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ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА
2017 Математика и механика № 46
УДК 512.742
DOI 10.17223/19988621/46/3
Peter V. Danchev
ON ^"Bext PROJECTIVE ABELIAN ^-GROUPS1
We introduce the concept of p"Bext projective abelian p-groups and show that they form a class which properly contains the class of all и-balanced projective p-groups. This somewhat enlarges a result due to Keef-Danchev in Houston J. Math. (2012).
Keywords: balancedprojectives, n-balancedprojectives, p"Bextprojectives.
1. Introduction and Background
Everywhere in the text of this brief paper our groups are /»-primary abelian, where p is a fixed prime for the duration of the article. The undefined explicitly below notions and notations are in agreement with [5]. For instance, a group G is called balanced projective if the equality Bext(G, X) = {0} holds for all groups X. In order to generalize this, imitating [3], for any integer n > 0, we say that the short exact sequence 0 ^ X ^ Y ^ G ^ 0 is n-balanced exact if it represents an element of pnBext(G, X). Thus we will say that a group G is n-balanced projective provided
every such n-balanced exact sequence splits. Evidently, these two notions coincide when n = 0.
It is worthwhile noticing that certain non-trivial properties of these groups are given in [3] (see also [4]). These ideas lead us to the next new concept:
Definition 1.1. Let n > 0 . A group G is said to be pnBext-projective if
(VX), pn - Bext(G, X) = {0}.
The aim of this note is to prove that each n-balanced projective group is pn yields Bext-projective but the converse fails. We close the work with a specific question arisen from unexpected difficulties in the proof of the central statement.
2. Main Result and Problem
Theorem 2.1. Suppose that G is a group and n <ra is a natural. If G is n-balanced projective, then it is pnBext-projective.
Proof. Letting the short exact sequence E defined by
f
0 ^ X ^ B ^ G ^ 0 is in pnBext(G, X), then there is another element E' of Bext(G, X) given by
f'
0 ^ X ^ B' ^ G ^ 0 such that the following pull-back diagram can be completed:
1 2010 Mathematics Subject Classication. 20 K10.
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Peter V. Danchev
f
0 ^ X ^ B ^ G ^ 0
|| i ipn
f'
0 ^ X ^ B'^ G ^ 0.
Let H(G) be the standard functorial group, depending on G, as defined in [7] (cf. [3] too). If nG : H(G) ^ G is our usual homomorphism used in defining EG, then
pn(nG(G[pn])) = {0}, so pnnG induces a homomorphism H(G)/G[pn] ^ G . Since E' is balanced exact and H(G)/ G[pn ] is totally projective, there is a homomorphism g' : H(G) ^ B' such that f ° g' = pnnG . Now the well-known universal properties of pull-back diagrams yield that there exists a homomorphism g : H(G) ^ B B such that f ° g = nG . However, this means that E is n-balanced exact. Since we are assuming G is n-balanced projective, it follows now that E splits, as wanted. □
Example 2.2. There is a pBext-projective group which is not 1-balanced projective. Proof. Referring to [6] there exists a summable Cffli -group A which is a
properp®1+1 -projective group (thus it is manifestly not totally projective by virtue of [5]). Moreover, since it is summable, it follows that it is also not 1-balanced projective. However, on the other side, since A is a C® -group, it follows that
Bext( A, X) = pffllExt(A, X) for all groups X (compare with [5]). And finally, because
it is a p®1+1 -projective as well, we can conclude that A has to be pBext-projective, as claimed. □
A reasonable query is whether or not for any n > 2 does there exist a pnBext-projective group that is not n-totally projective? Resuming, we have restricted our attention only on n = 1, though essentially the same argument works for larger values of n (see cf. [1] and [2] too). In fact, last argument stated above asserts that any element of pn Bext( A, X )will be n-balanced exact, so that every group which is projective with respect to the collection of n-balanced exact sequences will also be projective with respect to the functor pnBext. The second assertion then implies that there are n-balanced exact sequences that are not elements of pnBext(A, X).
Besides, notice that the totally projective (i.e., the balanced projective) groups are exactly p0Bext-projective groups, and there are an abundance of them. Nevertheless, it is actually not at all clear whether there are enoughpnBext-projectives whenever n > 0. So, the following homological question is of some interest:
Problem. Is it true that the collection of n-balanced exact sequences form the largest subfunctor of pnBext which does have this important homological property?
REFERENCES
1. Danchev P.V. (2012) Valuated pn-socles and C^ n-summable abelian p-groups. Pioneer J. Math. and Math. Sci. 6. pp. 233-249.
2. Danchev P.V. and Keef P.W. (2010) n-Summable valuated pn-socles and primary abelian groups. Commun. Algebra. 38. pp. 3137-3153.
On pnBext projective abelian p-groups
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3. Keef P.W. and Danchev P.V. (2012) On n-simply presented primary abelian groups. Houston J. Math. 38. pp. 1027-1050.
4. Keef P. and Danchev P. (2012) On properties of n-totally projective abelian p-groups. Ukrain. Math. J. 64. pp. 875-880.
5. Fuchs L. (2015) Abelian Groups. Switzerland: Springer.
6. Hill P. (1968) A summable Cn-group. Proc. Amer. Math. Soc. 23. pp. 428-430.
7. Richman F. and Walker E. (1979) Valuated groups. J. Algebra. 56. pp. 145-167.
Received: 08.02.2017
Peter V. DANCHEV (Professor, Mathematical Department, Plovdiv State University, Bulgaria) E-mail: [email protected]
Данчев П.В. (2017) О pnBext ПРОЕКТИВНЫХ АБЕЛЕВЫХ p-ГРУППАХ. Вестник Томского государственного университета. Математика и механика. № 46. С. 21-23
DOI 10.17223/19988621/46/3
Вводится понятие pnBext проективных абелевых p-групп и доказывается, что эти группы образуют собственный подкласс в классе всех n-сбалансированных проективных p-групп. Данное утверждение улучшает соответствующий полученный результат, опубликованный Кифом и Данчевым в журнале Houston J. Math. (2012).
Ключевые слова: сбалансированная проективность, n-сбалансированная проективность, pnBext проективность.
