Dr. Katz DEq Homework Solutions 27

# Dr. Katz DEq Homework Solutions 27 - 144tt...

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

Algebraic Solutions of Differential Equations 27 and the degeneration of the Hodge =~ De Rham spectral sequence gives us a surjection (2.3.4.1.2) R"f, (f2~/s (log D)) --~ R"f, (d~x) = E ~ Putting these together, we obtain the diagram which defines the Hasse- Witt operations: r "~ ~ R"f, (f2;/s (log D)) --~ E ~ (2,3.4.1.3) /l ]l Fg~ R"f, (ex) . .................. n_:__w_ ................ :,~ R"f, (6x) The matrix of this operation in a local base of R"f, ((_gx) is called the Hasse- Witt matrix of X/S in dimension n. The composite p-linear mapping (2.3.4.1.4) R,f,(Cx) - F*bs ,F~R,f,(Cgx ) n-w ,Rnf,(Cx) is the one induced by the p-th power endomorphism of d~ x. From the definition (2.2.4.1.3), it follows that we have: (2.3.4.1.5) Proposition. Hypotheses as in (2.3.2),
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 144tt operation (2.3.4.1.6) H-W: F* s R"f, (d~x) ~ R"f, ((gx) is none other than the intersection (F~"on ~ F 1) (R"f, (f2~/s(log D))). (2.3.4.1.7) Corollary (Assumptions as in (2.3.2)). In order to have a direct sum decomposition (2.3.4.1.8) R"f,(f2~/s(log D)), ~ F ~ GFc"on it is necessary and sufficient that the Hasse-Witt operation (2.3.4.1.9) H-W: F*s R"f,((~x)~ R"f, ff)x) be an isomorphism. Proof. Since the statements whose equivalence is asserted are both of the form "a certain homomorphism of locally free S-modules of finite rank is an isomorphism", and because the formation of these modules commutes with all changes of base S' --* S, we are immediately reduced to the case in which S is the spectrum of a field K. The isomorphisms ~g (2.3.4.1.10) co~E~ 'b ~,~ F~*~E~ 'a...
View Full Document

## This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.

Ask a homework question - tutors are online