## Covers induced by Ext.(English)Zbl 0981.16010

Let $${\mathbf C}\subseteq\text{Mod-}R$$ and $${\mathbf B}\subseteq R\text{-Mod}$$. Then $$^\bot{\mathbf C}$$ is defined to be the class $$\text{Ker Ext}(-,{\mathbf C})=\{D\mid\text{Ext}(D,C)=0$$ for all $$C\in{\mathbf C}\}$$ and $$\text{Ker Tor}(-,{\mathbf B})=\{A\mid\text{Tor}(A,B)=0$$ for all $$B\in{\mathbf B}\}$$. If $$M\in\text{Mod-}R$$, then $$\phi\in\text{Hom}(A,M)$$ ($$A\in{\mathbf C}$$) is called a $$\mathbf C$$-pre-cover of $$M$$ if the induced map $$\text{Hom}(A',A)\to\text{Hom}(A',M)$$ is surjective for all $$A'\in{\mathbf C}$$. If each $$\psi\in\text{Hom}(A,A)$$ satisfying $$\phi=\phi\psi$$ is an automorphism of $$A$$, then $$\phi$$ is said to be a $$\mathbf C$$-cover. It is proved that each right $$R$$-module has a $$^\bot{\mathbf C}$$-cover if $$\mathbf C$$ is any class of pure-injective modules. On the other hand, for any class $$\mathbf B$$ each module has a $$\text{Ker Tor}(-,{\mathbf B})$$-cover. These results therefore provide a generalization of the flat cover conjecture.
$$\mathbf C$$-pre-covers and $$\mathbf C$$-covers are studied intensively for modules over hereditary Dedekind domains. For Dedekind domains $$^\bot{\mathbf C}$$ is described explicitly for any class of cotorsion modules $$\mathbf C$$. The paper ends with three open problems.

### MSC:

 16D90 Module categories in associative algebras 16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras 16D40 Free, projective, and flat modules and ideals in associative algebras 16D50 Injective modules, self-injective associative rings 16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
Full Text:

### References:

 [1] Hügel, L.Angeleri; Tonolo, A.; Trlifaj, J., Tilting preenvelopes and cotilting precovers, Algebras and representation theory, 3, (2000) · Zbl 0999.16007 [2] Auslander, M.; Smalø, S., Preprojective modules over Artin algebras, J. algebra, 66, 61-122, (1980) · Zbl 0477.16013 [3] Becker, T.N.; Fuchs, L.; Shelah, S., Whitehead modules over domains, Forum math., 1, 53-68, (1989) · Zbl 0651.20061 [4] L. Bican, R. El Bashir, and, E. Enochs, All modules have flat covers, Bull. London Math. Soc. to appear. · Zbl 1029.16002 [5] Colpi, R.; D’Este, G.; Tonolo, A., Quasi-tilting modules and counter equivalences, J. algebra, 191, 461-494, (1991) · Zbl 0876.16004 [6] Cartan, H.; Eilenberg, S., Homological algebra, (1956), Princeton Univ. Press Princeton · Zbl 0075.24305 [7] Eklof, P., Homological algebra and set theory, Trans. amer. math. soc., 227, 207-225, (1977) · Zbl 0355.02047 [8] Eklof, P., Set theoretic methods in homological algebra and abelian groups, (1980), Presses Univ. de Montreal Montreal · Zbl 0488.03029 [9] Eklof, P.; Shelah, S., On Whitehead modules, J. algebra, 141, 492-510, (1991) · Zbl 0743.16004 [10] P. Eklof, and, J. Trlifaj, How to make Ext vanish, Bull. London Math. Soc, to appear. · Zbl 1030.16004 [11] Enochs, E., Injective and flat covers, envelopes and resolvents, Israel J. math., 39, 189-209, (1981) · Zbl 0464.16019 [12] Fuchs, L.; Salce, L., Modules over valuation domains, Lecture notes in pure and applied mathematics, 96, (1985), Dekker New York [13] Göbel, R.; Trlifaj, J., Cotilting and a hierarchy of almost cotorsion groups, J. algebra, 224, 110-122, (2000) · Zbl 0947.20036 [14] Jensen, C.; Lenzing, H., Model theoretic algebra, Algebra, logic and applications, 2, (1989), Gordon & Breach New York [15] Lambek, J., A module is flat if and only if its character modules is injective, Canad. math. bull., 7, 237-243, (1964) · Zbl 0119.27601 [16] Matlis, E., Injective modules over Noetherian rings, Pacific J. math., 8, 511-528, (1958) · Zbl 0084.26601 [17] Matsumura, H., Commutative ring theory, (1986), Cambridge Univ. Press Cambridge [18] Sabbagh, G., Aspects logiques de la pureté dans LES modules, C. R. acad. sci Paris ser. A-B, 271, A909-A912, (1970) · Zbl 0202.00901 [19] Trlifaj, J., Non-perfect rings and a theorem of eklof and Shelah, Comment. math. univ. carolinae, 32, 27-32, (1991) · Zbl 0742.16001 [20] Trlifaj, J., Whitehead test modules, Trans. amer. math soc., 348, 1521-1554, (1996) · Zbl 0865.16006 [21] Wisbauer, R., Foundations of module and ring theory, (1991), Gordon & Breach New York [22] Xu, J., Flat covers of modules, Lecture notes in mathematics, 1634, (1996), Springer-Verlag New York/Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.