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\begin{document}

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\section*{differential algebraic K-theory}

\begin{quote}%
under construction


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\hypertarget{arithmetic_geometry}{}\paragraph*{{Arithmetic geometry}}\label{arithmetic_geometry}

[[!include arithmetic geometry - contents]]

\hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{absolute_hodge_cohomology}{Absolute Hodge cohomology}\dotfill \pageref*{absolute_hodge_cohomology} \linebreak
\noindent\hyperlink{AlgebraicKTheorySheafOfSpectra}{Algebraic K-theory sheaf of spectra}\dotfill \pageref*{AlgebraicKTheorySheafOfSpectra} \linebreak
\noindent\hyperlink{RefinedBeilinsoRegulator}{The refined Beilinson regulator}\dotfill \pageref*{RefinedBeilinsoRegulator} \linebreak
\noindent\hyperlink{differential_algebraic_ktheory}{Differential algebraic K-theory}\dotfill \pageref*{differential_algebraic_ktheory} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Differential algebraic K-theory is the [[differential cohomology]]-refinement of [[algebraic K-theory]].

In (\hyperlink{BunkeTamme12}{Bunke-Tamme 12}) this is realized effectively as the [[schreiber:differential cohomology in a cohesive topos]] of the [[tangent (∞,1)-topos]] of the [[cohesive (∞,1)-topos]]

\begin{displaymath}
\mathbf{H} \coloneqq Sh_\infty\left(SmthMfd, \mathbf{B} \right)
  \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
  \mathbf{B}
\end{displaymath}
of [[∞-stacks]] on the [[site]] of [[smooth manifolds]] with values in turn in [[∞-stacks]] over a [[site]] of [[arithmetic schemes]], hence by [[smooth ∞-groupoids]] but over a [[base (∞,1)-topos]]

\begin{displaymath}
\mathbf{B}
  \coloneqq
  Sh_\infty\left(Sch_{\mathbb{Z}}\right)
\end{displaymath}
of algebraic [[∞-stacks]].

This may be regarded as sitting inside the [[smooth E-∞-groupoids]].

It is observed in this context that the [[Beilinson regulator]] in [[algebraic K-theory]] is naturally understood as a [[Chern character]] in this perspective of [[differential cohomology]] (\hyperlink{BunkeTamme12}{Bunke-Tamme 12}), which helps with studying it.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{absolute_hodge_cohomology}{}\subsubsection*{{Absolute Hodge cohomology}}\label{absolute_hodge_cohomology}

\begin{defn}
\label{HodgeComplexOverComplexNumbers}\hypertarget{HodgeComplexOverComplexNumbers}{}
Write

\begin{displaymath}
\Omega^\bullet_{\mathbb{C}} \in Stab(Sh_\infty(Sch_{\mathbb{C}}))
\end{displaymath}
for the [[chain complex]] of [[abelian sheaves]] (regarded as a [[sheaf of spectra]] under the [[stable Dold-Kan correspondence]]) which computes [[absolute Hodge cohomology]] of [[complex varieties]].

\end{defn}
(\hyperlink{BunkeTamme12}{Bunke-Tamme 12, section 3.1})

\begin{defn}
\label{HodgeComplexOverIntegers}\hypertarget{HodgeComplexOverIntegers}{}
Write

\begin{displaymath}
\Omega^\bullet_{\mathbb{Z}} \coloneqq compl^\ast \Omega^\bullet_{\mathbb{C}} \in Stab(\mathbf{B})
\end{displaymath}
for the [[inverse image]] of $\Omega^\bullet_{\mathbb{C}}$ under the [[base change]] given by

\begin{displaymath}
compl
  \coloneqq
  (-)\times_{\mathbb{Z}}Spec(\mathbb{C}) 
  \;\colon\; 
  Sch_{\mathbb{Z}}\longrightarrow Sch_{\mathbb{C}}
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{BunkeTamme12}{Bunke-Tamme 12, section 3.2})

There is a resolution of $\Omega^\bullet_{\mathbb{Z}} \in Stab(\mathbf{B}) \stackrel{Disc}{\hookrightarrow} Stab(\mathbf{H})$ by a sheaf of complexes of differential forms on smooth manifolds tensored with $\Omega^\bullet_{\mathbb{Z}}$

\begin{displaymath}
\Omega^\bullet \in Stab(\mathbf{H})
\end{displaymath}
(\hyperlink{BunkeTamme12}{Bunke-Tamme 12, (47)}).

\begin{defn}
\label{CyclesForHodgeComplexOverComplexNumbers}\hypertarget{CyclesForHodgeComplexOverComplexNumbers}{}
While $\Omega^\bullet \simeq \Omega^\bullet_{\mathbb{Z}}$, below we use the chain-level truncation $\Omega^{\bullet \geq 0}$ which is no longer in the image of $Disc$, hence no longer a [[flat modality]]-[[modal type]].

