Abstract
In this note we prove two theorems. In theorem 1 we prove that if M andN are two non-zero reflexive modules of finite projective dimensions over a Gorenstein local ring, such that Hom (M, N) is a third module of syzygies, then the natural homomorphismM* ⊗N → Hom (M, N) is an isomorphism. This extends the result in [7]. In theorem 2, we prove that projective dimension of a moduleM over a regular local ringR is less than or equal ton if and only if ExtR n (M, R) ⊗M → ExtR n (M, M) is surjective; in which case it is actually bijective. This extends the usual criterion for the projectivity of a module.
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Bose, S. The homomorphismM*⊗N → Hom (M,N). Proc. Indian Acad. Sci. (Math. Sci.) 94, 23–26 (1985). https://doi.org/10.1007/BF02837253
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DOI: https://doi.org/10.1007/BF02837253