Habiro lecture 7

V5A2-Habiro-Rings/Habiro_Rings_Notes.tex

@@ -58,6 +58,7 @@ \section*{Preliminaries}
 \include{Lecture 4}
 \include{Lecture 5}
 \include{Lecture 6}
+\include{Lecture 7}
 \newpage
 \printbibliography
 \end{document}

V5A2-Habiro-Rings/Lecture 3.tex

@@ -109,7 +109,7 @@ \section{Lecture 3 -- 8th November 2024}\label{sec: lecture 3}
     where $(\Ocal_{\KK})_{p}^{\wedge}$ is the $p$-adic completion of $\Ocal_{\KK}$ and $\varphi_{p}$ the Frobenius lift on $(\Ocal_{\KK})_{p}^{\wedge}$. 
 \end{definition}
 \begin{remark}
-    $(\Ocal_{\KK})_{p}^{\wedge}$ is a finite \'{e}tale $\ZZ_{p}$-algebra, and the Frobenius lifts uniquely to an endomorphism on the $p$-adic completion. This Frobenius lift explains the 
+    $(\Ocal_{\KK})_{p}^{\wedge}$ is a finite \'{e}tale $\ZZ_{p}$-algebra, and the Frobenius lifts uniquely to an endomorphism on the $p$-adic completion. \'{E}taleness of the $\ZZ_{p}$-algebra explains uniqueness of the Frobenius.  
 \end{remark}
 While this definition is the appropriate generealization of the Habiro ring, it is yet unclear how explicit elements of this ring can be constructed. In fact, there is no map in general $\Ocal_{\KK}\to\Hcal_{\Ocal_{\KK}[\frac{1}{\Delta}]}$ -- the constant map to power series does not satisfy the identity on the Frobenius lift. 
 \begin{remark}

