dule of

a pro duct of value reduced cyclic and quasicyclic torsion c-mo dules.

Pro of. Let A b e an o-mo dule of value length . Let F b e the set of all o-mo dule

maps

f : A ! C

where C is I for some ,or I if A is not value reduced. Then the evaluation

f 1

map

' : A ! P = C

f 2F

is de ned by 'x = f x. We shall show that ' is an emb edding

Supp ose x 2 A. If x=2 A , for some , then there is f 2 F taking A

to I such that f x generates I [p], so f x 2= I = 0, whence 'x 2= P .

Contrap ositively,if 'x 2 P , then x 2 A . To complete the pro of that ' is an

emb edding, we need only verify that it is one-to-one. If 'x = 0, then x 2 A. If

x 2 A is nonzero, in which case A is not value reduced, then there is f : A ! I

with f x 6=0,so 'x 6=0.

So an o-injective I is a summand of a pro duct of copies of the c-mo dules I .It

turns out that they are also pro ducts, but not necessarily pro ducts of copies of I .

To show this we rst prove a general theorem ab out writing mo dules as pro ducts.

Theorem 14. Let A b e a mo dule with a continuous descending ltration A indexed

by ordinals , such that A = A and A =0 for some limit ordinal . Supp ose A is

an absolute direct summand of A for each <, and set

P = A =A

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 15

for each <. Then there is a monomorphism ' : A ! P such that 'A =

'A \ P for each , and the comp osite of ' with the -th co ordinate map from P to

A =A restricts to the natural pro jection map on A . Moreover, if P = 'A K ,

+1 0

with P = 'A P \ K for each , then 'A= P .

Pro of. Inductively construct an ascending chain B of submo dules of A so

that A = B A for each ordinal , using the fact that A is an absolute direct

summand. Map A = B A to A =A by taking B to zero and using the natural

pro jection map on A . This de nes a map ' : A ! P taking A into P , such that

the comp osite of ' with the -th co ordinate map from P to A =A restricts to the

0 +1

natural pro jection map on A .Ifx is a nonzero elementof A, then, by the continuity

of the ltration, there exists such that x 2 A n A ,so'x 2 P n P . This

+1 +1

shows that ' is a monomorphism and that 'A ='A \ P .

Now supp ose P = 'A K , with P = 'A P \ K for each .Wewant

to show that K = 0. Note that ' induces an isomorphism from A =A to P =P .

+1 +1

P A P \ K

+1 +1 +1

from which it follows that P \ K = P \ K for all .SoK =0.

Corollary 15. An o-mo dule is injective if and only if it is a pro duct of a reduced

algebraically compact mo dule, a divisible mo dule, and an o-mo dule of the form K ,

where the pro duct is over a set of nonzero limit ordinals, and K is a divisible c-

mo dule such that the underlying mo dule of K is reduced algebraically compact,

and K =K .

Pro of. The reduced algebraically compact mo dules, and the o-mo dules K ,

are injective o-mo dules by Lemma 11. The divisible mo dules are clearly injective

o-mo dules. So the pro duct is injective.

Conversely, supp ose A is an injective o-mo dule. As A1 is a divisible summand

of A,wemay assume that A1 = 0. Set A = A! for each ordinal . Note

that this is a continuous ltration of A. Each A including A = A is an injective

o-group by Lemma 10. To see that A is an absolute summand of A, supp ose K A

and K \ A = 0. Then K A is a direct sum of o-mo dules b ecause A is comparable

to A for each ordinal . So the pro jection of K A onto A is an o-mo dule map,

hence extends to A.Thus A satis es the hyp otheses of Theorem 14.

So there is a monomorphism ' : A ! P such that 'A ='A \ P for each ,

and the comp osite of ' with the -th co ordinate map from P to A =A restricts

0 +1

to the natural pro jection map on A . It is easy to see that ' is an emb edding, so

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 16

'A is a summand of P , and therefore equal to P . The factors D = A =A are

0 0 +1

injective o-mo dules, hence c-mo dules by Lemma 10. As p D = D for each

and p D ! =D ! + ! = 0, the underlying mo dule of D ! is algebraically

compact by Lemma 11.

