14. FIELD THEORIES ON NON-COMPACT MANIFOLDS 89 14.11. Now we can prove Proposition 14.8.1 and Theorem 14.9.1. The proof is easy once we know that counterterms exist. Let us start by regarding the set of theories T (∞) (E , A ) as just a set, and not as arising from a sheaf on M. (After all, we have not yet proved Proposition 14.8.1, so we do not know that we have a presheaf of theories). Lemma 14.11.1. Let us choose a renormalization scheme, and a fake heat kernel. Then the map of sets O+ loc (Ec, A )[[ ]] → T (∞) (E , A ) I → {I[L] = lim ε→0 W ( P (ε, L),I − ICT (ε) ) } is a bijection. Proof. This is proved by the usual inductive argument. Now we can prove Proposition 14.8.1, which we restate here for conve- nience. Proposition 14.11.2. Let U ⊂ M be an open subset. Given any theory {I[L] ∈ Op + (E , A )[[ ]]} on M, for any fake heat kernel, U, there is a unique theory {IU[L] ∈ O+(Γ(U, p E |U), A )[[ ]]} with the property that the small L asymptotics of IU[L] is the restriction to U of the small L asymptotics of I[L]. Proof. Uniqueness is obvious, as any two theories on U with the same small L asymptotic expansions must coincide. For existence, we will use the bijection between theories and Lagrangians which arises from the choice of a renormalization scheme. We can as- sume that the theory {I[L]} on M arises from a local action functional I ∈ O+ loc (Ec, A )[[ ]]. Then, we define IU[L] to be the theory on U associated to the restriction of I to U. It is straightforward to check that, with this definition, I[L] and IU[L] have the same small L asymptotics. It follows from the proof of this lemma that the map O+ loc (Ec, A )[[ ]] → T (∞) (E , A ) of sets actually arises from a map of presheaves on M. Since O+ loc (Ec, A )[[ ]] is actually a sheaf on M, it follows that T (∞) (E , A ) is also a sheaf, and similarly for T (n) (E , A ). The remaining statements of Theorem 14.9.1 are proved in the same way as the corresponding statements on a compact manifold.

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