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\section*{algebraic K-theory}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{constructions}{Constructions}\dotfill \pageref*{constructions} \linebreak
\noindent\hyperlink{symmetric_monoidal_ktheory}{Symmetric monoidal K-theory}\dotfill \pageref*{symmetric_monoidal_ktheory} \linebreak
\noindent\hyperlink{quillen_qconstruction_for_exact_categories}{Quillen Q-construction (for exact categories)}\dotfill \pageref*{quillen_qconstruction_for_exact_categories} \linebreak
\noindent\hyperlink{waldhausen_construction_for_waldhausen_categories}{Waldhausen $S_\bullet$-construction (for Waldhausen categories)}\dotfill \pageref*{waldhausen_construction_for_waldhausen_categories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{for_rings}{For rings}\dotfill \pageref*{for_rings} \linebreak
\noindent\hyperlink{plus_construction}{Plus construction}\dotfill \pageref*{plus_construction} \linebreak
\noindent\hyperlink{direct_sum_ktheory}{Direct sum K-theory}\dotfill \pageref*{direct_sum_ktheory} \linebreak
\noindent\hyperlink{exact_ktheory}{Exact K-theory}\dotfill \pageref*{exact_ktheory} \linebreak
\noindent\hyperlink{for_schemes}{For schemes}\dotfill \pageref*{for_schemes} \linebreak
\noindent\hyperlink{quillen_ktheory}{Quillen K-theory}\dotfill \pageref*{quillen_ktheory} \linebreak
\noindent\hyperlink{thomasontrobaugh_ktheory}{Thomason-Trobaugh K-theory}\dotfill \pageref*{thomasontrobaugh_ktheory} \linebreak
\noindent\hyperlink{for_smooth_manifolds}{For smooth manifolds}\dotfill \pageref*{for_smooth_manifolds} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{chern_characters}{Chern characters}\dotfill \pageref*{chern_characters} \linebreak
\noindent\hyperlink{regulators_and_relation_to_ordinary_cohomology}{Regulators and relation to ordinary cohomology}\dotfill \pageref*{regulators_and_relation_to_ordinary_cohomology} \linebreak
\noindent\hyperlink{cyclotomic_trace_and_relation_to_topological_hochschild_homology}{Cyclotomic trace and relation to topological Hochschild homology}\dotfill \pageref*{cyclotomic_trace_and_relation_to_topological_hochschild_homology} \linebreak
\noindent\hyperlink{comparison_map_and_relation_to_topological_ktheory}{Comparison map and Relation to topological K-theory}\dotfill \pageref*{comparison_map_and_relation_to_topological_ktheory} \linebreak
\noindent\hyperlink{Descent}{Descent}\dotfill \pageref*{Descent} \linebreak
\noindent\hyperlink{zariski_and_nisnevich_descent}{Zariski and Nisnevich descent}\dotfill \pageref*{zariski_and_nisnevich_descent} \linebreak
\noindent\hyperlink{etale_descent}{Etale descent}\dotfill \pageref*{etale_descent} \linebreak
\noindent\hyperlink{AsTheKTheoryOfAlgebraicVectorBundles}{Description of the K-theory sheaf via algebraic vector bundles}\dotfill \pageref*{AsTheKTheoryOfAlgebraicVectorBundles} \linebreak
\noindent\hyperlink{RelationToKKAndMotives}{Relation to non-commutative topology and non-commutative motives}\dotfill \pageref*{RelationToKKAndMotives} \linebreak
\noindent\hyperlink{redshift_conjecture}{Red-shift conjecture}\dotfill \pageref*{redshift_conjecture} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{OnMonoidalStacks}{On monoidal stacks}\dotfill \pageref*{OnMonoidalStacks} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{introductions}{Introductions}\dotfill \pageref*{introductions} \linebreak
\noindent\hyperlink{classical}{Classical}\dotfill \pageref*{classical} \linebreak
\noindent\hyperlink{ReferencesAlgebraicKTheoryForQuotientStacks}{Algebraic K-theory of quotient stacks}\dotfill \pageref*{ReferencesAlgebraicKTheoryForQuotientStacks} \linebreak
\noindent\hyperlink{algebraic_ktheory_of_ring_spectra}{Algebraic K-theory of ring spectra}\dotfill \pageref*{algebraic_ktheory_of_ring_spectra} \linebreak
\noindent\hyperlink{via_stable_categories}{Via stable $(\infty,1)$-categories}\dotfill \pageref*{via_stable_categories} \linebreak
\noindent\hyperlink{via_symmetric_monoidal_categories}{Via symmetric monoidal $(\infty,1)$-categories}\dotfill \pageref*{via_symmetric_monoidal_categories} \linebreak
\noindent\hyperlink{ktheory_stacks}{K-theory stacks}\dotfill \pageref*{ktheory_stacks} \linebreak
\noindent\hyperlink{examples_3}{Examples}\dotfill \pageref*{examples_3} \linebreak


