Galois theory of thick subcategories in modular representation theory
by Mark Hovey and John Palmieri
Suppose B is a finite-dimensional cocommutative Hopf algebra over a
field k. Define a thick subcategory to be a full subcategory of the
category of finite-dimensional B-modules that is closed under summands
and, if two out of three modules in a short exact sequence are in it, so
is the third. Define a thick subcategory to be tensor-closed if it is
closed under tensoring with any finite-dimensional module.
The classification of these tensor-closed thick subcategories, analogous
to the Hopkins-Smith classification of thick subcategories in the stable
homotopy category, has been carried out for B=k[G], where G is a finite
group and k is an algebraically closed field of positive characteristic,
by Benson-Carlson-Rickard. A similar classification has been obtained
by the current authors when B is a finite subalgebra of the mod 2
Steenrod algebra, with scalars extended to the algebraic closure of
Z/2.
In the present paper, we eliminate the annoying requirement that the
field be algebraically closed. We show that, if the expected
classification of tensor-closed thick subcategories holds for B tensor
L, where L is a normal extension field of k, then it holds for B as
well. The proof involves importing the basic ideas of Galois theory
into axiomatic stable homotopy theory.