The homomorphisms between the Dickson-Mùi algebras as modules over the Steenrod algebra

摘要

The Dickson-Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, A. We determine explicitly all the A-module homomorphisms between the (reduced) Dickson-Mùi algebras and all the A-module automorphisms of the (reduced) Dickson-Mùi algebras. The algebra of all A-module endomorphisms of the (reduced) Dickson-Mùi algebra is claimed to be isomorphic to a quotient of the polynomial algebra on one indeterminate. We prove that the reduced Dickson-Mùi algebra is atomic in the meaning that if an A-module endomorphism of the algebra is non-zero on the least positive degree generator, then it is an automorphism. This particularly shows that the reduced Dickson-Mùi algebra is an indecomposable A-module. The similar results also hold for the odd characteristic Dickson algebras. In particular, the odd characteristic reduced Dickson algebra is atomic and therefore indecomposable as a module over the Steenrod algebra.