process

196 P. Carrillo Rousewhere P W R p R q R q ! R p R q is the canonical projection (as above) andP.U/ D U 0 . This is possible since m is a surjective submersion. Now, we apply theDNC construction to the diagram above to obtain, thanks to the functoriality of theconstruction, the following commutative diagram:D U VD.m/ D U0V 0 Q 0QDV U DU 0D.P / V0‰ ‰ 0 QP 0,where ‰ and Q are as in Section 3. Let g 2 S c .DV U / and define8 Rˆ< Rg.x; ; ; 0/ d q if t D 0;P c .g/.x; ; t/ Dˆ: Rf2R q W.x;;;t/2 U V g g.x; ; ; t/t q d if t ¤ 0:Then, from the last commutative diagram, we get thatP c .g/ D PQc .g ı ‰ / ı .‰ 0/ 1 ;hence, thanks to Lemma 4.11, we can conclude that we have a well-defined linear mapP c W S c .DV U / ! S c.D U 0V 0 /:We now use the Proposition 4.2 to writeS c .D U V / Dfh 2 C 1 .D U V / W h ı Q 1 2 S c .D U V /g;and so for h 2 S c .D U V/ we see thatP c .h ı Q 1 / ı Q 0 2 S c .D U0V/: 0We use again the second commutative diagram to see thatWe then have a well-defined linear mapm c .h/ D P c .h ı Q 1 / ı Q 0 :m c W S c .D U V / ! S c.D U0V 0 /:To pass to the global case we only have to use the decomposition of S c .D G .2/G .0/ / andof S c .D G G .0/ / as in (5), and of course the invariance under diffeomorphisms (Proposition4.2).