\end{defn}
\hypertarget{AlgebraicKTheorySheafOfSpectra}{}\subsubsection*{{Algebraic K-theory sheaf of spectra}}\label{AlgebraicKTheorySheafOfSpectra}

Write

\begin{displaymath}
\mathbf{Vect}_{lc} \in \mathbf{H}
\end{displaymath}
for the [[stack]] which to $X\times S \in SmthMfd \times Sch_{\mathbb{Z}}$ assigns the groupoid of locally free and locally finitely generated $pr_S^\ast \mathcal{O}_S$-[[modules]] (modules over the [[inverse image]] of the [[structure sheaf]] of $S$ under the [[projection]] map $X \times S \to S$).

This is a [[commutative monoid]] object with respect to [[direct sum]]. Write

\begin{displaymath}
K \coloneqq \mathcal{K}(\mathbf{Vect}_{lc}^{\oplus})
\end{displaymath}
for the stackification of the objectwise [[K-theory of a symmetric monoidal (∞,1)-category]]-construction.

This is the ordinary [[algebraic K-theory]] of [[schemes]], as in (\hyperlink{ThomasonTrobaugh90}{Thomason-Trobaugh 90}) (\hyperlink{BunkeTamme12}{Bunke-Tamme 12, section 3.3}), see at \emph{\href{algebraic%20K-theory#AsTheKTheoryOfAlgebraicVectorBundles}{algebraic K-theory -- as the K-theory of algebraic vector bundles}}.

\hypertarget{RefinedBeilinsoRegulator}{}\subsubsection*{{The refined Beilinson regulator}}\label{RefinedBeilinsoRegulator}

There is a refinement of the [[Beilinson regulator]] to a smoothly parameterized version $\mathbf{K}$ of [[algebraic K-theory]]:

\begin{defn}
\label{TheBeilinsonRegulator}\hypertarget{TheBeilinsonRegulator}{}
(\ldots{})

\begin{displaymath}
r^{Beil} 
  \;\colon\;
  K 
    \longrightarrow
  \Omega^\bullet_{\mathbb{Z}}
\end{displaymath}
\end{defn}
As a homomorphism of [[spectrum objects]] this is (\hyperlink{BunkeTamme12}{Bunke-Tamme 12, def. 4.26}). As a homomorphism of [[E-∞ ring]] objects, this is (\hyperlink{BunkeTamme13}{Bunke-Tamme 13, def. 2.18}).

\hypertarget{differential_algebraic_ktheory}{}\subsubsection*{{Differential algebraic K-theory}}\label{differential_algebraic_ktheory}

Recall the discussion at \emph{[[differential cohomology hexagon]]}.

\begin{defn}
\label{HodgeComplexOverComplexNumbers}\hypertarget{HodgeComplexOverComplexNumbers}{}
\emph{Differential algebraic K-theory} is the [[homotopy fiber product]]

\begin{displaymath}
\hat K \coloneqq K \underset{\Omega^\bullet}{\times} \Omega^{\bullet \geq 0}  \in Stab(\mathbf{H})
\end{displaymath}
of the inclusion of the non-negative degree truncation of the de Rham resolution of the absolute Hodge complex, def. \ref{CyclesForHodgeComplexOverComplexNumbers}, with the refined Beilinson regulator, def. \ref{TheBeilinsonRegulator}

\begin{displaymath}
\itexarray{
    \hat K &\longrightarrow& \Omega^{\bullet \geq 0}
    \\
    \downarrow &(pb)& \downarrow
    \\
    K &\underset{r^{Beil}}{\longrightarrow}& \Omega^\bullet
  }
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{BunkeTamme12}{Bunke-Tamme 12, def. 5.1}, \hyperlink{BunkeTamme13}{Bunke-Tamme 13, def. 3.1}).

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item differential refinement of [[Becker-Gottlieb transfer]] and its image under [[Borel regulators]]. See at \emph{[[transfer index conjecture]]}.

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[differential K-theory]]


\item [[differential form with logarithmic singularities]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Differential algebraic K-theory as above is introduced and studied in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Georg Tamme]], \emph{Regulators and cycle maps in higher-dimensional differential algebraic K-theory} (\href{http://arxiv.org/abs/1209.6451}{arXiv:1209.6451})


\item [[Ulrich Bunke]], [[David Gepner]], \emph{Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory} (\href{http://arxiv.org/abs/1306.0247}{arXiv:1306.0247})


\item [[Ulrich Bunke]], [[Georg Tamme]], \emph{Multiplicative differential algebraic K-theory and applications} (\href{http://arxiv.org/abs/1311.1421}{arXiv:1311.1421})



\end{itemize}
Relevant references in ordinary [[algebraic K-theory]] include

\begin{itemize}%
\item [[Robert Thomason]] and Thomas Trobaugh, \emph{Higher algebraic K-theory of schemes and of derived categories}, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhauser Boston, Boston, MA, 1990, pp. 247-435. MR 1106918 (92f:19001)

\end{itemize}
[[!redirects differential algebraic K-theory]]

[[!redirects differential algebraic K-theories]]

[[!redirects differential algebraic K-theories]]



\end{document}