{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Dr. Katz DEq Homework Solutions 43

# Dr. Katz DEq Homework Solutions 43 - Algebraic Solutions of...

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

Algebraic Solutions of Differential Equations 43 (3.2.3) Corollary. Hypotheses and notations as in (3.2.2), the indecom- posable central idempotents in K [G], which all lie in (P [G], have reductions modulo t~ in k [G] which remain indecomposable. Proof. The number of e i is equal to the number off~, both being the number of conjugacy classes in G. Hence every e i lifts an ft. Q.E.D. In terms of representations, this gives: (3.2.3 bis) Corollary. Hypotheses as in (3.2.2bis), the reduction mod p of any K-irreducible representation of G in a free C-module of finite rank is irreducible (and every irreducible representation of G in a finite-dimen- sional k-space arises this way). (3.2.3.1) Corollary. Hypotheses as in (3.2.2his), every irreducible repre- sentation of G in a finite-dimensional k-space is absolutely irreducible. Proof. This follows from (3.3.0bis), applied to arbitrary finite exten- sions K' of K, and arbitrary extensions of the valuation of K to K', by which we can realize arbitrary finite extensions k' of k as residue field.
This is the end of the preview. Sign up to access the rest of the document.
• Fall '11
• NormanKatz
• De Rham cohomology, irreducible representation, finite-dimensional k-space, Hodge =~ De Rham, absolutely irreducible repre

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern