On rank filtrations of algebraic K-theory and Steinberg modules

TL;DR

The work resolves Rognes' connectivity conjecture for the common basis complex in the field case by linking $CB_n(F)$ to higher Tits buildings and an iterated bar-construction framework on Steinberg modules. It develops a cohesive algebraic and combinatorial setup, proving $BD^{k}(R) o D^{k+1}(R)$ and showing $D^{a,b}(M) to ext{Σ}^{a+b+1}T^{a,b}(M)$, which yields high connectivity through inductive Morse arguments. The paper also identifies the Koszul dual of the Steinberg monoid for fields, establishing precise Tor-group computations and a Koszul dual description, thereby clarifying the $E_k$- indecomposables and their relation to the stable Steinberg module. The results have significant implications for the unstable connectivity patterns in the stable rank filtration of $K$-theory and for understanding the representation theory of ${ m GL}_n(F)$ in terms of Steinberg data.

Abstract

Motivated by his work on the stable rank filtration of algebraic K-theory spectra, Rognes defined a simplicial complex called the common basis complex and conjectured that this complex is highly connected for local rings and Euclidean domains. We prove this conjecture in the case of fields. Our methods give a novel description of this common basis complex of a PID as an iterated bar construction on an equivariant monoid built out of Tits buildings. We also identify the Koszul dual of a certain equivariant ring assembled out of Steinberg modules.

Figures (2)

• Figure 1: Let $e_1, e_2, e_3$ denote the standard basis of $R^3$. Two examples are shown of collections of summands $U_1, U_2, U_3$ of $R^3$ and their associated posets. Fibers of $\Phi_{\mathcal{U}}$ are color-coded according to their image. The values of the corank functions are written next to each element in gray.
• Figure 2: In this example, the set $S$ consists of the edge $\sigma_1$, the 2-simplex $\sigma_2$, and the vertex $\sigma_3$ shown in dark gray in the first figure. The subcomplex $Y \subseteq X$ is highlighted in orange in the second figure. The subdivided complex $X'$ is shown in the third, with the newly added simplices shown in black.

Theorems & Definitions (107)

• Conjecture 1: Rognes' connectivity conjecture
• Theorem 1
• Lemma 1
• Corollary 1
• Theorem 2
• Remark 1
• Lemma 2
• Definition 1
• Lemma 3
• proof