ABELIAN GROUPS IN A TOPOS OF SHEAVES : TORSION AND ESSENTIAL EXTENSIONS

We investigate the properties of torsion groups and their essential extensions in the category AbShL of Abellan groups in a topos of sheaves on a locale. We show that every torsion group is a direct sum of its p-prlmary components and for a torsion group A, the group [A,B] is reduced for any BeAbShL.. We give an example to show that in AbShL the torsion subgroup of an inJectlve group need not be inJectlve. Further we prove that if the locale is Boolean or finite then essential extensions of torsion groups are torsion. Finally we show that for a first countable hausdorff space X essential extensions of torsion groups in AbSh0(X) are torsion iff X

some more results about essential extensions of torsion groups, we conclude our paper by showing that for a first countable Hausdorff space X, essential extensions of torsion groups in AbShX are torsion groups iff X is discrete (Theorem 4.8).For basic facts about about abelian groups with which this paper is concerned see [3] and [4].
Details concerning presheaves and sheaves on a locale can be found in [5], category theory in [6] and topos theory in [7]. 2. BACKGROUND (2.1) Recall that a locale denoted by L is a complete lattice satisfying the following distribution law; u ^for all U, and any family (Ui}is I in L. The zero (= bottom) of L will be denoted by O, and the unit (= top) of L by E. A m0.ghism of locales h: L + M (also called local lattice homomorphism) is a map which preserves arbitrary Joins and finite meets (hence preserves the zero and the unit).
An obvious example of a locale is the topology OX (that is the lattice of open sets) of any topolocial space X with Joins as unions and meets as intersections.
REMARKS.A locale L satisfies both the Ascending and Descending Chain Conditions iff L is finite.
To prove the non-trivial implication note that such an L is spatial [8] and if L 0(X) and X is TO, one has the following observations concerning X: Each x s X has a smallest open neighbourhood W and for the partial x order g given, such that x g y(x,ysX) iff 0(x)0(y) (hence iff WyEWx), Wx /x {yly x}.t4reover DCC for L then implies that /x is finite, and since X is compact by ACC, X itself is finite.It follows that L is also finite.
(2.2) ABELIAN GROUPS IN A CATEGORY If E is any finitely complete category then by AbE one means a category with objects as abelian groups in E and maps as homomorphisms between them [9].For A E AbE and 0nN, (1) The diagonal map AA: 1A for all f=l,2,, ,n, where pf: / A fs the fth proJectfon +A qi

+A (if) The sum A n
A is the unique map such that A + A n + A 1A where qi A + A n is the fth injection for f=l,2...n.The composition +A AA: A A n A f s denoted by n A and the kernel of n A shall be denoted by :n" AbE is called a t_pr_s_fon !i fiE n A fs a monomorphfsm for all 0 n s N. (2) A s AbE is called a torsion group_ fff all k 0 n N are jointly epic, that is, for any two homomorphisms f and g with domain A, if fk =gk for all 0nsN then n n f;g.
2.4.Recall that by AbPShL and AbShL one means the categories of Abelfan groups fn the.topos PShL and ShL of presheaves and sheaves, respectively, on a locale L with values in the category Ab of abelian groups.For any U, V L and A e AbShL,AU will denote the component of A at U and if V U the restriction map AU / AV will be written as a alV.If A is the sheaf reflection of the given presheaf B (also denoted by A=) then we shall write AU BU.Also if h: A B is a morphism in AbShL then its component at U e L is denoted by AU BU.

