Remarks on equivalences of additive subcategories

We study category equivalences between some additive subcategories of module categories. As its application, we show that the group of autofunctors of the category of reflexive modules over a normal domain is isomorphic to the divisor class group. 1 A necessary condition for equivalences of additive subcategories Let R be a commutative ring. We denote the category of all finitely generated Rmodules by R-mod, and the full subcategory of R-mod consisting of all reflexive modules by ref(R). If R is a Cohen-Macaulay local ring, we denote the category of maximal Cohen-Macaulay modules by CM(R) as a full subcategory of R-mod. By an additive subcategory we always mean a full subcategory which is closed under finite direct sums and direct summands. Theorem 1. Let A and B be commutative rings. Let C (resp. D) be an additive full subcategory of A-mod (resp. B-mod) which contains a nontrivial free module. If there is a category equivalence between C and D, then A ∼= B as a ring. Moreover, if F and G are the functors which give the equivalences above, then F and G are of the forms F (X) ∼= HomA(G(B), X) and G(Y ) ∼= HomB(F (A), Y ) for each X ∈ C, Y ∈ D. proof. Let F : C → D and G : D → C be functors satisfying F · G ∼= 1D and G · F ∼= 1C. We denote the B-module F (A) by M and the A-module G(B) by N. Since F and G are fully faithful functors, there exist isomorphisms as rings EndB(M) ∼= EndA(A) = A and EndA(N) ∼= EndB(B) = B. Thus there are natural maps as follows: B β −−−→ EndB(M) ∼= −−−→ A α −−−→ EndA(N) ∼= −−−→ B (1.1) b −−−→ bM −−−→ a −−−→ aN −−−→ b′, where bM (resp. aN) denotes the multiplication map on M (resp. N) by b (resp. a). The title of the talk had been changed. First of all, we claim that b − b′ ∈ AnnBM for such b and b′ as above. Since M is finitely generated B-module, we can take a finite free cover of M and get the following diagram. B⊕n −−−→ M  bB  bM B⊕n −−−→ M Applying the functor G to this diagram, we have a diagram N⊕n −−−→ A  aN  aA