Fuchs [2] shows that injectivevaluated vector spaces are pro ducts. He do es this

in the context where the values lie in a linearly ordered set, and the pro duct is the

Hahn product|elements with well-ordered supp ort. In our situation, every element

has well-ordered supp ort.

Although o-exactness is not the right concept in the category of c-mo dules, as

it was in the category of v-mo dules, we can still investigate what happ ens in the

category of c-mo dules relative to that notion of exactness.

Theorem 16. The pro jectives in the category of c-mo dules relative to o-exactness

are the KT-mo dules. There are enough.

Pro of. Let C = S T b e a rank-one KT-mo dule and 0 ! A ! B ! C ! 0

an exact sequence of o-mo dules with A a c-mo dule so B is also a c-mo dule. Let x

be a node in T such that v x = + n for each nonnegativeinteger n. Then there

is b 2 B such that 'b=x, whence vp b = + n.Sowe can lift the tree X

generated by x and 0 backto B . If there is no such x, just let X =0.

Now supp ose wehave lifted back a subtree X of T to B , and t isanodeinT n X

such that pt = x 2 X .Wewant to extend the lifting to t. Supp ose vt = < = vx

0 0

and b 2 B is the lift of x. Then there is b 2 B such that 'b =t. As

0 0 0

'pb b=0wehave pb b 2 A + 1, so pb b = pa for some a 2 A . So

0 0 0 0

pb a= b and b a 2 B . Moreover, 'b a='b =t,sowe can extend

the lifting to t.

Corollary 8 shows that there are enough pro jectives.

6. Stable sequences of c-modules

In a preab elian category, the short exact sequences 0 ! A ! B ! C ! 0 are

generally not suitable for forming the functor ExtC; A. The reason is that the

pushout of a kernel need not b e a kernel, and the pullback of a cokernel need not

b e a cokernel. Those short exact sequences for which these prop erties do hold are

the ones that constitute ExtC; A, the stable exact sequences. In the category of o-

mo dules, every short exact sequence is stable. In the category of v-mo dules, the stable

short exact sequences are those that are short exact in the category of o-mo dules [6,

Theorem 6]. This do es not carry over to the category of c-mo dules.

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 17

A map f : A ! B in a preab elian category is a semistable kernel if, for any

pushout diagram

A ! B

0 0

A ! B

the map f is a kernel. Dually,a semistable cokernel is one for which each pullback

is a cokernel.

It follows from Theorem 3 that, in the category of o-mo dules, every kernel is

semistable: pushouts of kernels are kernels. This fails for the category of v-mo dules:

if A B is a semistable kernel in that category, then every coset of nite order in

B=A contains an element of maximum value but not conversely [6, Theorem 7].

Stanton [7] characterized semistable kernels for v-mo dules.

If A is an o-mo dule, we de ne the core ection K = F A, a c-mo dule, to have

the same underlying mo dule as A with the ltration given inductively by

K +1 = pK

K = A \ K for a limit.

The identity is an o-mo dule map F A ! A, and any o-mo dule map from a c-mo dule

to A factors uniquely through this map. Note that F dep ends only on A for a

limit.

Cokernels in the category of c-mo dules are cokernels in the category of o-mo dules.

Kernels in the category of c-mo dules are obtained by applying F to the kernel in the

category of o-mo dules. If ' : B ! C is a map of c-mo dules, and K = fb 2 B :

'b= 0g, then the kernel of ' in the category of c-mo dules is K equipp ed with the

ltration K , for limit ordinals , given inductively by

K = K \ B \ K p

where ranges over limit ordinals although you could let range over all ordinals.

This is the biggest c-group structure on K that resp ects the inclusion K B .

