Equivariant equivalence between Rognes’s common basis complex and the suspension of the minimal spanning poset nerve

Establish a GL_k(R)-equivariant weak equivalence between the common basis complex Σ^{-1} D^V(R^k) and the unreduced suspension Σ N P^∘, where R is a commutative ring, D^V(R^k) is Rognes’s common basis complex for R^k, and P^∘ is the poset of nontrivial minimal spanning posets in the subobject structure Sub_{R^k}.

Background

In Section 4.3, the authors reinterpret Rognes’s poset filtration for the spectrum D(R^k) by introducing a valuation taking simplices to their submodule configurations, which form minimal spanning posets in Sub_{R^k}. They identify D(R^k) ≃ Σ^∞ Σ N P^∘ in a GL_k(R)-equivariant manner, where P^∘ consists of nontrivial minimal spanning posets.

Rognes also showed D(R^k) ≃ Σ^∞ Σ (Σ^{-1} D^V(R^k)), with D^V(R^k) the common basis complex. Comparing these decompositions leads to a conjectured equivalence between Σ^{-1} D^V(R^k) and Σ N P^∘, refined to hold GL_k(R)-equivariantly.