This post is part of a series (started here) of posts on the structure of the category of unstable modules over the mod Steenrod algebra , which plays an important role in the proof of the Sullivan conjecture (and its variants).
In the previous post, we introduced some additional structure on the category .
- First, using the (cocommutative) Hopf algebra structure on , we got a symmetric monoidal structure on , which was an algebraic version of the Künneth theorem.
- Second, we described a “Frobenius” functor
which was symmetric monoidal, and which came with a Frobenius map .
- We constructed an exact sequence natural in ,
where was the suspension and the left adjoint. In particular, we showed that all the higher derived functors of (after ) vanish.
The first goal of this post is to use this extra structure to prove the following:
Theorem 39 The category is locally noetherian: the subobjects of the free unstable module satisfy the ascending chain condition (equivalently, are finitely generated as -modules).
In order to prove this theorem, we’ll use induction on and the technology developed in the previous post as a way to make Nakayama-type arguments. Namely, the exact sequence (4) becomes
as we saw in the previous post. Observe that is clearly noetherian (it’s also not hard to check this for ). Inductively, we may assume that (and therefore ) is noetherian.
Fix a subobject ; we’d like to show that is finitely generated. (more…)