Theorem 17. Akernel A B , in the category of c-mo dules, is semistable if and

only if A = A \ B for each limit ordinal .

0 0 0 0 0

Pro of. Note that if A B are c-mo dules, and A =A \B for each limit

0 0 0 0

B is a kernel in the category of c-mo dules. Conversely,ifA B ordinal , then A

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 18

0 0 0 0 0

is a kernel in the category of c-mo dules, and A =A , then A =A \ B

by the inductive de nition of the ltration on a kernel.

Let b e a limit ordinal and consider the pushout diagram

A B

0 0

A B

0 0 0 0

where B is a quotientof A B ,soB is the image of A B . Supp ose

0 0 0

A = A \ B . We will show that A =A \ B , which suces to prove the

0 0

\if " part of the theorem. If x 2 A \ B , then

x; 0+g a; a 2 A B

0 0

for some a 2 A.Soa 2 B , whence a 2 A and g a 2 A . Thus x 2 A .

0 0 0 0

with A =A and A =0. Conversely, supp ose a 2 A \ B . Let A be R

0 0 0 0 0 0

Then g a 2 B , so if A B is a kernel, then g a 2 A =A \ B , so

g a = 0. Hence any mo dule homomorphism from A to R that kills A kills a,

whence a =0.

It is easy to see that the condition of the theorem is equivalentto A \ B + !

p A \ B for each limit ordinal .Itisinteresting to compare this characterization

with that of semistable cokernels f : B ! C in the category of v-mo dules. There the

condition is that f be semi-nice: f B = C for each nonlimit ordinal [6,

Lemma 5].

Corollary 18. If A B is a semistable kernel in the category of c-mo dules, then

the sequence

0 ! A1 ! B 1 ! 1 ! 0

is a split short exact sequence of divisible mo dules.

Pro of. Let b e a limit ordinal greater than the value lengths of A and B .

Then the ab ove sequence is the same as

! 0 0 ! A ! B !

hence is short exact. Moreover, as A1= A +1 = pA = pA1, it follows

that A1 is divisible, and similarly for B and B=A.

A stable kernel A B is a semistable kernel such that the asso ciated cokernel

B ! B=A is also semistable. That is, the short exact sequence A B ! B=A is

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 19

stable. Similarly for stable cokernels.We will characterize the stable cokernels,

hence the stable exact sequences, without characterizing the semistable cokernels.

Wesay that a homomorphism f : B ! C with kernel A is isotop if for each n

there exist m such that either of the two equivalent conditions of Lemma 1 are met:

m n n

p C [p ] f B [p ]

m+n n

A \ p B p A:

The name comes from the fact that the map f is isotop exactly when the p-adic

top ology on A is induced by the p-adic top ology on B . This condition played a role

as a condition in [5, Theorem 15]. Here it gives a sucient condition for a cokernel

to b e semistable, that turns out to b e necessary for stability.

Theorem 19. Let f : B ! C b e a cokernel in the category of c-mo dules such that

the restriction

f : B ! C

is isotop for each limit ordinal . Then f is a semistable cokernel.

Pro of. Let

0 0

! C B

B ! C

0 0 0 0

b e a pullback diagram, with B = fb; c 2 B C : f b=g c g.We will show that

0 0 0

B = B \ B C

0 0 0 0

for each limit ordinal .Thus f is a cokernel b ecause f B = C for each

0 0 0

limit ordinal ,so f B = C for each ordinal .

0 0 0

Induct on the limit ordinal . Clearly B B \ B C so supp ose

0 0 0 0 ! 0

b; c 2 B C and f b=g c . Wewanttoshow that b; c 2 p B for

0 n 0

each limit ordinal < , so it suces to show that b; c 2 p B for each p ositive

integer n.