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

Algebraic K-theory is about natural constructions of [[cohomology theories]]/[[spectra]] from [[algebra|algebraic]] data such as [[commutative rings]], [[symmetric monoidal categories]] and various [[homotopy theory|homotopy theoretic]] refinements of these.

From a modern perspective, the algebraic K-theory [[spectrum]] $\mathbf{K}(R)$ of a [[commutative ring]] is simply the [[K-theory of a symmetric monoidal category|∞-group completion]] of [[algebraic vector bundles]] on $Spec(R)$; this will be discussed in more detail \hyperlink{AsTheKTheoryOfAlgebraicVectorBundles}{below}. In particular there is a natural concept of algebraic K-theory of ``[[brave new rings]]'', i.e. of [[ring spectra]]/[[E-∞ rings]].

Historically, the \emph{algebraic K-theory} of a [[commutative ring]] $R$ (what today is the ``0th'' algebraic K-theory group) was originally defined to be the [[Grothendieck group]] of its [[symmetric monoidal category]] of [[projective modules]] (under [[tensor product]] of modules). Under the \href{module#RelationToVectorBundlesInIntroduction}{relation between modules and vector bundles}, this is directly analogous to the basic definition of [[topological K-theory]], whence the common term. (In fact when applied to the \emph{[[stack]]} of [[vector bundles]] then algebraic K-theory subsumes [[topological K-theory]] and also [[differential K-theory]], see \hyperlink{OnMonoidalStacks}{below}).

There are canonical maps $K_0(R)\to Pic(R)$ from the 0th algebraic K-theory of a ring to its [[Picard group]] and $K_1(R)\to GL_1(R)$ from the first algebraic K-theory group of $R$ to its [[group of units]] which are given in components by the [[determinant]] functor. This fact is sometimes used to motivate algebraic K-theory as a ``generalization of [[linear algebra]]'' (see e.g. \href{http://mathoverflow.net/a/171369/381}{this MO discussion}). This is also how the traditional [[regulator of a number field]] relates to [[Beilinson regulators]] of algebraic K-theory.

More generally, following the axiomatics of \emph{[[generalized (Eilenberg-Steenrod) cohomology]]} any \emph{algebraic K-theory} should be given by a sequence of [[functors]] $K_i$ from some suitable class of [[categories]] of ``algebraic nature'' to [[abelian groups]], satisfying some natural conditions. Moreover, following the [[Brown representability theorem]] these groups should arise as the [[homotopy groups]] of a [[spectrum]], the algebraic \emph{[[K-theory spectrum]]}. Classical constructions producing this by combinatorial means are known as the \emph{[[Quillen Q-construction]]} defined on \emph{[[Quillen exact categories]]} and more generally the \emph{[[Waldhausen S-construction]]} defined on \emph{[[Waldhausen categories]]}.

For more on the history of the subject see (\hyperlink{Arlettaz04}{Arlettaz 04}, \hyperlink{Grayson13}{Grayson 13}) and see at at \emph{[[Algebraic K-theory, a historical perspective]]}.

There are two ways to think of the traditional algebraic K-theory of a commutative ring more conceptually: on the one hand this construction is the [[group completion]] of the [[direct sum]] [[symmetric monoidal category|symmetric monoidal]]-structure on the [[category of modules]], on the other hand it is the group completion of the addition operation expressed by [[short exact sequences]] in that category. This leads to the two modern ways of expressing and viewing algebraic K-theory:

\begin{enumerate}%
\item \textbf{monoidal.} The [[core]] of a [[symmetric monoidal category]] or more generally of a [[symmetric monoidal (∞,1)-category]] has a [[universal construction|universal]] completion to an [[abelian ∞-group]]/[[connective spectrum]] optained by universally adjoining [[inverses]] to the symmetric monoidal operation -- the [[∞-group completion]]. This yields the concept of [[K-theory of a permutative category|algebraic K-theory of a symmetric monoidal category]] and more generally that of [[algebraic K-theory of a symmetric monoidal (∞,1)-category]];


\item \textbf{exact/stable.} Analogously, inverting the addition operation expressed by the [[exact sequences]] in an [[abelian category]] or more generally in a [[stable (∞,1)-category]] yields the [[algebraic K-theory of a stable (∞,1)-category]]. Explicit ways to express this are known as the \emph{[[Quillen Q-construction]]} and the \emph{[[Waldhausen S-construction]]}. This turns out to be a universal construction in the context of [[non-commutative motives]].



