Dr. Katz DEq Homework Solutions 51

Dr. Katz DEq Homework Solutions 51 - make use of it,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Algebraic Solutions of Differential Equations 51 and substituting via (3.5.1) and especially via (3.5.1.3), (3.5.2.1) becomes (3.5.2.2) Lie(P)P-lI(P-Q)(a(1))-h(1)= -g(1) p. A final substitution via (3.5.1) gives the assertion (3.5.2.3) PP-1 (f(1))- h(1)= -g(1) p. We now "decode" (3.5.2.3) by returning to the definitions (3.5.0.7-9) of f, h, g; (3.5.2.3) becomes the assertion (3.5.2.4) if 1~, PP-I(P(xOI\ xl ! PP(XO_x, (~0) p, (3.5.2.5) if 1r PP-1(x~-lp(xl))-x(-IPV(xO=-(P(Xl)) p. Both (3.5.2.4) and (3.5.2.5) are in fact true, and follow ((3.5.2.5) directly, (3.5.2.4) after dividing by x 0 from Hochschild's identity (cf. [20]), according to which, if P is any derivation of any commutative ring A of characteristic p>0, then for any element xeA, (3.5.2.6) Pp-l(xo-~ p(x))-xp-l Pp(x)= -(P(x)) p. This concludes the proof of (3.5.0.3) in case b = 1. In the following, we will make use of it, through the identity (3.5.2.3).
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: make use of it, through the identity (3.5.2.3). (3.5.3) Because our proof of (3.5.0.3) in the general case is so unen- lightening we first present the proof in the case b = 2, which is somewhat more intelligible. The assertion to be verified is Lie (p)k I (P -- Q) Lie (Q)e (~ (r (l) ^ z (2))) k+g'=p- 1 (3.5.3.0) - I (P p- Qp)(o~ (z (1) ^ z (2))) - o~ (I (P - Q) (, (1) ^ z (2))) modulo dF(V, (gv). Substituting via (3,5.1.1) and noting that by (3.5.1.3) the terms under the summation sign vanish for (~0, (3.5.3.0) becomes Lie (P)P-x (I (P - Q))(~r (1) ^ a (2)) - I(Pp - QO)(a (1) ^ a (2)) (3.5.3.1) - ~----~(I(P- Q)(,(I) ^ ,(2))) modulo dF(V, (9~,). Expanding the interior products and substituting via (3.5.1), (3.5.3.1) becomes Lie (P)P-1 (f(1) a (2) - f(2) a (1)) - (h (1) a (2) - h (2) a (1)) (3.5.3.2) -- ~(g(1)z(2)-g(2)z(1))modulodr(V, Cv). 4*...
View Full Document

Ask a homework question - tutors are online