# Jack: Rognes' Rank Filtration and Stable Buildings (Friday, June 17)

## Combinatorics on $\sdot$

:::{.remark}
Recall the Waldhausen $\sdot$ construction: let $\cC$ be a category with cofibrations $\mathsf{cC}$ and weak equivalences $\mathsf{wC}$ where $S_q\cC$ has objects ladders of length/width $q$, regarded as elements of $\Fun(\Ar[q], \cC)$.
Note that $\sdot\cC$ is again Waldhausen, so this construction can be iterated and we define
\[
\K(\cC) = \realize{w\sdot^k \cC}
.\]
Define
\[
r: [q] \to \Ar[q] \\
j &\mapsto (0\mapsto j)
.\]
Since $r$ is a diagram, it can be pulled back to the functor category, so let $r^*$ take a diagram $F$ on $\Ar[q]$ to a digram $\sigma = r^* F$ where $\ell \mapsto \sigma(0) \to \cdots \to \sigma(\ell)$.

> ?

By the pushout condition, we can reduce $\sdot \cC$ to a functor on $[q]$.
:::

:::{.lemma title="?"}
For each $q$ and $n\geq 1$ there are equivalences of categories
\[
(r^*)^n: 
S_q^n \cC \to (r^* S_q)^n \cC
\cof S_q^n \cC \to (r^* \cof S_q)^n \cC \\
wS_q^n \cC \to (r^* wS_q)^n \cC
.\]
:::

:::{.proof title="?"}
View an object of $S_q \cC$. as a diagram $\sigma = r^* F$ in $\cof\cC$ on $[*]$ so that $\sigma(0) = *$ together with a choice of subquotient $\sigma(k)/\sigma(j)$:

\begin{tikzcd}
	{\sigma(j)} && {\sigma(k)} \\
	\\
	{*} && {\sigma(k)/\sigma(i)}
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXHNpZ21hKGopIl0sWzIsMCwiXFxzaWdtYShrKSJdLFsyLDIsIlxcc2lnbWEoaykvXFxzaWdtYShpKSJdLFswLDIsIioiXV0=)

> Todo: fix.

A cofibration $\sigma\to \tau$ is a commutative ladder such that ...?

Choosing subquotients gives a canonical quotient fibration extending the diagram to $\Ar[q]$.
Any consistent choice of subquotients provides an inverse $r^* F\to F$, producing an equivalence.

:::

## Rank filtration on \(\K\dash\)theory 

:::{.remark}
If $R$ is a ring with strong invariant dimensions, so if there exists a split injection $R^n\to R^m$ then $n\leq m$.
Note that $\K(\rmod^{\free,\fg})$ agrees with $\K(\rmod^{\prof, \fg})$ in all positive dimensions, essentially using that $\rmod^{\free} \injects \rmod^{\proj}$ induces a covering on \(\K\dash\)theory spectra.
Note that these spectra are connective!
:::

:::{.definition title="Rank filtration"}
Let $F(R) = \rmod^{\free,\fg}$; there is a filtration by rank $\Fil F(R)$.
Define $\Fil_k \K(R)_n \leq \K(R)_n$ to be the subcomplex realizing the simplicial full subcategory of $[q] \to wS_q^n F(R)$ of diagrams that factor through $\Fil_k F(R)$.
:::

:::{.remark}
The connecting maps $\Sigma \K(R)_n \injects \K(R)_{n+1}$ restrict to the filtration, forming a prespectrum.
The connecting map takes a suspended $q\dash$simplex $F$ to the $(1, q)$ bisimplex $(0\surjects F)$ in $w\sdot \sdot^n F(R)$ which preserves rank.
This yields the $k$th unstable \(\K\dash\)theory of $R$, which approximates $\K(R)$:
\[
\colim_k \pi_k F_k\K(R) \iso \pi_i \K(R)
.\]
:::

:::{.proposition title="?"}
There is an equivalence of spaces
\[
\F_k \K(R)_n \modulo F_{n-1} \K(R)_n \homotopic D(R^k)_n\modulo_h \GL_k(R)
.\]
and
\[
F_k \K R \modulo F_{k-1} \K R \homotopic D(R^k) \modulo_h \GL_k(R)
.\]
:::

:::{.remark}
Idea: $F_k \K(R)_n \modulo F_{k-1} \K(R)_n$ is the realization of some simplicial category $X_\cdot'$, whose objects are diagrams on $\Ar[q]^n$ in $\Sq^n F(R)$ where the top (staircases going up!) module has rank exactly $k$, together with a base object $X_q$, morphisms are isomorphisms of such diagrams.
Let $X_1$ be the category with objects lattices on $[q]^n$ (having zeros and pushouts in the right places) of free \(R\dash\)modules.
By the previous lemma, $r^*: X'_q \to X_q$ induces an equivalence of categories.
Assembling the $X_q$ yields a simplicial category, and (claim) $r^*$ is an equivalence of simplicial categories after defining a choice of $\bd_0$.
:::

:::{.remark}
We now study $X_\cdot$, which has two simplicial structures: $D(R^k)_n \leq Y_\cdot \leq X_\cdot$ where $Y_\cdot$ has objects lattices where the top module is equal to $R^k$ (not just up to isomorphism) and inclusions as cofibrations.
$D(R^k)_n$ has the same objects as $Y_\cdot$, but only the identity morphisms.
Every object in $Y_\cdot$ is isomorphic to an object of $X$, so choosing isomorphism givs a deformation retract on realizations $\realize{X_\cdot} \to \realize{Y_\cdot}$.
Morphisms in $Y_\cdot$ are determined by some action on $R^k$, i.e. an element of $\GL_k(R)$, so $Y_\cdot$ is the simplicial based translate category for the $\GL_k(R)$ action on $D(R^k)_n$.
This yields an equivalence on spaces, and it only remains to lift to spectra -- the inclusions $D(R^k)\injects Y_\cdot \injects X_\cdot$ respect the structure maps on $\K(R)$.
:::

## Barratt-Priddy-Quillen

:::{.remark}
Note that $\Finset_*$ is filtered by cardinality, and there is a functor
\[
\Finset_* &\to \F(R) \\
I &\mapsto R^I
,\]
the free \(R\dash\)module on the set $I$.
:::

:::{.definition title="Axial submodules"}
The **axial submodules** of $R^k$ are free modules of the form $R^I$ for any $I \subseteq K$.
Let $D^*(K) \injects D(R^k)_n$ be the subcomplex of lattice diagrams in axial submodules of $R$.
$D(R^k)_n$ is referred to as a **building**, and let $A_{n, k} = D(K)n \leq D(R^k)_n$ denote the **standard apartment**.
:::

:::{.theorem title="Barratt-Priddy-Quillen"}
$A_{n, k} \cong S^{nk}$, and as a corollary $\K(\Finset_*) = \SS$.
:::

:::{.proof title="?"}
\[
F_k \K(\Sinset_*)_n \modulo F_{k-1} \K(\Finset)_n \iso D(K)_n \modulo_h \Sigma_K \iso S^{nk} \modulo_h \Sigma_k
.\]
When $k=1, S^n \modulo_h \Sigma_1 \cong S^n$, and for $k\geq 2$ the quotient $S^{nk} \modulo_h \Sigma_k$ is $(2n-1)\dash$connected which gives a stable equivalence $\SS = \F_1 \K(\Finset_*)$.
:::