The global dimension of certain proper classes

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The global dimension of certain proper classes

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Communications

of the Moscow Mathematical

205

Society

The global dimension of certain proper classes R.G. Alizade Let 31 be a proper class of short exact sequences of R-modules, where Λ is a ring with identity (see [1 ] , [3] , [4]). The projective dimension of a module C over Μ is given by pr. d i m - . C — min{»i| E i t ' ^ J " 1 (C, *4) = Ofor any module A }• If there is no such n, we put .9? .71 p r . d i m _. C = oo. The injective dimension i n j . d i m ^ ^ l is defined similarly. Finally, the global dimension of .·# is given by: gl. d i m Si = s u p p r . d i m _. C = s u p i n j . d i m -. A. C "H Α - E s t M C -4)}, where the union is taken over all °^ P.a J Ρ 6 3 and all homomorphisms a : C -> P. Then ah' is a proper class. Proposition. If for the above class άΚ proper submodules of coprojective modules arc coprojective, then g l . d i m Sfi are similar). Let Λ·32

5 = £ 1 3 o£ - 2 °£'i be any (^-proper sequence of length 3. Since Εχ 6 if, it follows that A-£j = Jc^E]) = £ [ 6 # for some A: # 0. Since fc3 is a monomorphism, it is sufficient to prove that kJ(S) 3 ί Έ £ , ο £ , ο ί ; = O.£ 2 € if, and so m^fj) = E'2 6 ^ for some m. Since E'\ € # , it is divisible by m, that is, miiE'{) = E\ for some E'{. It is clear that E\ € c/ a nd S' = S" E f j i i J . f j ' . Since m1 {E'{) = miii 1 ]') = ^ ί , it follows from the injectivity that we only need to prove that m3(S") = £"3°£2 °£"ϊ = 0. But this is so, because Ei, E[ € if and gl. dim of = 1. α The short exact sequence 0 -*~ Ζ ρ -ν Ζ ρ β ο -»- 2 ρ 0 ο —*- O ( i n ^ ) induces an exact sequence: . . . - E x t ^ ( Z p 0 0 , Z) J$

E x t ^ (Zp,

Z) - E x t ^ ( 2 ρ β 0 ,

Ζ) - . . .

l

It is easy to verify that Ext (Zpooi Z) is torsion-free, so that E x t ^ Z p » , Z) = Ext^iZpoo.Z) = ZHExtHZpoo.Z)), and E x t 1 * (Zp«,, Z ) = 0. Clearly, Exfi (Z p ,Z) = Ext^Zp, Z) is a non-trivial ι educed group. Then otl is always the null homomorphism. Hence, it follows from the exact sequence that Ext= (Zpoo, Ζ) φ 0. Thus, gl. dim 31= 2, where 31 is any of the classes ifB, d°, or if. In conclusion, the author expresses his deep gratitude to E.G. Sklyarenko for his help with this work. References

[1] S. Maclane, Homology, Springer-Verlag, Berlin-Gottingen-Heidelberg 1963. MR 28 # 122. Translation: Gomologiya, Mir, Moscow 1966. [2] L. Fuchs, Infinite Abelian Groups. I, Academic Press, New York-London 1970. MR 41 # 333. Translation: Beckonechnye abelevy gruppy. I, Mir, Moscow 1972. [3] A.P. Mishina and L.A. Skornyakov, Abelevy gruppy i moduli (Abelian Groups and Modules), Nauka, Moscow 1969. MR 43 # 1960. [4] E.G. Sklyarenko, Relative homological algebra in the category of modules, Uspekhi Mat. Nauk 33:3 (1978), 85-120. MR 58 # 16777. = Russian Math. Surveys 33:3 (1978), 97-137. [5] T. Kirk, On one class of purities, Comment. Math. Univ. Carol. 14:1 (1973), 139-154. [6] R.G. Alizade, The inheritance of the properties of coprojectivity and co-injectivity, Izv. Akad. Nauk Azerbaidzhan. SSR 5 (1983), 7-12. [7] S.N. Fedin, The pure dimension of rings and modules, Uspekhi Mat. Nauk SSSR 37:5 (1982), 203-204. MR83m:16024. = Russian Math. Surveys 37:5 (1982), 170-171. [8] F. Okoh, Hereditary algebras that are not pure-hereditary, Lecture Notes in Math. 834 (1983), 432-437. [9] C.L. Walker, Properties of Ext and quasi-splitting of Abelian groups, Acta Math. Acad. Sci. Hung. 15 (1964), 157-160. MR 29 # 4798. [10] N. Hart, Two parallel relative homological algebras, Acta Math. Acad. Sci. Hung. 25 (1974), 321-327. MR 51 # 753. Inst. Math. Mech. Acad. Sci. Azerbaidzhan SSR

Received by the Board of Governors 22 March 1984