\end{enumerate}
Here the second construction may be understood as first [[split exact sequence|splitting]] the [[exact sequences]] and then applying the first construction to the resulting [[direct sum]] monoidal structure. Typically the first construction here contains more information but is harder to compute, and vice versa (see also MO-discussion \href{http://mathoverflow.net/a/98602/381}{here} and \href{http://mathoverflow.net/a/102583/381}{here}).

Both of these constructions produce a [[spectrum]] (hence [[Brown representability theorem|representing]] a [[generalized (Eilenberg-Steenrod) cohomology theory]]) -- called the \emph{[[K-theory spectrum]]} -- and the algebraic K-theory groups are the [[homotopy groups]] of that spectrum.

The classical case of the algebraic K-theory of a commutative ring $R$ is a special case of this general concept of algebraic K-theory by either forming the [[symmetric monoidal category]] $(Mod(R), \oplus)$ and applying the [[abelian ∞-group]]-completion to that, or else forming the [[stable (∞,1)-category of chain complexes]] of $R$-modules and applyong the [[Waldhausen S-construction]] to that. In both cases the result is a [[spectrum]] whose degree-0 [[homotopy group]] is the ordinary algebraic K-theory of $R$ as given by the [[Grothendieck group]] and whose higher homotopy groups are its \emph{higher algebraic K-theory} groups.

\hypertarget{constructions}{}\subsection*{{Constructions}}\label{constructions}

\hypertarget{symmetric_monoidal_ktheory}{}\subsubsection*{{Symmetric monoidal K-theory}}\label{symmetric_monoidal_ktheory}

For a [[symmetric monoidal category]] $C$, K-theory may be defined by taking

\begin{itemize}%
\item the [[maximal subgroupoid]] $i C \subset C$,
\item the [[classifying space]] $B(i C)$ (a [[topological monoid]]),
\item the [[group completion]] $\Omega B B (i C)$.

\end{itemize}
See at

\begin{itemize}%
\item [[K-theory of a symmetric monoidal (∞,1)-category]]


\item [[K-theory of a permutative category]]


\item [[K-theory of a bipermutative category]]



\end{itemize}
\hypertarget{quillen_qconstruction_for_exact_categories}{}\subsubsection*{{Quillen Q-construction (for exact categories)}}\label{quillen_qconstruction_for_exact_categories}

Given an [[Quillen exact category]] $E$, one defines $K(E)$ by applying

\begin{itemize}%
\item the [[Quillen Q-construction]] $Q(E)$,
\item the [[group completion]] $\Omega B Q(E)$.

\end{itemize}
See at

\begin{itemize}%
\item [[Quillen Q-construction]]

\end{itemize}
\hypertarget{waldhausen_construction_for_waldhausen_categories}{}\subsubsection*{{Waldhausen $S_\bullet$-construction (for Waldhausen categories)}}\label{waldhausen_construction_for_waldhausen_categories}

Given a [[Waldhausen category]] $(C, w C)$, one defines its $K$-theory by applying

\begin{itemize}%
\item the [[Waldhausen S-construction]] $w S_\bullet C$ (a [[simplicial category]]),
\item the [[nerve]] (a [[bisimplicial set]]),
\item the [[colimit]] (a [[simplicial set]]),
\item the [[geometric realization]] (a [[space]]).

\end{itemize}
There is also a [[Waldhausen S-construction]] for [[stable (infinity,1)-categories]] and, most generally, for [[Waldhausen (infinity,1)-categories]].

See at

\begin{itemize}%
\item [[Waldhausen S-construction]]
\item [[K-theory of a stable (infinity,1)-category]]

\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{for_rings}{}\subsubsection*{{For rings}}\label{for_rings}

We recall several constructions of the [[algebraic K-theory]] of a [[ring]]. See (\hyperlink{Weibel}{Weibel, IV.4.8, IV.4.11.1}) for details.

