A nilpotence theorem for modules over the mod 2 Steenrod algebra
The authors give a characterization of the nilpotent elements of \(\text{Ext}_ A^{**}({\mathbf F}_ 2,{\mathbf F}_ 2)\), where \(A\) denotes a sub-Hopf algebra of the mod 2 Steenrod algebra, thereby proving a conjecture of Adams. Call a sub-Hopf algebra of \(A\) elementary if it is an exterior algebra. The main result of the paper is that an element of \(\text{Ext}_ A^{**}({\mathbf F}_ 2,{\mathbf F}_ 2)\) is nilpotent if its restrictions to all elementary sub-Hopf algebras of \(A\) vanish. This result is part of a program for proving structure theorems for Steenrod algebra modules analogous to the results of Devinatz, Hopkins and Smith for spectra.
scientific article; zbMATH DE number 550342