By David Bailin; Alexander Love

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5. Lemma. For a module RP θP (f ⊗ x)g = f ϕP (x ⊗ g). and with S = End(R P ), (a) ϕP : P ⊗S P ∗ −→ R RR is surjective iff (b) θP : P ∗ ⊗R P −→ S SS is surjective iff RP RP is a generator; is finitely generated projective. 52 6 THE MORITA CHARACTERIZATION OF EQUIVALENCE. Proof. (a) Clearly, Im ϕP = TrR (P ). (b) Note that θP is surjective iff 1S ∈ Im θP iff there exist f1 , . . , fn ∈ P ∗ and x1 , . . , xn ∈ P with n n (fi ⊗ xi ) = θP ( (−)fi xi = 1S . i=1 i=1 But this last simply means that for each x ∈ P n x = (x)1S = (x)fi xi , i=1 or that there exists a dual basis for R P .

A covariant functor F : C −→ D is faithful in case it is one-to-one on the class of morphisms of C. The functor F is full in case for each pair C, C in C the map F : MorC (C, C ) −→ MorD (F (C), F (C )) is surjective. Finally, the functor F is dense in case for each object D in D there is an object C in C with F (C) ∼ = D. It turns out that these three properties characterize category equivalences. 1. Theorem. Let C and D be (additive) categories and let F : C −→ D be a covariant (additive) functor.

Since F is dense, there is an H in RMod with F (H) ∼ = K. Since F is also full and faithful, there exist R-homomorphisms for which the diagram 0 ❄ ✲ M 0 f✲ g✲ M = = ❄ ❄ g ✲ M ✲ M h ❄ ✲ H 0 M ✲ 0 ✲ 0 commutes and has exact rows and columns. Thus, h is an isomorphism, so that F (h) : F (M ) −→ K is an isomorphism, and Ker F (g) = Im F (f ). A property of modules is said to be a Morita invariant if it is preserved by additive equivalences. Our next result is the key to identifying a whole lot of important Morita invariants.

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