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Lifting Endo-trivial Modules J.L. Alperin
 

Summary: Lifting Endo-trivial Modules
J.L. Alperin 
Endo-permutation modules arise in a number of ways in the representation theory of
nite groups, in particular in connection with equivalences between blocks and as sources of
simple modules. Great progress has been made recently towards a complete classi cation of
such modules in characteristic p. The question of lifting such modules to valuation rings has
been open now for some time and is now quite relevant to the classi cation of such modules
over such rings. In this paper, we solve the problem in the key case of endo-trivial modules.
Let R be a complete discrete valuation ring of characteristic zero with maximal ideal
J and residue class eld k of prime characteristic p. For a xed nite p-group P , all the
kP-modules considered will be nite dimensional and all the RP-modules will be nitely
generated and free as R-modules. An RP-module M (and similarly for kP-modules) is an
endo-trivial module if the tensor product M

M is the direct sum of the trivial RP-module
R and a free RG-module. The standard results of the theory of such modules may be found
in the references [4, 5, 8].
In order to state the result of this paper, recall that an RP-module M lifts a kP-module
U if M=JM is isomorphic with U .
Theorem. Any endo-trivial kP-module lifts to an endo-trivial RP-module.

  

Source: Alperin, Jon L. - Department of Mathematics, University of Chicago

 

Collections: Mathematics