The Auslander-Reiten conjecture is the following. Let \(R\) be a commutative Noetherian ring and let \(M\) be a finitely generated \(R\)-module. If \(\mathrm{Ext}^i_R(M,M) = 0 = \mathrm{Ext}^i_R(M,R)\) for all \(i \geq 1\) then \(M\) is projective. Several special cases of the conjecture are known to hold. Although it was initially proposed over Artin algebras, it is still meaningful for arbitrary commutative Noetherian rings. The paper has two main results, involving exact sequences and isomorphisms of Ext-modules. Both of these theorems extend (and imply, for \(d \geq 2\)) the Auslander-Reiten duality theorem: Let \((R,\mathfrak m)\) be a \(d\)-dimensional Gorenstein local ring. Let \(M, N\) be maximal Cohen-Macaulay \(R\)-modules such that \(NF(M) \cap NF(N) \subseteq \{ \mathfrak m \}\). Then for all \(i \in \mathbb Z\), there is an isomorphism of Tate cohomology modules \(\widehat{\mathrm{Ext}}^i_R(N,M)^\vee \cong \widehat{\mathrm{Ext}}_R^{(d-1)-i} (M,N)\). (Here \(NF(M)\) refers to the non-free locus of the \(R\)-module \(M\), i.e. the set of prime ideals \(\mathfrak p\) such that \(M_{\mathfrak p}\) is not \(R_{\mathfrak p}\)-free.) In fact, one of the results of the paper even completely recovers this duality result for all \(d \geq 0\). The authors give a number of applications and corollaries of their results.