\hypertarget{plus_construction}{}\paragraph*{{Plus construction}}\label{plus_construction}

Given an associative unital [[ring]] $R$, one may define the algebraic K-theory [[space]] $K(R) = BGL(R)^+$ by taking

\begin{itemize}%
\item the [[general linear groups]] $GL_n(R)$ for $n \ge 0$,
\item their [[classifying spaces]],
\item the [[colimit]] $BGL(R) = colim_n BGL_n(R)$,
\item the [[Quillen plus construction]].

\end{itemize}
\hypertarget{direct_sum_ktheory}{}\paragraph*{{Direct sum K-theory}}\label{direct_sum_ktheory}

Consider the category $P(X)$ of [[finitely generated]] [[projective]] (right) $R$-modules. It has a [[symmetric monoidal structure]] given by [[direct sum]]. The algebraic K-theory $K(R)$ may be described as the [[K-theory of a symmetric monoidal (infinity,1)-category]] of $P(R)$. That is, it is the [[group completion]] $K(R) = \Omega B B (i P(X))$ where $i P(X)$ denotes the [[maximal subgroupoid]]. See (\hyperlink{Weibel}{Weibel, IV.4.8, IV.4.11.1}).

\hypertarget{exact_ktheory}{}\paragraph*{{Exact K-theory}}\label{exact_ktheory}

Consider the category $P(X)$ of [[finitely generated]] [[projective module|projective]] (right) $R$-modules. This is an [[exact category]] and the K-theory $K(R)$ may be described via the [[Quillen Q-construction]]:

\begin{displaymath}
K(R) = \Omega B (Q(P(R)).
\end{displaymath}
\hypertarget{for_schemes}{}\subsubsection*{{For schemes}}\label{for_schemes}

For [[schemes]], there are two constructions which do not agree in full generality. See \hyperlink{ThomasonTrobaugh90}{Thomason-Trobaugh 90}.

\hypertarget{quillen_ktheory}{}\paragraph*{{Quillen K-theory}}\label{quillen_ktheory}

The Quillen K-theory of a [[scheme]] $X$ is defined as the algebraic K-theory of the [[exact category]] $Vect(X)$ of [[vector bundles]] on $X$ (using the [[Quillen Q-construction]]).

\hypertarget{thomasontrobaugh_ktheory}{}\paragraph*{{Thomason-Trobaugh K-theory}}\label{thomasontrobaugh_ktheory}

Let $Perf(X)$ be the category of [[perfect complexes]] on $X$. This admits the structure of a [[Waldhausen category]], and the Thomason-Trobaugh K-theory of $X$ is defined via the [[Waldhausen S-construction]].

It may also be defined as the [[K-theory of a stable (infinity,1)-category]] of $Perf(X)$ viewed as a [[stable (infinity,1)-category]].

Thomason-Trobaugh K-theory coincides with Quillen K-theory for schemes that admit an ample family of [[line bundles]], but has the advantage of better global descent properties.

\hypertarget{for_smooth_manifolds}{}\subsubsection*{{For smooth manifolds}}\label{for_smooth_manifolds}

Discussion of algebraic K-theory as a [[smooth spectrum]] $SmoothMfd^{op} \longrightarrow Spectra$ via $X \mapsto K(C^\infty(X))$ is in (\hyperlink{BunkeNikolausVoelkl13}{Bunke-Nikolaus-Voelkl 13}, \hyperlink{Bunke14}{Bunke 14}).

For more on this see at

\begin{itemize}%
\item [[algebraic K-theory of smooth manifolds]]


\item [[differential cohomology hexagon]], section \emph{\href{differential+cohomology+diagram#SoothVectorBundlesWithConnectionAndEInvariant}{Algebraic K-theory of smooth manifolds and the e-invariant}}



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{chern_characters}{}\subsubsection*{{Chern characters}}\label{chern_characters}

\hypertarget{regulators_and_relation_to_ordinary_cohomology}{}\paragraph*{{Regulators and relation to ordinary cohomology}}\label{regulators_and_relation_to_ordinary_cohomology}

See at \emph{[[Beilinson regulator]]}.

