Proceedings of the 44th Symposium on Ring Theory and ...

Example 10. Let R be a domain but not a field and let Q be a field <strong>of</strong> fractions <strong>of</strong> R.
We denote by S T or a subcategory consisting <strong>of</strong> torsion R-modules, that is
Then we shall see that one has
S T or = {M ∈ R-Mod | M ⊗ R Q = 0}.
(S T or , S f.g. ) (S f.g. , S T or ) = {M ∈ R-Mod | dim Q M ⊗ R Q < ∞}.
Therefore, a subcategory (S f.g. , S T or ) is a Serre subcategory by Corollary 8, but a subcategory
(S T or , S f.g. ) is not closed under extensions by Theorem 7.
First <strong>of</strong> all, we shall show that <strong>the</strong> above equality holds. We suppose that M is in
(S f.g. , S T or ). Then <strong>the</strong>re exists a short exact sequence
0 → X → M → Y → 0
<strong>of</strong> R-modules where X is in S f.g. and Y is in S T or . We apply an exact functor − ⊗ R Q to
this sequence. Then we see that one has M ⊗ R Q ∼ = X ⊗ R Q and this module is a finite
dimensional Q-vector space.
Conversely, let M be an R-module with dim Q M ⊗ R Q < ∞. Then we can denote
M ⊗ R Q = ∑ n
i=1 Q(m i ⊗ 1 Q ) with m i ∈ M and <strong>the</strong> unit element 1 Q <strong>of</strong> Q. We consider a
short exact sequence
n∑
n∑
0 → Rm i → M → M/ Rm i → 0
i=1
<strong>of</strong> R-modules. It is clear that ∑ n
i=1 Rm i is in S f.g. and M/ ∑ n
i=1 Rm i is in S T or . So M is
in (S f.g. , S T or ). Consequently, <strong>the</strong> above equality holds.
Next, it is clear that M ⊗ R Q has finite dimension as Q-vector space for an R-module
M <strong>of</strong> (S T or , S f.g. ). Thus, one has (S T or , S f.g. ) ⊆ (S f.g. , S T or ).
Finally, we shall see that a field <strong>of</strong> fractions Q <strong>of</strong> R is in (S f.g. , S T or ) but not in
(S T or , S f.g. ), so one has (S T or , S f.g. ) (S f.g. , S T or ). Indeed, it follows from dim Q Q⊗ R Q =
1 that Q is in (S f.g. , S T or ). On <strong>the</strong> o<strong>the</strong>r hand, we assume that Q is in (S T or , S f.g. ). Since
R is a domain, a torsion R-submodule <strong>of</strong> Q is only <strong>the</strong> zero module. It means that Q
must be a finitely generated R-module. But, this is a contradiction.
i=1
References
[1] M. Aghapournahr and L. Melkersson, Local cohomology and Serre subcategories, J. Algebra 320
(2008), 1275–1287.
[2] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.
[3] T. Yoshizawa, Subcategories <strong>of</strong> extension modules by Serre subcategories, To appear in Proc. Amer.
Math. Soc.
Graduate School <strong>of</strong> Natural Science and Technology
Okayama University
3-1-1 Tsushima-naka, Kita-ku, Okayama 700-8530 JAPAN
E-mail address: tyoshiza@math.okayama-u.ac.jp
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