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14. FIELD THEORIES ON NON-COMPACT MANIFOLDS 87 again defined using any fake heat kernel on U, with the property that the small L asymptotic expansion of IU[L] is the restriction to U of the small L asymptotic expansion of I[L]. The existence of this restriction map shows that there is a presheaf on M which assigns to an open subset U ⊂ M the set T (n) (Γ(U, EU)), A ) of theories on U. We will denote this presheaf by T (n)(E , A ). 14.9. Now we are ready to state the main theorem. Theorem 14.9.1. (1) The presheaves T (n) (E , A ) (of theories de- fined modulo n+1 ) and T (∞) (E , A ) (of theories defined to all order in ) on M are sheaves. (2) There is a canonical isomorphism between the sheaf T (0) (E , A ) and the sheaf of local action functionals I ∈ Oloc(Ec, A ) which are at least cubic modulo the ideal Γ(X, m) ⊂ A . (3) For n 0, T (n) (E , A ) is, in a canonical way, a torsor over T (n−1) (E , A ) for the sheaf of abelian groups Oloc(Ec, A ), in a canonical way. (4) Choosing a renormalization scheme leads to a map T (n) (E , A ) → T (n+1) (E , A ) of sheaves for each n, which is a section. Thus, the choice of a renormalization scheme leads to an isomorphism of sheaves T (∞) (E , A ) ∼ O+ loc (Ec, A )[[ ]] on M. 14.10. The proof of the theorem, and of Proposition 14.8.1, is along the same lines as before, using counterterms. Proposition 14.10.1. Let us fix a renormalization scheme, and a fake heat kernel Kt on M. Let I = i Ii,k ∈ O+ loc (Ec, A )[[ ]]. Then: (1) there is a unique series of counterterms ICT (ε) = i ICT i,k (ε) where Ii,k CT (ε) ∈ Oloc(E , A ) ⊗alg P((0, ∞)) 0 is purely singular as a function of ε, with the property that the limit lim ε→0 W ( P (ε, L),I − ICT (ε) ) exists, for all L.