\hypertarget{cyclotomic_trace_and_relation_to_topological_hochschild_homology}{}\paragraph*{{Cyclotomic trace and relation to topological Hochschild homology}}\label{cyclotomic_trace_and_relation_to_topological_hochschild_homology}

Given a ring $R$, then there is a natural morphism of [[spectra]]

$\backslash$begin\{tikzcd\} \& TC(R) $\backslash$arrowd$\backslash$ K(R) $\backslash$arrowru $\backslash$arrowr \& THH(R)$\backslash$ $\backslash$end\{tikzcd\}

from the algebraic K-theory spectrum to the [[topological Hochschild homology]] spectrum and factoring through the [[topological cyclic homology]] spectrum called the \emph{[[cyclotomic trace]]} which much like a [[Chern character]] map for algebraic K-theory.

\hypertarget{comparison_map_and_relation_to_topological_ktheory}{}\subsubsection*{{Comparison map and Relation to topological K-theory}}\label{comparison_map_and_relation_to_topological_ktheory}

\begin{itemize}%
\item [[comparison map between algebraic and topological K-theory]]

\end{itemize}
\hypertarget{Descent}{}\subsubsection*{{Descent}}\label{Descent}

See also

\begin{itemize}%
\item Moritz Kerz, Florian Strunk, [[Georg Tamme]], \emph{Algebraic K-theory and descent for blow-ups} (\href{https://arxiv.org/abs/1611.08466}{arXiv:1611.08466})

\end{itemize}
\hypertarget{zariski_and_nisnevich_descent}{}\paragraph*{{Zariski and Nisnevich descent}}\label{zariski_and_nisnevich_descent}

The algebraic K-theory spectrum $\mathbf{K}$ satisfies [[descent]] to give a [[sheaf]] of [[connective spectra]] on the [[Zariski site]]. For regular noetherian schemes this statement is due to (\hyperlink{BrownGersten73}{Brown Gersten 73}). The generalization to finite dimensional noetherian schemes is due to (\hyperlink{ThomasonTrobaugh90}{Thomason-Trobaugh 90}).

Moreover, $\mathbf{K}$ satisfies descent with respect to the [[Nisnevich site|Nisnevich topology]] (which lies between Zariski and \'e{}tale). This is due to (\hyperlink{Nisnevich89}{Nisnevich 89}) and was generalized in turn to finite dimensional noetherian schemes in the same paper of Thomason.

Further generalization of the descent result to finite dimensional quasi-compact quasi-separated schemes is due to (\hyperlink{Rosenschon06}{Rosenschon 06}).

\hypertarget{etale_descent}{}\paragraph*{{Etale descent}}\label{etale_descent}

The question of descent of $\mathbf{K}$ over the [[étale site]] is closely related to the [[Lichtenbaum-Quillen conjecture]], see also (\hyperlink{Thomason85}{Thomason 85}). This is now a theorem of Rost and Voevodsky and it implies that K-theory does satisfy etale descent in sufficient large degrees.

\begin{quote}%
\href{http://mathoverflow.net/a/180265/381}{MO comment}


\end{quote}
\hypertarget{AsTheKTheoryOfAlgebraicVectorBundles}{}\paragraph*{{Description of the K-theory sheaf via algebraic vector bundles}}\label{AsTheKTheoryOfAlgebraicVectorBundles}

Let $Sch$ denote the gros [[Zariski site]] of regular, separated, noetherian schemes. It is explained in (\hyperlink{BunkeTamme12}{Bunke-Tamme 12, section 3.3} that the presheaf of [[spectra]] on $Sch$ defined by algebraic K-theory admits the following description.

Regard the [[stack]] $\mathbf{Vect}^\oplus$ of [[algebraic vector bundles]] on $Sch$ as taking values in [[symmetric monoidal (∞,1)-categories]], via the [[direct sum]] of vector bundles. Then apply the [[K-theory of a symmetric monoidal (∞,1)-category]]-construction $\mathcal{K}$ to this, yielding a [[sheaf of spectra]]. This identifies with the usual Thomason-Trobaugh K-theory sheaf,a fact that follows from

\begin{enumerate}%
\item Zariski descent for Thomason-Trobaugh K-theory,


\item the Zariski-local equivalence between Thomason-Trobaugh K-theory, Quillen K-theory, and direct sum K-theory.



