𝑇-faithful subcategories and localization

Beachy, John A.

doi:10.1090/S0002-9947-1974-0364322-4

$T$-faithful subcategories and localization
HTML articles powered by AMS MathViewer

by John A. Beachy PDF

Trans. Amer. Math. Soc. 195 (1974), 61-79 Request permission

Abstract:

For any additive functor from a category of modules into an abelian category there is a largest Giraud subcategory for which the functor acts faithfully on homomorphisms into the subcategory. It is the largest Giraud subcategory into which the functor reflects exact sequences, and under certain conditions it is just the largest Giraud subcategory on which the functor acts faithfully. If the functor is exact and has a right adjoint, then the subcategory is equivalent to the quotient category determined by the kernel of the functor. In certain cases, the construction can be applied to a Morita context in order to obtain a recent theorem of Mueller. Similarly, the functor defines a certain reflective subcategory and an associated radical, which is a torsion radical in case the functor preserves monomorphisms. Certain results concerning this radical, when defined by an adjoint functor, can be applied to obtain two theorems of Morita on balanced modules.

References

Gorô Azumaya, Some properties of $\textrm {TTF}$-classes, Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972) Lecture Notes in Math. Vol. 353, Springer, Berlin., 1973, pp. 72–83. MR 0338073
John A. Beachy, Generating and cogenerating structures, Trans. Amer. Math. Soc. 158 (1971), 75–92. MR 288160, DOI 10.1090/S0002-9947-1971-0288160-3
John A. Beachy, Cotorsion radicals and projective modules, Bull. Austral. Math. Soc. 5 (1971), 241–253. MR 292879, DOI 10.1017/S0004972700047122
Kent R. Fuller, Density and equivalence, J. Algebra 29 (1974), 528–550. MR 374192, DOI 10.1016/0021-8693(74)90088-X
A. G. Heinicke, Triples and localizations, Canad. Math. Bull. 14 (1971), 333–339. MR 318229, DOI 10.4153/CMB-1971-061-2
J. P. Jans, Some aspects of torsion, Pacific J. Math. 15 (1965), 1249–1259. MR 191936
Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, Lecture Notes in Mathematics, Vol. 177, Springer-Verlag, Berlin-New York, 1971. With an appendix by H. H. Storrer on torsion theories and dominant dimensions. MR 0284459
Joachim Lambek, Localization and completion, J. Pure Appl. Algebra 2 (1972), 343–370. MR 320047, DOI 10.1016/0022-4049(72)90011-4
Barry Mitchell, Theory of categories, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. MR 0202787
Kiiti Morita, Localizations in categories of modules. I, Math. Z. 114 (1970), 121–144. MR 263858, DOI 10.1007/BF01110321
Kiiti Morita, Flat modules, injective modules and quotient rings, Math. Z. 120 (1971), 25–40. MR 286833, DOI 10.1007/BF01109715

The quotient category of a Morita context

Bo Stenström, Rings and modules of quotients, Lecture Notes in Mathematics, Vol. 237, Springer-Verlag, Berlin-New York, 1971. MR 0325663
R. G. Swan, Algebraic $K$-theory, Lecture Notes in Mathematics, No. 76, Springer-Verlag, Berlin-New York, 1968. MR 0245634
Hiroyuki Tachikawa, On splitting of module categories, Math. Z. 111 (1969), 145–150. MR 246923, DOI 10.1007/BF01111195