As f is isotop, there is m such that

m n n

p C [p ] f B [p ]:

0 0 n m

Because b 2 B and c 2 C , we can write b = p b with b 2 p B , and

0 0

0 n 0 0 m 0

c = p c with c 2 p C . Then

0 0

n 0 0 0 m

p f b g c = f b g c =0 and f b g c 2 p C

0 0

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 20

n 0 n

so there exists b 2 B [p ] such that f b =f b g c . Then p b b =

1 1 0 0 1

n 0 n 0 0

p b = b and f b b =g c . So p b b ;c =b; c and

0 0 1 0 1

0 0

0 0 0

b b ;c 2 B \ B C

0 1

which is equal to B by induction.

Corollary 20. If f : B ! C is a cokernel in the category of c-mo dules, with C

b ounded, then f is semistable. If, in addition, B ! = 0, then f is stable.

Corollary 21. Any exact sequence of b ounded mo dules is stable in the category of

c-mo dules.

The condition of Theorem 19 do es not characterize semistable cokernels. Let B

and C be R with B ! =B [p] and C ! = 0, and consider the map B ! C which

is multiplication by p. Clearly f is not isotop, but it is semistable b ecause of the

following theorem.

Theorem 22. If B is a c-group, and B ! is of nite length, then the map f : B !

C = B=B! is a semistable cokernel.

Pro of. Consider the pullback diagram

0 0

B ! ! B ! C

k g

B ! ! B ! C

0 0 0 0 0 0

We rst show that f B ! = C ! , so f B ! + m = C ! + m for each

0 0

p ositiveinteger m. Let c 2 C ! . Because B ! has nite length, B ! is nice as a

0 0 ! 0 ! 0 0 ! 0

submo dule of B ,sof p B =p C .Thus there exists b 2 B such that b; c 2 p B .

0 0 0 ! 0 0

As f b= g c = 0, wehave b 2 B ! , so b; c 2 B ! C ! \ p B = B !

0 0 0

and f b; c =c .

n 0 0

Nowcho ose n so that p B ! =0. We claim that B ! + n=C ! + nin

0 0 0 0 0

B C ,so f B = C for each ! + n, and therefore the map f is a

0 0 0 n 0

where cokernel. To see this, supp ose b; c 2 B ! + n. Then b; c =p b; c

n 0 0 0

= 0, wehave b 2 B ! , so b = p b = 0. Clearly 2 B ! . As f b =g c b ;c

0 0 0 0

0 0

C ! + n B ! + n.

Wewanttoshow that the converse of Theorem 19 holds for stable cokernels.

First we lo ok at some conditions that assure that f is isotop. Note that pB [p ] =

pB [p ].

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 21

Lemma 23. Let A B ! C b e a short exact sequences of mo dules. Then f is

isotop if and only if

! n k n

1. p C [p ] f p B [p ] for each k and n, and

2. if K C is a countable direct sum of torsion cyclics, then for each n there

m n n

exists m such that p K [p ] f B [p ].

m n+k

Pro of. Supp ose f is isotop. Then there exists m such that p C [p ]

n+k k m+k n k n

f B [p ]. Multiplying by p we get p C [p ] f p B [p ], so 1 holds. Condi-

m n n

tion 2 holds for any submo dule K of C : use the m such that p C [p ] f B [p ].

Conversely, supp ose 1 and 2 hold. Wewant to show that for each n there

m n n

exists m such that p C [p ] f B [p ]. First we show that, for n>1, there exists

m n n1 ! n

m such that if c 2 p C [p ] and p c 2 p C , then c 2 f B [p ]. By induction on n

there exists m such that

m n1 n1

p C [p ] f B [p ]:

Now

n1 ! m+n1

p c 2 p C [p] f p B [p]

n1 n1 m m+n

by 1. So p c = p f p b with b 2 B [p ]. Then

m m n1 n1

c f p b 2 p C [p ] f B [p ],

m n

and clearly f p b 2 f B [p ].

