Support varieties for the Steenrod algebra (Q1127711)

Let \(\mathcal A\) be the mod \(p\) Steenrod algebra. Algebraic topologists would like to understand the cohomology of \(\mathcal A\), \(\text{Ext}_{\mathcal A}^*({\mathbb{F}}_p, {\mathbb{F}}_p)\), and one approach is to approximate \(\mathcal A\) by finite dimensional sub-Hopf algebras and compute their cohomology. For technical reasons, the authors work over \(k\), the algebraic closure of \({\mathbb{F}}_p\) so let \(A= {\mathcal A} \otimes k\), let \(B\) be a finite dimensional sub-Hopf algebra of \(A\) and let \(M\) be a finite dimensional \(B\)-module. Define the support variety of \(M\), \(| B| _M\), to be the (maximal ideal) spectrum of \(\text{Ext}_B^*(k, k) / J_B(M)\), where \(J_B(M)\) is the annihilator ideal of \(\text{Ext}_B^*(M,M)\), considered as a module over \(\text{Ext}_B^*(k, k)\) via the map \( - \otimes M : \text{Ext}_B^*(k,k) \rightarrow \text{Ext}_B^*(M,M)\). The authors identify \(| B| _M\) for all such \(B\) and \(M\), for example, \(| B| _k\) is a union of affine subvarieties, one for each quasi-elementary sub-Hopf algebra of \(B\), and they conclude that \(\text{Ext}_B^*(M,M)\) is more or less determined, modulo nilpotence.