V5A2-Habiro-Rings/Lecture 6.tex

@@ -2,7 +2,7 @@ \section{Lecture 6 -- 29th November 2024}\label{sec: lecture 6}
 We make some recollections from algebraic number theory, largely following Milne's texts. 
 \begin{definition}[Dedekind Zeta Function]\label{def: Dedekind zeta}
     Let $\KK$ be a number field. The Dedekind zeta function is given by 
-    $$\zeta_{\KK}(s)=\sum_{\afrak\subseteq\Ocal_{\KK}}\frac{1}{\Nm(\afrak)}=\prod_{\pfrak\subseteq\Ocal_{\KK}}(1-\Nm(\pfrak)^{-s})^{-1}.$$
+    $$\zeta_{\KK}(s)=\sum_{\afrak\subseteq\Ocal_{\KK}}\frac{1}{\Nm(\afrak)^{s}}=\prod_{\pfrak\subseteq\Ocal_{\KK}}(1-\Nm(\pfrak)^{-s})^{-1}.$$
 \end{definition}
 As in the case of the Riemann zeta function, the Dedekind zeta function can be analytically continued to a meromorphic function on $\CC$ with simple pole at $s=1$. The class number formula gives the behavior of the function at the simple pole $s=1$. We state an adapted variant. Recall the definition of the $\Gamma$-function. 
 \begin{definition}[$\Gamma$-Function]\label{def: gamma function}
@@ -59,7 +59,7 @@ \section{Lecture 6 -- 29th November 2024}\label{sec: lecture 6}
     In the case $i=0$, this recovers the Grothendieck group of vector bundles on $\spec(R)$, the group completion of isomorphism classes of vector bundles modulo the scissor relation. 
 \end{remark}
 A result of Quillen-Borel shows the $K$-groups of rings of integers of number fields are finitely generated. 
-\begin{theorem}[Quillen-Borel, Soul\'{e}; {\cite[I.5, Thm. 6, 7]{KThyHandbook}}]
+\begin{theorem}[Quillen-Borel, Soul\'{e}; {\cite[I.5, Thm. 6, 7]{KThyHandbook}}]\label{thm: ranks of K-groups of number fields}
     Let $\KK$ be a number field. Then 
     $$\mathrm{rank}(K_{2n-1}(\Ocal_{\KK}))=\begin{cases}
         r_{2} & n\equiv0\pmod{2} \\ 
@@ -69,13 +69,13 @@ \section{Lecture 6 -- 29th November 2024}\label{sec: lecture 6}
 \end{theorem}
 We focus on the case $n=2$ considering $K_{3}$ of the number field. More generally, we have the following result of Borel relating the value of the $L$-series of (\ref{eqn: L-series}) can be related to certain Borel regulators. 
 \begin{definition}[Borel Regulator]\label{def: Borel regulator}
-    Let $K$ be a number field. The $n$-th Borel regulator $\Reg_{\KK}(n)$ is the volume of the quotient $(P_{n}/\Lambda)/(P_{n}/\Lambda')$ where $P_{n}$ is the space of primitives in $H_{n}(\mathrm{SL}(R),\RR)$, $\Lambda$ the image of $K_{n}(\Ocal_{\KK})$ in $H_{n}(\mathrm{SL}(R),\RR)$, and $\Lambda'$ the image of the symmetric space. 
+    Let $\KK$ be a number field. The $n$-th Borel regulator $\Reg_{\KK}(n)$ is the volume of the quotient $(P_{n}/\Lambda)/(P_{n}/\Lambda')$ where $P_{n}$ is the space of primitives in $H_{n}(\mathrm{SL}(R),\RR)$, $\Lambda$ the image of $K_{n}(\Ocal_{\KK})$ in $H_{n}(\mathrm{SL}(R),\RR)$, and $\Lambda'$ the image of the symmetric space. 
 \end{definition}
 \begin{remark}
     See \cite[\S IV.1.18.1]{Weibel} for an expanded discussion.
 \end{remark}
 These are related to values of the $L$-series as follows:
-\begin{theorem}[Borel; {\cite[\S IV.1.18.1]{Weibel}}]\label{thm: Borel regulator and }
+\begin{theorem}[Borel; {\cite[\S IV.1.18.1]{Weibel}}]\label{thm: Borel regulator and L-function}
     Let $\KK$ be a number field. The $L$-series satsifies the asymptotic formula 
     \begin{equation}\label{eqn: L-series asymptotic formula}
         L_{\KK}(n)\sim\Reg_{\KK}(n).
@@ -93,7 +93,7 @@ \section{Lecture 6 -- 29th November 2024}\label{sec: lecture 6}
 The relation of \Cref{def: Bloch group} is closely related to Milnor $K$-theory, in particular related in the following way, which can be deduced from \cite[\S 1.5, Thm. 8]{KThyHandbook}.
 \begin{theorem}[Bloch]\label{thm: Bloch rational computation of K3}
     Let $\KK$ be a number field. Then there are isomorphisms
-    $$K_{3}(\KK)\otimes_{\ZZ}\QQ\cong K_{3}(\Ocal_{\KK})\otimes_{\ZZ}\QQ\cong B(\KK).$$
+    $$K_{3}(\KK)\otimes_{\ZZ}\QQ\cong B(\KK).$$
 \end{theorem}
 \begin{remark}
     Conjecturally for $n\geq 2$, we would expect that we could inductively define $\wp_{n}(F)$ to be the quotient of $\QQ[F^{\times}\setminus\{1\}]$ by the $\QQ$-vector subspace generated by functional equations of the $n$th polylogarithm and relate the weight $n$ Goncharov-Zagier complex