\end{enumerate}
\hypertarget{RelationToKKAndMotives}{}\subsubsection*{{Relation to non-commutative topology and non-commutative motives}}\label{RelationToKKAndMotives}

[[!include noncommutative motives - table]]

\hypertarget{redshift_conjecture}{}\subsubsection*{{Red-shift conjecture}}\label{redshift_conjecture}

\begin{itemize}%
\item [[red-shift conjecture]]

\end{itemize}
[[!include chromatic tower examples - table]]

\hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2}

\hypertarget{OnMonoidalStacks}{}\subsubsection*{{On monoidal stacks}}\label{OnMonoidalStacks}

Algebraic K-theory is traditionally applied to single [[symmetric monoidal (∞,1)-categories|symmetric monoidal]]/[[stable (∞,1)-category|stable]] [[(∞,1)-categories]], but to the extent that it is [[(∞,1)-functor|functorial]] it may just as well be applied to [[(∞,1)-sheaves]] with values in these.

Notably, applied to the monoidal [[stack]] of [[vector bundles]] (with [[connection on a bundle|connection]]) on the [[site]] of [[smooth manifolds]], the [[K-theory of a symmetric monoidal (∞,1)-category|K-theory of a monoidal category]]-functor produces a [[sheaf of spectra]] which is a form of [[differential K-theory]] and whose [[geometric realization]] is the [[topological K-theory]] spectrum. For more on this see at \emph{\href{differential%20cohomology%20diagram#DifferentialKTheory}{differential cohomology hexagon -- Differential K-theory}}.

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

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

\end{itemize}
Types of categories for which a theory of algebraic K-theory exist include notably the notions

\begin{itemize}%
\item [[Quillen exact category]],


\item [[abelian category]],


\item [[Waldhausen category]],


\item [[triangulated category]].



\end{itemize}
Concrete examples of interest include for instance

\begin{itemize}%
\item the category of finitely generated [[projective object]]s over a unital $k$-[[associative unital algebra|algebra]],


\item the category of [[coherent sheaf|coherent sheaves]] over a [[noetherian ring|noetherian]] [[scheme]],


\item the category of locally free sheaves over a scheme,


\item [[Gersten resolution]]


\item [[Milnor's K2]] ([[Steinberg group]], [[universal central extension]])


\item [[higher algebraic K-theory]] [[Quillen exact category]], [[Quillen's Q-construction]], [[Waldhausen S-construction]], [[Volodin spaces]];


\item [[topological cyclic homology]], [[algebraic K-theory of operator algebras]]


\item [[Beilinson regulator]]


\item [[differential algebraic K-theory]]


\item [[non-connective algebraic K-theory]]


\item [[real algebraic K-theory]]


\item [[bivariant algebraic K-theory]]


\item [[K-motive]]


\item [[red-shift conjecture]]


\item [[filtrations on algebraic K-theory]]



\end{itemize}
[[!include noncommutative motives - table]]

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

\hypertarget{introductions}{}\subsubsection*{{Introductions}}\label{introductions}

Surveys with accounts of the historical development include

\begin{itemize}%
\item [[Dominique Arlettaz]], \emph{Algebraic K-theory of rings from a topological viewpoint} (\href{http://www.math.uiuc.edu/K-theory/0420/Arlettaz-survey.pdf}{pdf})


\item [[Daniel Grayson]], \emph{Quillen's work in algebraic K-theory}, J. K-Theory 11 (2013), 527--547 \href{http://www.math.uiuc.edu/~dan/Papers/qs-published-final.pdf}{pdf}



\end{itemize}
An introductory textbook account is in

\begin{itemize}%
\item [[Charles Weibel]], \emph{The K-Book: An introduction to algebraic K-theory} (\href{http://www.math.rutgers.edu/~weibel/Kbook.html}{web})

\end{itemize}
Further review includes

\begin{itemize}%
\item Olivier Isely, \emph{Algebraic $K$-theory}, 2005-06 ([[IselyKTheory.pdf:file]])


\item [[Teena Gerhardt]], \emph{Computations in algebraic K-theory}, talk at \href{http://qcpages.qc.cuny.edu/~swilson/cunyworkshop14.html}{CUNY Workshop on differential cohomologies 2014} (\href{http://videostreaming.gc.cuny.edu/videos/video/1800/in/channel/55/}{video recording})



\end{itemize}
Review of the relation to [[Dennis trace]], [[topological cyclic homology]] and [[topological Hochschild homology]] is in

\begin{itemize}%
\item [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], \emph{The local structure of algebraic K-theory}, Springer 2013 (\href{http://math.mit.edu/~nrozen/juvitop/dundas.pdf}{pdf})