NOTE. AbSh2
Ab for the two-element locale 2 and if X is a discrete topological space then AbShX Ab IXI.Further AbSh3 for the three-element locale is the same as AbPSh2 that is the arrow category of Ab.Further AbSh3 is also AbShS for the Sierpinski space S with points 0 and and non-trivlal open set {I}. 2.5.Recall that for any local lattice homomorphism #: L M we get a pair of , adjoint functors AbshM #/ AbShL where (#,A)U A(#(U)) for U L, and for any V e M /, , # , (# C)V tCW (W e L).Then # is left exact, left adJolnt to #,.As a special (w)v case we get for each U L a pair of adJolnt functors :AbShL AbSh+U and Eu:AbSh+U AbShL defined by (A)W A(WAU) and is left adjoint left exact to .We shall also denote RuA by Further preserves all limits and co-llmlts.[A,B]V (V U), given by h (hw)w U hlV (hw)w v [I0].
2.7.In (2.3) we described what we mean by torsion free and torsion groups in AbE.For the case E ShL, we have the following: (I) A AbShL is a torsion free group iff each AU is torsion free in Ab.
(2) A AbShL is a torsion group iff A t Kern A That is for a AU, there 0#hEN exists a cover U ViiUi, and 0 m.1 e Z, such that mialU i, 0 for all iel.
PROPOSITION 2.8.For any Ue L the functors and E,, p erve torsion groups.
PROOF.Let A e AbShL which is a torsion group.Then A it Kern A since On:N R U preserves all co-limlts and limits (2.5), it follows RuA t Ru(Ker n A) t Ker nAIU, hence AIU is torsion in AbSh +U.By a similar 0#neN OneN argument it can be shown that E U preserves torsion groups.
3. TORSION GROUPS.THEOREM 3.1.A AbShL is a torsion group iff there is a cover E VielUi such that AIU i, is torsion in AbSh+U i for all il.PROOF.
(/) Clear by taking the trivial cover of E. On the other hand if sll AIU i are torsion groups in AbSh+Ui, we claim A is torsion.So consider any be AU, U L. Then U Viel(U A UI) and b (U A U i) eA(U A Ui)= AIUi(UAUi for all ie I. all ie I.
But A[U i is torsion in AbSh+Ui, and so for each i I, there is a cover U U i =VjEJi Wji and 0 nji N such that njlb[Wji 0, j e Ji" Hence for b AU, we can find a cover U =JeJi iel Wji such that njiblWi 0 for all i, J, 0 ni N, which shows that A is a torsion group in AbShL.
Proposition 3.1 shows that torsi6n is a local property.However it is not a global property as we shall see from the following counter example: Consider L +I and A AbShL given by n By Proposition 3.1, A is torsion, since for the cover m Vn<mn the group Aln kn Z/kZ is torsion in AbSh+n for all n < m.But Am nmZ/nZ is not torsion in Ab, as the element (l+nZ)n< m does not have a finite order.DEFINITION 3.3.For a given prime p, by the p-primary component of a group A e n AbShL we mean the subgroup of A given by U 0nN Ker p A We denote the p-primary component of A by A AeAbShL is called a p-primary group if A=A P P DEFINITION 3.4.By the torsion subgroup B of an any group AAbShL we mean the subgroup of A given by B UonENKer n A.
THEOREM 3.5.Every torsion group is a direct sum of its p-prlmary components.PROOF.Let A be a torsion group and denote by B the presheaf BU t(AU) the torsion subgroup of AU.Then A is the sheaf reflection of ,0".--.>.Now BU t(AU)--(t(AU))p where (t(AU))p denotes the p-prlmary component of t(aU).A and hence we get A A P P P P P DEFINTION 3.6.By the torsion t_of a group A we mean the set of all prime numbers p such that A # 0. P PROPOSITION 3.7.
If A is a torsion group and B A is an essential extension then B and A have the same torsion type.
PROOF.Hence B 0 iff A 0 which means that A and B have the same torsion P P type.
DEFINITION 3.8.We call an A AbShL to be a reduced group if it has no non zero inJectlve subgroups.Recall that in the category Ab, for any torsion group B the group Hom(B,K) is reduced for all K Ab.
We shall prove the analogue of thls for the Ab-valued hom-functor H and the internal hom-functor [-,-] of AbShL(2.6).
LEMMA 3.9.If A AbShL is a torsion group then H(A,P) is reduced in Ab for all P AbShL.
PROOF.Let 0 C, H(&,P) be an injective subgroup.Consider any 0 C, then for some U L and A AU, =u(a) O. Since A is torsion and a AU there exists a cover U VII Ui and 0 n i N such that nialUi 0 for all i I.But u(a) 0 implies that k(a[Uk) 0 for some k I. Consider now 0 nkE N, then C an inJectlve hence divisible group in Ab implies that there exists some 8 C such that nk8 .Therefore nkUk (alU k) 8Uk(nkalU k) SUe(0) 0, which means Uk(alU ) 0, a contradiction, hence C 0 which shows that AbShL(A,P)=H(A,P) is reduced in tIe category Ab.
If A Is a torsion group in AbShL, then [A,P] is reduced in AbShL for all P AbShL. PROOF.
Let 0 B [A,P] be an Injectlve subgroup.
Then for ome U E L, BU 0 is an injectlve subgroup of [A,P]U H+u(AIU,PIU).Since A is torsion, it follows AIU is torsion (2.8) in AbSh+U.andso by last lemma H+u(AIU,PIU) is reduced in Ab.Thus BU 0 for all U g L, hence B 0 which means [A,P] is reduced in AbShL.

