Abstract
As is well-known, the infinite module theory of certain types of finite-dimensional algebras has a strong similarity to abelian group theory. (See [1] for the case of Kronecker modules, [28] for the representations of tame hereditary algebras). The same phenomenon occurs for k-linear representations of (suitably restricted) ordered sets [18, 19]. In all these cases there is a primary decomposition for torsion-modules (cf. Cor. 2.6), the torsion submodule is always a pure submodule (cf. Prop. 3.1) and torsion-free modules behave in some respect like flat modules (cf. Prop. 2.4). Further, divisible modules are algebraically compact (cf. Thms. 4.4 and 4.6), in some cases even injective [19].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aronszajn, N., Fixman, U.: Algebraic spectral problems. Studia Math. 30, 273–338 (1968).
Auslander, M.: On the dimension of modules and algebras, III. Global dimension. Nagoya Math. J. 9, 67–77 (1955).
Auslander, M., Reiten, I.: Representation theory of Artin algebras III. Comm. Algebra 3, 239–294 (1975).
Auslander, M., Platzeck, M.I.: Representation theory of hereditary Artin algebras, in: Representation theory of algebras (R. Gordon, ed.) Proc. Philadelphia Conf. 1976. Marcel Dekker, New York: 1978.
Bass, H.: Finitistic dimension and a homological generalization of semi-primary rings. Trans Amer. Math. Soc. 95, 466–488 (1960).
Baer, D.: Zerlegungen von Moduln und Injektive über Ringoiden. Arch. Math. 36, 495–501 (1981).
Baer, D., Lenzing, H.: A homological approach to representations of algebras I: The wild case. J. Pure Appl. Algebra 24, 227–233 (1982).
Baer, D., Brune, H., Lenzing, H.: A homological approach to representations of algebras II: Tame hereditary algebras. J. Pure Appl. Algebra 26, 141–153 (1982).
Brune, H.: Some left pure semisimple ringoids which are not right pure semi-simple. Comm. Algebra 7, 1795–1803 (1979)
Brune, H.: On a theorem of Kulikov for artinian rings. Comm. Algebra 10, 433–448 (1982).
Brune, H.: On the global dimension of the functor category ((modR)p, Ab) and a theorem of Kulikov, J. Pure Appl. Algebra, to appear.
Dlab, V., Ringel, C.M.: Indecomposable representations of graphs and algebras. Mem. Amer. Math. Soc. 173 (1976).
Faith, C., Walker, E.A.: Direct-sum representations of injective modules. J. Algebra 5, 203–221 (1967).
Fuchs, L.: Modules over valuation domains, in: Algebraic structures and applications. Lecture Notes in Pure Appl. Math. 82. Marcel Dekker, New York: 1982.
Gabriel, P.: Indecomposable representations II. Ist. Naz. Alta Mat. Symp. Math. 11, 81–107 (1973).
Gruson, L.: Dimension homologique des modules plats sur un anneau commutatif noetherien. Ist. Naz. Alta Mat. Symp. Math. 11, 243–254 (1973).
Gruson, L., Jensen, C.U.: Dimensions cohomologiques reliées aux foncteurs lim(i), in: Séminaire d’Algèbre P. Dubreil et M.-P. Malliavin, Lecture Notes in Mathematics 867, Springer-Verlag: Berlin-Heidelberg-New York 1981.
Höppner, M.: How to use abelian group theory for the study of diagrams over posets. Preprint 1982.
Höppner, M., Lenzing, H.: Diagrams over ordered sets: A simple model of abelian group theory, in: Abelian group theory. (R. G öbel, E.A. Walker, eds.). Lecture Notes in Mathematics 874, Springer-Verlag: Berlin-Heidelberg-New York 1981.
Jensen, C.U.: On the global dimension of the functor category (mod R, Ab).J. Pure Appl, Algebra 11, 45–51 (1977).
Jensen, C.U., Lenzing, H.: Algebraic compactness of reduced products and applications to pure global dimension. Comm. Algebra, to appear.
Lazard, D.: Autour de la platitude, Bull. Soc. Math. France 97, 81–128 (1968).
Lenzing, H.: Direct sums of projective modules as direct summands of their direct product. Comm. Algebra 4, 681–691 (1976).
Lenzing, H.: The pure-projective dimension of torsion-free divisible modules. Comm. Algebra, to appear.
Mac Lane, S.: Categories for the working mathematician. Springer-Verlag: Berlin-Heidelberg-New York 1971.
Mitchell, B.: Rings with several objects. Advances in Math. 8, 1–161 (1972).
Okoh, F.: Some properties of purely simple Kronecker modules I. J. Pure Appl. Algebra, to appear.
Ringel, C.M.: Infinite dimensional representations of finite dimensional hereditary algebras. Ist. Naz. Alta Mat. Symp. Math. 23, 321–412 (1979).
Ringel, C.M.: Report on the Brauer-Thrall conjectures, in: Representation theory I (V. Dlab and P. Gabriel, eds.) Proc. Ottawa 1979. Lecture Notes in Math. 831. Springer: 1980.
Ringel, C.M.: Tame algebras, in: Representation theory I (see [29]).
Sharpe, D.W., Vâmos, P.: Injective modules. Cambrigde University Press 1972.
Goblot, R.: Sur les dérivés de certaines limites projectives. Applications aux modules. Bull. Sci. Math. 94, 251–255 (1970).
Osofsky, B.: Upper bounds on homological dimension. Nagoya Math. J. 32, 315–322 (1968).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lenzing, H. (1983). Homological Transfer from Finitely Presented to Infinite Modules. In: Göbel, R., Lady, L., Mader, A. (eds) Abelian Group Theory. Lecture Notes in Mathematics, vol 1006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21560-9_53
Download citation
DOI: https://doi.org/10.1007/978-3-662-21560-9_53
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-12335-4
Online ISBN: 978-3-662-21560-9
eBook Packages: Springer Book Archive