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Abstract. We Consider a Uni Ed Setting for Studying Lo Cal Valuated 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 1 . b e denoted by R p 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 0 inside the group B in the sense that if the subgroups A and A are isomorphic as v- 0 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 n 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 = < T +1 S . In particular, S = S , and, if is a limit ordinal, then S = < +1 If X = X , then X = X for each > . The length of X is the least +1 such that X = X . A forest is torsion if for each x there is n such that n x =0.Ifx is a no de in a forest, then the order,or exp onent,of x is the smallest n 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- ;n structed from ordinals and n, where n ! . A no de of the forest F is either ;n 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 0 de ned by a a :::a = a :::a if m 1, 1 2 m 2 m t = t . k minn;k +1 Clearly F is a forest of length + n.If1 n<!, then F is torsion with the ;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 1 . 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. +1 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 ;n 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. ;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. f 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 . f 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 T O - ltration. Set G1= G .
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