m n n

Supp ose now, bywayofcontradiction, that p C [p ] is not contained in f B [p ]

i n n1

for any m. Then we could construct c 2 p C [p ] inductively, with ht p c <

i i

n1 n

ht p c

i+1 i

Wewant to go from stability to condition 2 of Lemma 23. For n 1 de ne

i i1

B = Rx where x has order p and let P be B with all the p x identi ed.

n i i n n+1 i

Then p P is cyclic of order p, and there is a natural map taking P onto B

n n n n

! n1

with kernel p P .We can think of B as b eing the submo dule p B of B .Any

n 1 n n

automorphism of B extends to an automorphism of B . The mo dule B is the p -

1 n n

extension of B and may b e thoughtofasfx 2 D : p x 2 B g where D is a divisible

1 1

hull of B .

n1 n m n

Lemma 24. If B [p ] S B [p ], and S do es not contain p B [p ] for any m,

n n n

then there is an automorphism of B such that S P [p ].

n n n

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 22

Pro of. Consider the case n = 1, endowing B [p] with the top ology given by

the submo dules p B [p]. First we nd a prop er dense submo dule D of B [p] that

1 1

contains S . Let S b e the closure of S in B [p]. As B [p] is countably generated,

1 1

B [p]= S T as a valuated vector space. If S 6= S , then set D = S + T .IfS = S ,

then T cannot b e nitely generated b ecause S contains no p B [p]. Thus T contains

0 0

a prop er dense submo dule T . In this case set D = S + T .

Let e = p x ,so e ;e ;::: is the standard basis for B [p]. There is a nonzero

i i 1 2 1

linear functional ' on B [p] with 'D = 0. Because D is dense and prop er, it cannot

contain p B [p] for any m, so the set K = fk : 'e 6=0g is in nite. For each k in

1 k

K ,cho ose a unit u 2 R such that 'u e = 1. De ne an automorphism of B

k k k 1

k j

by setting x =u x if k 2 K , and x =x + p u x if j=2 K , and k is the

k k k j j k k

smallest elementof K bigger than j .

We see that ' e = 1 for all j ,so' P [p]=0as P [p] is generated by

j 1 1 1 1

the elements e e . But P [p] has co dimension 1 in B [p], so Ker ' = P [p],

i j 1 1 1 1 1

whence D P [p].

1 1

n1 n1 m+n1

For n>1 note that p S B [p], and that if p S p B [p], then

1 1

n1 m n

S = S + B [p ] p B [p ]

which is not so. So the case n = 1 obtains for p S , so there is an automorphism

of B such that p S P [p]. Extend to an automorphism of B , and

1 1 1 1 1 1 n

consider the commutative diagram.

! B ! B P

n n n

1 1

P ! B ! B

1 1 1

where the maps down are multiplication by p . Because p ker =0,wehave

n n1

P [p ] B [p ], so, as

n n

n1 n n1

p P [p ]= P [p] p S

n n 1 1 1

wehave S P [p ].

n n

Wenow derive a couple of consequences of the stability of a short exact sequence.

The rst is a bit technical.

Lemma 25. Let A B ! C b e a stable short exact sequence in the category of

c-mo dules, and g : C ! C . Let b e an ordinal, and n and k p ositiveintegers. If

0 0 n

+ ! and g x=0, then there exists y in C + k such that p y = x and x 2 C

g y 2 f B + k [p ].

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 23

Pro of. Consider the pullback diagram

0 0

A B ! C

k g

A B ! C

0 0 0 0 0

with B = fb; c 2 B C : f b=g c g.Asf is a semistable cokernel, B + !

0 0 k +n 0

maps onto C + ! , so there is an elementa; x 2 B + ! p B . Thus

n 0

a; x= p b; y with y in C + k and b 2 B + k . As g x=0,wehave a 2 A,

n 0 0

and since A B is a semistable kernel, a 2 A + ! , so a = p a with a 2 A + k .

n 0 0 0 n

Thus 0;x=p b a ;y and f b a =g y . But b a 2 B + k [p ].

