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Dr. Katz DEq Homework Solutions 84

Dr. Katz DEq Homework Solutions 84 - 84 N M Katz ordv(i~i...

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84 N.M. Katz: ordv (i~i fip ti) =mini(ordv(ff ti)) (6.0.6.1) = min i (i + p ordvo (f~P)). Let Kv (resp. (KV)~o) denote the completion of K (resp. K P) with respect to the valuation v (resp. Vo). Then (6.0.6.0) gives (6.0.6.2) Kv ~- (KP)vo 0"" (KV)vo~- K | (KV),, o. ~ KP primes Since (KP)vo = KW, we have (6.0.6.3) K~= K | K~. KP Let Soln(K) (resp. Soln(Ko)) denote the K v (resp. K p) vector space of solutions of the differential equation (6.0.5.1) which lie in K (resp. in Kv). Then (6.0.6.4) Soln (Ko) ~ Soln (K) | K~. Kp This is because the differential operator (6.0.6.5) -~- - ~ ai -~- : K-* K i=0 is KP-linear, and the differential operator d , ,-1 d i is deduced from (6.0.6.5) by the (flat !) extension of scalars K p ~ KW. (6.0.6.7) Because we can "clear denominators" by multiplying by p-th powers, the spaces Soln(K) and Soln(K~) are in fact
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