Geodesic
The problem of the existence of sufficiently many injective Frechet modules over non-normed Frechet algebras
Izvestiya. Mathematics , Tome 62 (1998) no. 4, pp. 773-788.

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The main aim of this paper is to show that in the category of Frechet modules over certain Frechet algebras there cannot exist sufficiently many injective objects. In particular, we show that over Frechet algebras of formal power series there are no non-zero injective Frechet modules. We describe a class of Frechet algebras, which includes algebras of holomorphic functions over irreducible Stein spaces, over which there is no injective metrizable hypermodule. We also study the property of divisibility for Frechet modules and its relationship with the property of injectivity. We also show that every separable divisible Frechet module has periodic elements and prove a theorem on the non-existence of divisible Banach modules. 
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     title = {The problem of the existence of sufficiently many injective {Frechet} modules over non-normed {Frechet} algebras},
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A. Yu. Pirkovskii. The problem of the existence of sufficiently many injective Frechet modules over non-normed Frechet algebras. Izvestiya. Mathematics , Tome 62 (1998) no. 4, pp. 773-788. http://geodesic.mathdoc.fr/item/IM2_1998_62_4_a4/

[1] Akbarov S. S., “Dvoistvennost Pontryagina v teorii topologicheskikh modulei”, Funktsion. analiz i prilozh., 29:4 (1995), 68–72 | MR | Zbl

[2] Atya M., Makdonald I., Vvedenie v kommutativnuyu algebru, Mir, M., 1972 | MR

[3] Burbaki N., Gomologicheskaya algebra, Nauka, M., 1987 | MR

[4] Gamelin T., Ravnomernye algebry, Mir, M., 1973 | Zbl

[5] Ganning R., Rossi Kh., Analiticheskie funktsii mnogikh kompleksnykh peremennykh, Mir, M., 1969 | MR

[6] Grauert G., Remmert R., Analiticheskie lokalnye algebry, Nauka, M., 1988 | MR | Zbl

[7] Grauert G., Remmert R., Teoriya prostranstv Shteina, Nauka, M., 1989 | MR | Zbl

[8] Grotendik A., “O prostranstvakh $(F)$ i $(DF)$”, Matematika, 2:3 (1958), 81–127

[9] Felps R., Lektsii o teoremakh Shoke, Mir, M., 1968

[10] Khelemskii A. Ya., Gomologiya v banakhovykh i topologicheskikh algebrakh, Izd-vo MGU, M., 1986 | MR

[11] Khelemskii A. Ya., Banakhovy i polinormirovannye algebry: obschaya teoriya, predstavleniya, gomologii, Nauka, M., 1989 | MR

[12] Shefer Kh., Topologicheskie vektornye prostranstva, Mir, M., 1971 | MR

[13] Engelking R., Obschaya topologiya, Mir, M., 1986 | MR

[14] Bungart L., “Holomorphic functions with values in locally convex spaces and applications to integral formulas”, Trans. Amer. Math. Soc., 111 (1964), 317–344 | DOI | MR | Zbl

[15] Eschmeier J., Putinar M., Spectral Decompositions and Analytic Sheaves, Clarendon Press, Oxford, 1996 | MR

[16] Helemskii A. Ya., “31 problems of the homology of the algebras of analysis”, Linear and complex analysis: problem book 3, part I, Lecture Notes in Math., 1573, eds. Havin V. P., Nikolski N. K., 1994, 54–78 | MR

[17] Mallios A., Topological Algebras: Selected Topics, North Holland, Amsterdam, 1986 | MR | Zbl

[18] Perez Carreras P., Bonet J., Barreled Locally Convex Spaces, North Holland, Amsterdam, 1987 | MR | Zbl

[19] “Séminaire de géométrie analytique”, Astèrisque, 16, eds. A. Douady, J. L. Verdier, Soc. Math. France, Paris, 1974 | MR

[20] Taylor J. L., “Homology and cohomology for topological algebras”, Adv. Math., 9 (1972), 137–182 | DOI | MR | Zbl