REMARK.
Recall that in the category Ab, the torsion subgroup of an inJectlve group is always inJectlve.We show in the following example that, for an arbitrary L, the torsion subgroup of an inJectlve group need not be inJective, except for some special locales which we shall discuss in the next section.where the Pi are finite groups with increasing exponent.By one of our previous results ([I], proposition 2.3) the inJective hull of A is given by the group B n<E (Pn) d H n<E(en) +...+ E(P2 x E(P I) E(P I) 0 where E(P I) denotes the fnJective hull of group PI in Ab.If TB is the torsion subgroup of B, then (TB)n Bn all n < , and so E(P ).
and since A TB, it follows TB is not inJective since B, being the inJective hull of A, is the minimal inJectlve extension of A, hence the result.

ESSENTIAL EXTENSIONS OF TORSION GROUPS.
If for any torsion group A AbShL, all essential extensions of A are torsion, then we say that essential extensions in AbShL preserve torsion.
The followlng proposition shows "essential extensions preserve torsion" is a local property.PROPOSITION 4.1.Essentlal extenslons preserve torslon in AbShL iff there exists a cover E =VlIUi such that essential extensions preserve torsion in AbSh/U I for all i I.
PROOF.(+) Clear by taking the trivial cover of E. For the converse, consider any essential extension B of the torsion group A in AbShL.Since for each i I the AbShL / AbSh+U i, preserves essential extensions and torsion [I] it functor i: follows BIU I is an essential extension of the torsion group AIU i in AbSh+U I.By hypothesls, BIU i is torsion in AbSh+U i all i e I, hence by Theorem 2.1, B is torsion in AbShL.
PROPOSITION 4.2.For any L, essenltal extensions in AbShL preserve torsion iff every injectlve group splits into a direct sum of a torsion group and a torsion free group.PROOF.(/) Let B denote the torsion subgroup of an injectlve group A AbShL.

If C
B is any essential extension, then by hypothesis C i a torsion group.Since A is injectlve we may assume that C c__ A, so C torsion implies C c__ B and hence C B.
Thus B has no proper essential extensions whlch means that B is injective.Therefore A BeE for some subgroup E of A. If TE denotes the torsion subgroup of E, then TE c__B and so TEC_B E 0, hence TE 0. Thus E Is torsion free.
(+) Let P be a torsion group and H the inJectlve hull of P. By hypothesis H T e F where T is a torsion group and F is a torsion free group.If F # O, then since H is an essential extension of P it follows that P F 0, a contradiction, since P is torsion.
Hence F 0 which shows that H is a torsion group.Since every essential extension of P has an embedding into H, it follows all essential extensions of P are torsion, hence the result.

PROOF.
Consider an essential extension B of a torsion group A in AbShL.Let C denote the torsion subgroup of B(3.4).For any U L consider an arbitrary element g U be the largest element in +U such that blW CW.We claim W is b BU.Let W dense in +U.If not, then there exists S +U, S 0 such that S A W 0. Now for V S, blV E CV gives V W and so V + V A W S A W 0 implies V 0. In any his # 0. Since B D_ A is an essential extension therefore there exists particular a S and m g Z such that 0 mblV g AV c__ CV.Now C is the torsion subgroup of B V and 0 # mblB CV implies blV g CV.But then V O, a contradiction, since 0 # mblV E AV. Hence W is dense in +U.Since L is Boolean we have W U, thus BU c_ CU for all U g L and so B C. Hnce B is torsion.

