88 2. THEORIES, LAGRANGIANS AND COUNTERTERMS (2) This limit defines a collection of elements I[L] ∈ Op + (E , A )[[ ]], satisfying the renormalization group equation and locality axiom, and so defines a theory. (3) The counterterms ICT i,k (ε) do not depend on the choice of a fake heat kernel. (4) If U ⊂ M is an open subset, then the counterterms for I restricted to U are the restrictions to U of the counterterms ICT i,k (ε) for I. Thus, counterterms define a map of sheaves O+ loc (Ec, A )[[ ]] → O+ loc (Ec, A ⊗alg P((0, ∞)ε) 0 )[[ ]]. Proof. The proof of the first two statements is identical to the proof of the corresponding statement on a compact manifold, and so is mostly omitted. One point is worth mentioning briefly, though: the counterterms are defined to be the singular parts of the small ε asymptotic expansion of the weight wγ(P (ε, L),I) attached to a graph γ. One can ask whether such asymptotic expansions exist when we use a fake heat kernel rather than an actual heat kernel. The asymptotic expansion, as constructed in Appendix 1, only relies on the small t expansion of the heat kernel Kt, and thus exists whenever the propagator is constructed from a kernel with such a small t expansion. The third clause is proved using the uniqueness of the counterterms, as follows. Suppose that Kt, Kt are two fake heat kernels, with associated fake propagators P (ε, L), P (ε, L). Let ICT (ε), ICT (ε) denote the counterterms associated to the two different fake heat kernels. We need to show that they are the same. Note that lim ε→0 W P (0,L) − P (0,L),W ( P (ε, L),I − ICT (ε) ) exists. We can write the expression inside the limit as W P (0,ε) − P (0,ε),W P (ε, L),I − ICT (ε) . Note that P (0,ε) − P(0,ε) tends to zero, with all derivatives, faster than any power of ε. Also, W P (ε, L),I − ICT (ε) has a small ε asymptotic expansion in terms of functions of ε of polynomial growth at the origin. From these two facts it follows that lim ε→0 W P (ε, L),I − ICT (ε) exists. Uniqueness of the counterterms implies that ICT (ε) = ICT (ε). The fourth clause can be proved easily using independence of the coun- terterms of the fake heat kernel.

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