\end{itemize}
\hypertarget{classical}{}\subsubsection*{{Classical}}\label{classical}

Original articles include

\begin{itemize}%
\item [[Daniel Quillen]], \emph{Higher algebraic K-theory}, in Higher K-theories, pp. 85--147, Proc. Seattle 1972, Lec. Notes Math. 341, Springer 1973. (\href{http://math.mit.edu/~hrm/kansem/quillen-higher-algebraic-k-theory.pdf}{pdf})

also: [[Daniel Grayson]], \emph{Higher algebraic K-theory II, after Daniel Quillen} (\href{http://www.math.illinois.edu/~dan/Papers/HigherAlgKThyII.pdf}{pdf})


\item [[Kenneth Brown]], Stephen M. Gersten, \emph{Algebraic K-theory as generalized sheaf cohomology}, Higher K-Theories, Lecture Notes in Mathematics Volume 341, 1973, pp 266-292.


\item [[F. Waldhausen]], \emph{Algebraic K-theory of spaces}, Alg. and Geo. Top., Springer Lect. Notes Math. 1126 (1985), 318-419, \href{http://www.maths.ed.ac.uk/~aar/surgery/rutgers/wald.pdf}{pdf}.


\item [[R. W. Thomason]], \emph{Algebraic K-theory and \'e{}tale cohomology}, Ann. Sci. Ecole Norm. Sup. 18 (4), 1985, pp. 437--552.


\item [[Yevsey Nisnevich]], \emph{The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory}, Algebraic K-theory: connections with geometry and topology, 1989, pp 241-341.


\item [[R. W. Thomason]], Thomas Trobaugh, \emph{Higher algebraic K-theory of schemes and of derived categories}, \emph{The Grothendieck Festschrift}, 1990, 247-435.


\item Andreas Rosenschon, P.A. Ostv\ae{}r, \emph{Descent for K-theories}, Journal of Pure and Applied Algebra 206, 2006, pp 141--152.



\end{itemize}
For [[complex varieties]]:

\begin{itemize}%
\item Claudio Pedrini, [[Charles Weibel]], \emph{The higher K-theory of complex varieties}, K-theory 21 (2001), 367-385 (\href{http://www.math.uiuc.edu/K-theory/0403/}{web})


\item Michael Paluch, \emph{Algebraic K-theory and topological spaces} (\href{http://www.math.uiuc.edu/K-theory/0471/alg-top.pdf}{pdf})



\end{itemize}
For [[ring spectra]]

\begin{itemize}%
\item [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], chapter VI of \emph{[[Rings, modules and algebras in stable homotopy theory]]}, AMS Mathematical Surveys and Monographs Volume 47 (1997) (\href{http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf}{pdf})

\end{itemize}
For discussion of stable phenomena in algebraic K-theory, see section 4 of

\begin{itemize}%
\item [[Ralph Cohen]], \emph{Stability phenomena in the topology of moduli spaces} (\href{http://arxiv.org/PS_cache/arxiv/pdf/0908/0908.1938v2.pdf}{pdf})

\end{itemize}
Discussion of the [[comparison map between algebraic and topological K-theory]] includes

\begin{itemize}%
\item [[Jonathan Rosenberg]], \emph{Comparison between algebraic and topological K-theory for Banach algebras and $C^\ast$-algebras} (\href{http://www2.math.umd.edu/~jmr/algtopK.pdf}{pdf})

\end{itemize}
For [[smooth manifolds]]:

\begin{itemize}%
\item [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], \emph{Differential cohomology theories as sheaves of spectra}, Journal of Homotopy and Related Structures October 2014 (\href{http://arxiv.org/abs/1311.3188}{arXiv:1311.3188})


\item [[Ulrich Bunke]], \emph{A regulator for smooth manifolds and an index theorem} (\href{http://arxiv.org/abs/1407.1379}{arXiv:1407.1379})



\end{itemize}
\hypertarget{ReferencesAlgebraicKTheoryForQuotientStacks}{}\subsubsection*{{Algebraic K-theory of quotient stacks}}\label{ReferencesAlgebraicKTheoryForQuotientStacks}

Discussion of algebraic K-theory for [[algebraic stacks]] (generalizing algebraic [[equivariant K-theory]]) is in

\begin{itemize}%
\item [[Robert Thomason]], \emph{Algebraic K-theory of group scheme actions}, Algebraic Topology and Algebraic K-theory, Ann. Math. Stud., Princeton, 113, (1987), 539-563.


