A functional characterization of Tor for noetherian rings of global dimension 1

Abstract

which the functor Tory (N, ): RM -- Ab is naturally equivalent to G. In this paper, we show that if R is a noetherian ring of global dimension 1 (for example, if R is a Dedekind domain, or a ring of triangular matrices over a field), then the class of Tor-functors is characterized as that class of functors G forwhich G is half exact, G preserves direct limits, and G(R) = 0. In [2], Auslander shows that if R is noetherian and F: RM-- Ab is a coherent half exact functor which preserves direct limits, then there is a right R-module N and an exact sequence of functors TorR (N, ) F(R) OR-- F->- TorR (N, ) 0. In ?1, we recall the definition of coherent functors and some of their properties from [2]. We also show that every half exact functor is a filtered of half exact coherent functors (filtered limit being our term for a generalization of direct limits defined in [1]). In ?2, we show that if R is noetherian and G: RM ->- Ab is a half exact functor which preserves direct limits, then there is an exact sequence for G similar to the one above, provided that a filtered of Tor functors is a Tor functor. (If R is of global dimension 1, this condition is satisfied, and our characterization of Tor functor follows immediately.) In ?3, we extend Auslander's exact sequence to the class of coherent rings, namely rings for which the dual of projective modules are flat. In ?4, we let R be a coherent ring and G: RM->- Ab be a half exact functor which preserves direct limits, and obtain a result similar to that of ?2. We also compare the coherent case with the noetherian case. The results in this paper are from the author's thesis done at Brandeis University under Professor Maurice Auslander, whom the author wishes to thank for his encouragement, suggestions, and inspiration.