V5A2-Habiro-Rings/Lecture 7.tex

@@ -0,0 +1,69 @@
+\section{Lecture 7 -- 6th December 2024}\label{sec: lecture 7}
+Let us consider the construction of algebraic $K$-theory as in \Cref{def: K-theory of a ring}. 
+
+For a commutative ring $R$, we can consider the anima $\Proj(R)$ consisting of finitely generated projective $R$-modules which lies in $\CMon(\Ani)$, the commutative monoid objects in the category of anima, under the direct sum operation. However, within $\Proj(R)$ we can consider the full subcategory spanned by the free $R$-modules $\Free(R)$ which can be obtained as $\coprod_{n\geq0}*/\GL_{n}(R)$ acting by automorphisms on free modules of rank $n$ for each $n$. The inclusion induces a map $\Free(R)^{\infty-\Grp}\to\Proj(R)^{\infty-\Grp}$ where $(-)^{\infty-\Grp}$ denotes group completion as an anima. In fact, this suffices to compute algebraic $K$-theory in strictly positive degrees. 
+\begin{proposition}\label{prop: K-theory on free modules}
+    Let $R$ be a ring. The map of anima $\Free(R)^{\infty-\Grp}\to\Proj(R)^{\infty-\Grp}$ is an isomorphism on homotopy groups for $i\geq 1$. 
+\end{proposition}
+This provides a way to compute the $K$-theory of $R$ via the homotopy, and in fact homology, of $*/\GL_{\infty}(R)$. And while \emph{a priori} this seems like an exceptionally daunting task, homological stability shows that it suffices to compute $H_{i}(*/\GL_{n}(R))$ since the map 
+$$H_{i}(*/\GL_{n})\longrightarrow H_{i}(*/\GL_{n+1})$$
+is an isomorphism for $n$ sufficiently large with respect to $i$. For $R=\Ocal_{\KK}$ for $\KK$ a number field, $K_{i}(\Ocal_{\KK})$ is a finitely generated Abelian group by a result of Borel \Cref{thm: ranks of K-groups of number fields}. Moreover, in these cases, the phenomena are well-studied as the (co)homology of arithmetic groups, which are closely connected to automorphic forms. 
+\begin{example}
+    $H_{i}(*/\SL_{2}(\ZZ))\cong H_{i}(\HH^{\pm}/\SL_{2}(\ZZ))$ where $\HH^{\pm}$ is the union of the upper and lower half plane and the action of $\SL_{2}(\ZZ)$ on $\HH^{\pm}$ by M\"{o}bius transformations. In particular, the desired homology group of $*/\SL_{n}(\ZZ)$ can be computed as the homology of an arithmetic locally symmetric space $\HH^{\pm}/\SL_{2}(\ZZ)$ which on passage to the Borel-Serre compactification is a manifold with corners. 
+
+    More generally for an arithmetic group $\Gamma$, the homology of the quotient $*/\Gamma$ can be computed as the homology of the Borel-Serre compactification of an associated arithmetic locally symmetric space which is a manifold with corners. 
+\end{example}
+We now make some recollections from condensed mathematics to the end of defining condensed $K$-theory which rseults from considering $\coprod_{n\geq0}*/\GL_{n}(\CC)$ as a condensed anima. The upshot of this approach is that it is able to preserve topological information such as local compactness, instead of treating $\GL_{n}(\CC)$ as a mere abstract group. 
+\begin{definition}[Condensed Set]\label{def: condensed set}
+    A condensed set is a sheaf of sets on the site of profinite sets and coverings given by finite families of jointly surjective maps. 
+\end{definition}
+Na\"{i}vely, one would expect that we could define the condensed $K$-theory anima of $\CC$ as the sheafification of the presheaf 
+$$S\mapsto K(\mathrm{Cont}(S,\GL_{n}(\CC)))$$
+where passing to homotopy groups recovers $K$-theory in some fixed degree as a condensed Abelian group. 
+\begin{remark}
+    As is typical in the condensed setting $S=*$ recovers $\mathrm{Cont}(S,\GL_{n}(\CC))=\GL_{n}(\CC)$ which is the ordinary $K$-theory anima.
+\end{remark}
+
+One notices, however that the desired homotopy groups $\CC/(2\pi i)^{n}\ZZ$ are quite similar to products of $\RR/\ZZ$ which are locally compact Abelian groups that satisfy Pontryagin duality. 
+\begin{definition}[Continuous $K$-Theory Anima]\label{def: continuous K-theory anima}
+    The continuous $K$-theory anima $K^{\cont}(\CC)$ is the Pontryagin bidual of the sheafification of the presheaf 
+    $$S\mapsto K(\mathrm{Cont}(S,\GL_{n}(\CC))).$$
+\end{definition}
+The homotopy groups of the continuous $K$-theory anima were computed by Clausen to be the following. 
+\begin{theorem}[Clausen]\label{thm: Clausen condensed k theory}
+    The homotopy groups of the continuous $K$-theory anima are given as follows:
+    $$\pi_{i}K^{\cont}(\CC)\cong\begin{cases}
+        \ZZ & i=0 \\
+        \CC/(2\pi i)^{n}\ZZ & i=2n-1 \\
+        0 & i\equiv0\pmod{2}, i>1.
+    \end{cases}$$
+\end{theorem}
+The proof of \Cref{thm: Clausen condensed k theory} boils down to the computation of condensed cohomology of $*/\GL_{n}(\CC)$ with $\RR$ or $\ZZ$-coefficients. With $\ZZ$-coefficients, the computation of the integral homology of $\coprod_{n\geq 1}*/|\GL_{n}(\CC)|$ gives topological $K$-theory $\ku$ whose homotopy groups are $\ZZ$ in all even degrees and zero otherwise. In the case of $\RR$-coefficients, this is a Lie algebra computation.
+\begin{remark}
+    Conjecturally, the $K$-theory of the liquid and gaseous complex numbers are given by 
+    $$K_{i}(\CC_{\Liq})=K_{i}(\CC_{\Gas})=\begin{cases}
+        \ZZ & i\leq 0, i\equiv0\pmod{2} \\
+        0 & i=0 \text{ or }i>0\text{ and }i\equiv0\pmod{2} \\
+        \CC/(2\pi i)^{n}\ZZ & i>0\text{ and }\equiv1\pmod{2}.
+    \end{cases}$$
+    Moreover, it is expected that this recovers periodic $K$-theory $\KU$. 
+\end{remark}
+
+One can also provide a condensed account of Beilinson's construction of $K$-theory by $\CC_{\Liq}$ or $\CC_{\Gas}$ the liquid and gaseous complex numbers, respectively. 
+
+Recall from the preceding discussion that for a field $F$, $K_{0}(F)\cong\ZZ$ classifing the dimension of finite dimensional vector spaces and $K_{1}(F)=\GL_{\infty}(F)^{\mathsf{ab}}\cong F^{\times}$. In the case of $F=\CC$, we can define $\CC^{\times}\cong K_{1}(\CC)=K_{1}^{\cont}(\CC)\cong\CC/(2\pi i)\ZZ$ where the maps are given by the exponential and the logarithm. More generally, there is a map 
+\begin{equation}\label{eqn: map to continuous k theory}
+    K_{2n-1}(\Ocal_{\KK})\to\left(\prod_{\tau:\KK\to\CC}\CC/(2\pi i)^{n}\ZZ\right)^{\Gal(\CC/\RR)}
+\end{equation}
+which is Galois equivariant under the action of complex conjugation on both $\CC$ and $(2\pi i)^{n}\ZZ$ agree, and hence lands in the Galois invariant part of the product. In particular, for a complex place the value of one embedding is already determined by the other, but for a real place there is only one term in the product which already lies in the Galois invariant part of the quotient. Now, on taking the real part if $n$ is odd, and the imaginary part if $n$ is even, the map (\ref{eqn: map to continuous k theory}) above extends to a map 
+\begin{equation}\label{eqn: extended map to continuous k-theory}
+    K_{2n-1}(\Ocal_{\KK})\to\left(\prod_{\tau:\KK\to\CC}\RR\right)^{\Gal(\CC/\RR)}.
+\end{equation}
+Depending if $n$ is even or odd, the real places may or may not contribute since real places are invariant under the Galois action. Thus the target of (\ref{eqn: extended map to continuous k-theory}) is 
+$$\begin{cases}
+    \RR^{r_{1}+r_{2}} & n\equiv1\pmod{2} \\
+    \RR^{r_{2}} & n\equiv0\pmod{2}
+\end{cases}$$
+
+
+Given a number field $\KK$, each embedding $\tau:\KK\to\CC$ gives rise to a regulator map $K_{2n-1}(\Ocal_{\KK})\to K^{\cont}_{2n-1}(\CC)\cong\CC/(2\pi i)^{n}\ZZ$. 

