Dr. Katz DEq Homework Solutions 27

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

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

View Full Document Right Arrow Icon
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),
Background image of page 1
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