Lemma 26. Let A B ! C b e a stable short exact sequence in the category of

c-mo dules, an ordinal, and k and n nonnegativeintegers. If K C is a direct

m n n

sum of torsion cyclics, then there exists m such that p K [p ] f B + k [p ].

Pro of. If K is b ounded, then the conclusion clearly holds. Wemay assume

that K is standard : one cyclic summand of each length. Indeed, if the conclusion

fails for each m, then it fails on a standard submo dule of K .

By induction on n, there exists m such that

m n1 n1

p K [p ] f B + k [p ]:

Wemay drop the cyclics of lengths less than m + n from K , without a ecting the

m n1 n1

conclusion of the theorem. Thus wemay assume that p K [p ]= K [p ]so

n1 n

K [p ] f B + k [p ]:

m n n

Supp ose, byway of contradiction, that p K [p ] is not contained in f B + k [p ]

for any m. Then Lemma 24 gives a homomorphism g of P onto K , with kernel

! n n

p P = hxi, so that f B + k [p ] \ K g P [p ]. Write P as S F for a forest

n n n

0 0 0 0

F , and extend F to a forest F such that F = p F . Let C = S F . Then g is

an o-mo dule map from P = C , to the c-mo dule C .We can extend g from F

0 0

to F , b ecause C is a c-mo dule, and so to an o-mo dule map C ! C , By Lemma 25

0 n n

there exists y 2 C + k P such that p y = x and g y 2 f B + k [p ], so

n n n

g y 2 g P [p ]. But if g y =g t, and p t = 0, then y t 2hxi so p y =0,a

contradiction.

Theorem 27. Let A B ! C b e a stable short exact sequence in the category of

c-mo dules, and a limit ordinal. Then the restriction f : B ! C is isotop.

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 24

! n n

Pro of. We rst show that p C [p ] f B + k [p ] for all n and k , which

is condition 1 of Lemma 23. Supp ose the torsion submo dule of C has in nite

! n

nal rank, and c 2 p C [p ]. Then there exists a sequence c of indep endent

n n

elements of C [p ] such that c 2 C + i has order p , and hci\hc ;c ;:::i =0.

i 0 1

Construct a direct sum K = Re of torsion cyclics in C such that p e = c .By

i i i

Lemma 26, the sequence c is eventually in f B + k [p ]. The sequence c + c is

i i

0 0

such another such sequence, and we can construct another direct sum K = Re of

i 0 n

torsion cyclics in C so that p e = c + c. Then c 2 f B + k [p ] by Lemma 26.

! n n

That is, p C [p ] f B + k [p ].

Now supp ose the torsion submo dule of C has nite nal rank, so it is b ounded

! n 1 n

plus nite-rank divisible. This implies that p C [p ]=p C [p ]. We will show

1 n n 0

with c-mo dule that p C [p ] f B + k [p ]. To do this we let C be R

0 0 0 n+1 1 n

structure given by C + ! =C [p]. Let z generate C [p ]. Given c 2 p C [p ],

0 n

let g : C ! C with g z =c. It follows from Lemma 25, with x = p z , that

c 2 f B + k [p ] the y you get there is a unit times z . That completes the pro of

! n n

that p C [p ] f B + k [p ] for all n and k .

The result now follows from Lemmas 26 and 23.

Corollary 28. A short exact sequence A B ! C in the category of c-mo dules is

stable if and only if A =A \ B , and the restriction f : B ! C is isotop,

for each limit ordinal .

7. Projective and injective c-modules

The full characterization of stability in the category of c-mo dules is not needed to

describ e the pro jectives and injectives.

Theorem 29. A c-mo dule is pro jective in the category of c-mo dules exactly when

it is the direct sum of a free mo dule and a divisible mo dule.

Pro of. Clearly free mo dules are pro jective in the category of c-mo dules. To

see that divisible mo dules are pro jective, let A B ! C b e a stable exact sequence

of c-mo dules. Then A1 B 1 ! C 1 is a split exact sequence Corollary 18.

