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80 2. THEORIES, LAGRANGIANS AND COUNTERTERMS Suppose that we have a local action functional J = (i,k) (I,K) i J (i,k) ∈ O+ loc (E , A )[[ ]] with associated effective interactions J[L]. Suppose, by induction, that J(i,k)[L] = I(i,k)[L] for all (i, k) (I, K). We need to find some J (I,K) such that, if we set J = J + I J (I,K) , then J (I,K) [L] = I(I,K)[L]. We simply let J (I,K) = I (I,K) [L] − J (I,K) [L]. The renormalization group equation implies that J (I,K) is independent of L. It is automatic that J (I,K) [L] = I(I,K)[L]. Finally, the fact that both J(I,K)[L] and J(I,K)[L] satisfy the small L asymp- totics axiom of a theory implies that J (I,K) is local. 14. Field theories on non-compact manifolds A second generalization is to non-compact manifolds. On non-compact manifolds, we don’t just have ultraviolet divergences (arising from small scales) but infrared divergences, which arise when we try to integrate over the non-compact manifold. However, by imposing a suitable infrared cut-off, we will find a notion of theory on a non-compact manifold and a bijection between theories and Lagrangians. The infrared cut-off we impose is rather brutal: we multiply the propaga- tor by a cut-off function so that it becomes supported on a small neighbour- hood of the diagonal in M 2 . However, the notion of theory is independent of the cut-off chosen. We will also show that there are restriction maps, allowing one to restrict a theory on a non-compact manifold M to any open subset U. This allows one to define a sheaf of theories on any manifold. If we are on a compact manifold, global sections of this sheaf are theories in the sense we defined before. The sheaf-theoretic statement of our main theorem asserts that this sheaf is isomorphic to the sheaf of local action functionals. As always, this isomorphism depends on the choice of a renormalization scheme. The renor- malization scheme independent statement of the theorem is that the sheaf of theories defined modulo n+1 is a torsor over the sheaf of theories defined modulo n , for the sheaf of local action functionals on M.