V5A2-Habiro-Rings/refs.bib

@@ -6,11 +6,16 @@ @misc{stacks-project
   shorthand = {Stacks}
 }
 
+
 @misc{HabiroNumberField,
-  author = {Garoufalidis, Stavros and Scholze, Peter and Wheeler and Zagier, Don},
-  title = {The Habiro Ring of a Number Field},
-  note = {Forthcoming},
-  shorthand = {GS+24}
+      title={The Habiro ring of a number field}, 
+      author={Stavros Garoufalidis and Peter Scholze and Campbell Wheeler and Don Zagier},
+      year={2024},
+      eprint={2412.04241},
+      archivePrefix={arXiv},
+      primaryClass={math.NT},
+      url={https://arxiv.org/abs/2412.04241}, 
+      shorthand = {GS+24}
 }
 
 @Article{HabiroRing,

V5A2-Habiro-Rings/shortcuts.tex

@@ -105,9 +105,19 @@
 \newcommand{\Frob}{\mathrm{Frob}}
 
 \newcommand{\SL}{\mathrm{SL}}
+\newcommand{\GL}{\mathrm{GL}}
 \newcommand{\Li}{\mathrm{Li}}
 \newcommand{\sfPic}{\mathsf{Pic}}
 \newcommand{\img}{\mathrm{Im}}
 \newcommand{\Reg}{\mathrm{Reg}}
 \newcommand{\Dscr}{\EuScript{D}}
-\newcommand{\Ani}{\mathsf{Ani}}
+\newcommand{\Ani}{\mathsf{Ani}}
+\newcommand{\Proj}{\mathsf{Proj}}
+\newcommand{\Free}{\mathsf{Free}}
+\newcommand{\CMon}{\mathsf{CMon}}
+\newcommand{\cond}{\mathsf{cond}}
+\newcommand{\cont}{\mathsf{cont}}
+\newcommand{\Liq}{\mathsf{Liq}}
+\newcommand{\Gas}{\mathsf{Gas}}
+\newcommand{\ku}{\mathsf{ku}}
+\newcommand{\KU}{\mathsf{KU}}