As any map from a divisible mo dule into C go es into C 1, it follows that divisible

mo dules are pro jective.

For the converse, wemust show that anyvalue-reduced pro jective c-mo dule A is

a free mo dule. By Theorems 6, 17 and 19, A is a stable quotient of a mo dule, so A

is a summand of a mo dule, hence a mo dule. The torsion submo dule of A must b e

zero for otherwise A would have a torsion cyclic summand, and torsion cyclics are not

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 25

pro jective Corollary 21. So A is a reduced torsion-free mo dule, whence A! =0.

Any map from a free mo dule onto A is pure, hence a stable cokernel by Theorems 17

and 19, so A is free.

Note that there are not enough pro jectives in the category of c-mo dules. In fact,

C is the quotient of a c-pro jective exactly when C ! = C 1.

What are the c-injectives?

Theorem 30. If D is a divisible o-mo dule such that either D 1=D , or there is

a limit ordinal such that D =0and D =D , then D is injective in the

category of c-mo dules.

Pro of. Let A B b e a semistable kernel of c-mo dules, and f : A ! D .If

D 1=D ,we can extend f to a homomorphism B ! D b ecause D is divisible, and

the result will b e a map of o-mo dules. In the second case, f A =0.As A B is

semistable, A = A \ B . As D is divisible, we can extend f to a homomorphism

from B to D that kills B . Because D =D , this map is a map of o-mo dules.

Call the c-mo dules of Theorem 30 elementary c-injectives of typ e .We will

show that the c-injectives are pro ducts of elementary c-injectives.

Theorem 31. If G is an injective in the category of c-mo dules, then G 1 is

divisible for each nonzero ordinal including limit ordinals.

Pro of. Let = + n for a limit ordinal. For = 0 so n>0, consider the

inclusion pR R with the heightvaluation on b oth. This is a stable kernel b ecause

R! =0 and R=pR is b ounded Corollary 20. We can map p 2 pR to any element

of G, so when we extend to R we see that G = pG.

Now supp ose is a nonzero limit ordinal, and n>0. Set A = S F . Let F be

the forest obtained by taking F and F and letting the ro ot no de of height in

;! ;1

F b e another child of the no de x of height in F . This has the e ect of giving

;1 ;!

the no de p x height + i +1 in F for each i. Set B = S F . It is not hard to see

that A = A \ B for each ordinal < , so the inclusion A B is pure.

As B=A is b ounded, the inclusion A B is stable. As G is a c-mo dule, any

elementof G is the image of the no de x of value in A under some map A ! G,

n1 n1

so any elementof G 1 is the image of p x, hence is in G , as p x has value

in B .

If n = 0, construct A and B as ab ove, but value each p x, and the ro ot of F ,

. Nowany elementof G is the image of x under some map A ! G, with

hence is in pG b ecause x 2 pB .

FILTERED MODULES OVER DISCRETE VALUATION DOMAINS 26

Corollary 32. If G is injective in the category of c-mo dules, then G 1 is a

divisible summand for each nonzero ordinal .

Pro of. We can write G = G 1 K as a mo dule. But as G 1 is

comparable to G for each , this is a direct sum in the category of o-mo dules.

Theorem 33. Every c-injective is a pro duct of elementary c-injectives.

Pro of. Let A b e a c-injective. As A1= A for suciently large ,

we can split o A1by Corollary 32, so wemay assume that A is reduced. Set

A = A! for each ordinal .From Corollary 32, each A is a summand of A,

hence a c-injective and therefore an absolute summand of A.Thus A satis es the

hyp otheses of Theorem 14, so A is isomorphic to the pro duct A =A . Clearly

A =A is an elementary c-injectiveoftyp e ! .

There are not enough injectives in the category of c-mo dules. If A B is a stable

kernel, and B is a c-injective, then A is divisible for each limit ordinal .

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