REMARK.
On the other hand, one can see that if essential extensions preserve torsion in AbShL, then it does not necessarily follow that L is Boolean.Here is a counterexampl e: B Consider L 3. If B +lh is torsion in AbSh3, then both B and B 2 B 2 are torsion in Ab.By ([I], Proposition 2.3) the inJectlve hull of B is given by A E(B 2) x E(Ker h) E(B 2 which is torsion in AbSh3.Hence Hence all essential extensions of B are torsion, although L 3 is not Boolean.Of course the remark is a special case of the following more general result which shows that there are non-Boolean L such that essential extensions in AbShL preserve torsion.THEOREM 4.4.For any finite L essential extensions in AbShL preserve torsion. PROOF.
Let B be any essential extension of the torsion group A. Then for an arbitrary a e AU, U e L. A torsion implies that there is a cover U U V U2 V...V U k and 0 n i e N such that nalUi 0 for all i 1,2,...,k.If m nln2...nk then malU i 0 for all i and therefore ma 0 and m # 0. This shows for each U e L, AU is a torsion group in the category Ab.Now, if there are V e L such that BV is not a torsion group then let S be minimal such that BS is not torsion.
Then S 0 and for all U < S, BU is a torsic. ,..roup in Ab.If W U<S U, then since each BU is torsion it follows by prop.>:r 3.1 that B[W is torsion in AbSh+W.By the same argument as above it follows BW is torsion and hence But BW is torsion and so for some 0 n N, nblW--0.But 0 # nb BS is again of so by the same argument 0 # nblW a contradiction.Hence BS is a infinite order and torsion group which contradicts the definition of S. This shows B is a torsion group in AbShL.
REMARK 3.4.Recall from (2.1) that the finite locales L are exactly those L in which both ACC and DCC hold.It is therefore of interest to note that there exists an L which satisfies DCC but for which essential extensions in AbShL do not preserve torsion.
Here is an example which is actually the same as that considered in (3.11)   for a different purpose: If A and its injectlve hull B _ A are as in 3.11, then B Is not torsion because its torsion subgroup Is proper.
If essential extensions preserve torsion in AbShL, then for all U L, the following are true: (i) Essential extensions preserve torsion in AbSh+U.
PROOF.Let B be any essential extension of the torsion group A in AbSh+U.Since the functor : AbSh+U AbShL preserves essential extensions [I] and also torsion (2.9), it follows EuB is an essential extension of the torsion group A. By hypothesis EuB is torsion in AbShL.Therefore (EuB) B is again torsion since the functor preserves torsion (2.9), hence the result.
(ll) Consider the local lattice homomorphlsm : L U given by (W) W V U.
Then produces ,: AbSh/U AbShL(2.6)where (,A)W A(U V W), W L Let B be an essential extension of the torsion group A in AbSh+U.We claim that B is torsion.We first show that , preserves torsion.Let 0 a (,A)W A(U V W).
Since A is torsion, there is a cover (U V W) V iEl U I in /U, and 0 ni N such that nialUi 0 for all i I.So we can for a cover W (U V W) W VIEI (U i^W ) in L such that (nla) IUi ^W) 0 all i E I. Hence for 0 a e (,A)W, we can always find a cover W VII (U i^W ) in L, such that 0 nlal(U i ^W), and that proves ,A is torsion in AbShL.
To show that preserves essential extesnlons take 0 b in (,B)W B(W V U), W L. Since B _ A is essential in AbSh/U.there exists V W V U and m Z such that 0 mblV AV.But U < V and V WVU implies V (V A W) V U and therefore 0 mb I(VVW) V U e A((VA W) V U).Thus for 0 b E @,B)W, there is (V AW).<_ W such that 0 # mblV A W ,A) (vAw) for some m Z.This shows ,B is an essential extension of ,A in AbShL.Finally we show that , reflects torsion.
So, let ,P be a torsion group in AbShl for some P AbSh/U.If 0 a e PW, W U, then 0 a e (,P)W P(W U) PW, and so ,P being a torsion group implies, that there is a cover W VielWI in L, and 0 n.1 N such that nlalWI 0 all i g I, where alWle (,P)WI P(W i V U).If we consider the cover W Vig I (W i V U) in /I.then we get 0 nlal(W i V U) for all i I, which proves that P is torsion in AbSh+U.Thus, in order to prove (ll), we consider an essential extension D of the torsion group C in AbSh+U.Then by the above argument ,D is an essential extension of ,C in AbShL.But ,C is torsion since C is torsion, hence

2. 6 .
Besides the obvious external Ab valued hom-functor H HL: AbShL pp x AbShL Ab, AbShL also has an internal hom-functor [-,-]: AbShL pp x AbShL AbShL, for which [A,B]U H+u(A[U,B[U) with the restriction maps [A,B]U If B B P is the subpresheaf B U (t(AU)) then clearly B ,B in AbPShL.The Sheaf P P P reflection being a left adjoint preserves co-llmlts, in particular direct sums and so A follows A c B and therefore A c A O B for all W<S.Consider an arbitrary b BS of infinite order.Since B _ A is an essenltal there exists V S and 0 # m Z such that 0#mb[V e AV.Then V # extension, S, for otherwise 0 # mb AV has finite order and so b will have finite order, a since b has infinite order.Hence V W. This implies b[W # 0.contradiction,