Abstract
The method of differentiation for the category Λ-lat of lattices over an order Λ will be extended to integral almost Abelian categories A instead of Λ-lat. In particular, this yields a differentiation for finitely generated left modules over left Artinian rings.
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References
Anderson, F. W. and Fuller, K. R.: Rings and Categories of Modules, Springer, New York, 1974.
Auslander, M.: Coherent functors, In: Proc. Conf. Categorial Algebra (La Jolla, 1965), pp. 189-231.
Auslander, M.: Representation theory of Artin algebras, I, Comm. Algebra 1 (1974), 177-268.
Auslander, M. and Kleiner, M.: Adjoint functors and an extension of the Green correspondence for group representations, Adv. Math. 104 (1994), 297-314.
Bourbaki, N.: Commutative Algebra, Hermann, Paris, 1972.
Colby, R. R. and Fuller, K. R.: Tilting, cotilting, and serially tilted rings, Comm. Algebra 18 (1990), 1585-1615.
Colpi, R.: Tilting modules and *-modules, Comm. Algebra 21 (1993), 1095-1102.
Colpi, R., D'Este, G. and Tonolo, A.: Quasi tilting modules and counter equivalences, J. Algebra 191 (1997), 461-494.
Colpi, R. and Menini, C.: On the structure of *-modules, J. Algebra 158 (1993), 400-419.
Drozd, Ju. A. and Kiričenko, V. V.: On quasi-Bass orders, Izv. Akad. Nauk SSSR, Ser. Mat. 36 (1972), 328-370 and Math. USSR Izv. 6 (1972), 323-366.
Freyd, P.: Representations in Abelian categories, In: Proc. Conf. Categorial Algebra (La Jolla 1965), pp. 95-120.
Fuller, K. R.: Density and equivalence, J. Algebra 29 (1974), 528-550.
Gabriel, P.: Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323-448.
Gabriel, P.: Représentations indécomposables des ensembles ordonnés, Sém. Dubreil 1972–73, Paris, exposé 13, pp. 1-10.
Gabriel, P. and Zisman, M.: Calculus of Fractions and Homotopy Theory, Springer, Berlin, 1967.
Grätzer, G.: General Lattice Theory, Birkhäuser, Basel, 1978.
Green, J. A.: A transfer theorem for modular representations, J. Algebra 1 (1984), 73-84.
Grothendieck, A.: Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957), 119-221.
Klemp, B. and Simson, D.: Schurian sp-representation-finite right peak PI-rings and their indecomposable socle-projective modules, J. Algebra 134 (1990), 390-468.
Nazarova, L. A. and Roiter, A. V.: Representations of partially ordered sets, Zapiski Nauchn. Sem. LOMI 28 (1972), 5-31 and J. Soviet Math. 3 (1975), 585-606.
Rump, W.: Two-point differentiation of general orders, J. Pure Appl. Algebra 153 (2000), 171-190.
Rump, W.: Differentiation and splitting for orders, Colloq. Math., to appear.
Rump, W.: Almost Abelian Categories, Cahiers de topologie et géométrie différentielle catégoriques, to appear.
Rump, W.: *-Modules, tilting, and almost Abelian categories, Comm. Algebra, to appear.
Simson, D.: On vector space categories and differentiations of right peak rings, In: Proc. Internat. Conf. Repr. Algebras IV, Ottawa, Carleton Univ., 1984.
Simson, D.: Vector space categories, right peak rings, and their socle projective modules, J. Algebra 92 (1985), 532-571.
Simson, D.: Linear Representations of Partially Ordered Sets and Vector Space Categories, Gordon and Breach, Amsterdam, 1992.
Stenström, B.: Rings of Quotients, Springer, New York, 1975.
Zavadskij, A. G.: A differentiation with respect to a pair of points, In: Matrix Problems (Kiev, 1977), pp. 115-121 (in Russian).
Zavadskij, A. G. and Kiri¡ cenko, V. V.: Semimaximal rings of finite type, Mat. Sbornik 103(145) (1977), 323-345 and Math. USSR Sbornik 32 (1977), 273-291.
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Rump, W. Differentiation for Orders and Artinian Rings. Algebras and Representation Theory 7, 395–417 (2004). https://doi.org/10.1023/B:ALGE.0000042182.98997.7c
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DOI: https://doi.org/10.1023/B:ALGE.0000042182.98997.7c