An equivalence induced by Ext and Tor applied to the finitistic weak dimension of coherent rings

Wilkins, Elwood

Glasgow mathematical journal, Tome 40 (1998) no. 2, pp. 167-176

Résumé

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.

Détail

DOI : 10.1017/S0017089500032481

Bibliographie

[1] 1.Auslander, M., A functorial approach to representation theory, in Representations of algebras, eds. Auslander, M. and Lluis, E., Lecture Notes in Mathematics 944 (Springer-Verlag, 1982), 105–179. Google Scholar | DOI

[2] 2.Auslander, M., Isolated Singularities and almost split sequences, in Representation Theory II, eds. Dlab, V., Gabriel, P., Michler, G., Lecture Notes in Mathematics 1178 (Springer-Verlag, 1987), 194–242. Google Scholar

[3] 3.Auslander, M. and Reiten, I., Representation theory of Artin algebras III. Almost split sequences, Comm. Algebra 3 (1975), 239–294. Google Scholar

[4] 4.Auslander, M., Reiten, I. and Smalø, S. O., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36 (Cambridge University Press, 1995). Google Scholar

[5] 5.Bass, H., Finitistic dimension and a homological generalisation of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. Google Scholar | DOI

[6] 6.Jensen, L. Gruson et C. U., Dimension cohomologiques reliees aux foncteurs lim , in Sém. d'Algebre, Malliavin, P. Dubreil et M.-P., eds, Lecture Notes in Mathematics 867, (Springer-Verlag, 1981), 234–294. Google Scholar

[7] 7.Herzog, I., The Ziegler spectrum of a locally coherent Grothendieck category, Proc. London Math. Soc. (3) 74 (1997), 503–509. Google Scholar | DOI

[8] 8.Jensen, C. U. and Lenzing, H., Model theoretic algebra (Gordon and Breach, 1989). Google Scholar

[9] 9.Krause, H., The spectrum of a locally coherent category, J. Pure Appl. Algebra, 114 (1997), 259–271. Google Scholar | DOI

[10] 10.Krause, H., Finitistic dimension and the Ziegler spectrum, Proc. Amer. Math. Soc., to appear. Google Scholar

[11] 11.Prest, M., Model theory and modules, LMS Lecture Notes Series (Cambridge University Press, 1988). Google Scholar

[12] 12.Stenström, B., Coherent rings and FP-injective modules, J. London Math. Soc. 2 (1970), 323–329. Google Scholar

[13] 13.Warfield, R. B. Jnr, Serial rings and finitely presented modules, J. Algebra 37 (1975) 187–222. Google Scholar | DOI

[14] 14.Ziegler, M., Model theory of modules, Ann. Pure Appl. Logic 26 (1984), 149–213. Google Scholar