\item [[Amalendu Krishna]], Charanya Ravi, \emph{On the K-theory of schemes with group scheme actions} (\href{http://arxiv.org/abs/1509.05147}{arXiv:1509.05147})



\end{itemize}
See also at \emph{\href{universal+Chern-Simons+line+3-bundle#ForReductiveAlgebraicGroups}{universal Chern-Simons 3-bundle -- For reductive groups}}.

\hypertarget{algebraic_ktheory_of_ring_spectra}{}\subsubsection*{{Algebraic K-theory of ring spectra}}\label{algebraic_ktheory_of_ring_spectra}

(\hyperlink{ThomasonTrobaugh90}{Thomason-Trobaugh 90})

\hyperlink{BlumbergGepnerTabuada10}{Blumberg-Gepner-Tabuada 10}

Waldhausen's \emph{[[A-theory]]} is the algebraic K-theory of [[suspension spectra]] of [[loop spaces]].

\hypertarget{via_stable_categories}{}\subsubsection*{{Via stable $(\infty,1)$-categories}}\label{via_stable_categories}

The [[stable (∞,1)-category]] theory picture is discussed in

\begin{itemize}%
\item [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], \emph{A universal characterization of higher algebraic K-theory} (\href{http://arxiv.org/abs/1001.2282}{arXiv:1001.2282}).

\end{itemize}
(in terms of [[noncommutative motives]]) and in

\begin{itemize}%
\item [[Clark Barwick]], \emph{On the algebraic K-theory of higher categories, I. The universal property of Waldhausen K-theory} (\href{http://de.arxiv.org/abs/1204.3607}{arXiv:1204.3607})

\end{itemize}
\hypertarget{via_symmetric_monoidal_categories}{}\subsubsection*{{Via symmetric monoidal $(\infty,1)$-categories}}\label{via_symmetric_monoidal_categories}

The perspective of [[algebraic K-theory of a symmetric monoidal (∞,1)-category]] is developed in

\begin{itemize}%
\item [[Ulrich Bunke]], [[Georg Tamme]], section 2.1 of \emph{Regulators and cycle maps in higher-dimensional differential algebraic K-theory} (\href{http://arxiv.org/abs/1209.6451}{arXiv:1209.6451})


\item [[Thomas Nikolaus]] \emph{Algebraic K-Theory of $\infty$-Operads} (\href{http://arxiv.org/abs/1303.2198}{arXiv:1303.2198})


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


\item [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], def. 6.1 in \emph{Differential cohomology theories as sheaves of spectra} (\href{http://arxiv.org/abs/1311.3188}{arXiv:1311.3188})


\item [[David Gepner]], [[Moritz Groth]], [[Thomas Nikolaus]], \emph{Universality of multiplicative infinite loop space machines}, \href{http://arxiv.org/abs/1305.4550}{arXiv:1305.4550}.



\end{itemize}
\hypertarget{ktheory_stacks}{}\subsubsection*{{K-theory stacks}}\label{ktheory_stacks}

The system of [[infinite loop spaces]] of the algebraic K-theory spectrum regarded as an [[∞-stack]] on the [[Nisnevich site]] and the [[principal ∞-bundles]] over it is considered in

\begin{itemize}%
\item [[Sho Saito]], \emph{Higher Tate central extensions via K-theory and infinity-topos theory} (\href{http://arxiv.org/abs/1405.0923}{arXiv:1405.0923})

\end{itemize}
implementing a suggestion stated in

\begin{itemize}%
\item [[Vladimir Drinfeld]], \emph{Infinite-dimensional vector bundles in algebraic geometry (an introduction)} (\href{http://arxiv.org/abs/math/0309155}{arXiv:math/0309155})

\end{itemize}
\hypertarget{examples_3}{}\subsubsection*{{Examples}}\label{examples_3}

\begin{itemize}%
\item Alexey Ananyevskiy, \emph{On the algebraic $K$-theory of some homogeneous varieties} (\href{http://www.math.uni-bielefeld.de/lag/man/431.pdf}{pdf})

\end{itemize}
[[!redirects algebraic K-theory of a stable (∞,1)-category]] [[!redirects algebraic K-theory of a stable (infinity,1)-category]]



\end{document}