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\begin{document}

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\section*{Koszul complex}

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

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - 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{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

The \emph{Koszul complex} of a sequence of elements $(x_1, \cdots, x_d)$ in a [[commutative ring]] $R$ (or more generally of [[center|central]] elements in a [[non-commutative ring]]) is a [[cochain complex]] whose entry in degree $-n$ is the [[exterior power]] $\wedge^n R^d$ of the [[free module]] $R^d = R^{\oplus_d}$ over $R$ of [[rank]] $d$, and whose [[differential]] is given in each degree on the $k$th summand by multiplication with $x_k$.

The key property of the Koszul complex is that in good cases (namely if the sequence $(x_1, \cdots, x_d)$ is a [[regular sequence]] in $R$), it constitutes is a [[free resolution|free]] [[homological resolution]] of the [[quotient]] $R/(x_1, \cdots, x_d)$ of $R$ by the [[ideal]] generated by these elements (see prop. \ref{KoszulComplexOfRegularSequenceIsFreeResolutionOfQuotientRing} below).

In cases where the Koszul complex fails to be a [[homological resolution]] of the [[quotient ring]], it may be augmented by further generators to yield a resolution after all then called a \emph{[[Koszul-Tate resolution]]}.

From the perspective of [[derived algebraic geometry]] the Koszul complex may be interpreted as the [[formal duality|formal dual]] of the [[derived critical locus]] of the elements $(x_1, \cdots, x_d)$, regarded as functions on the [[spectrum of a ring|spectrum]] $Spec(R)$.

In this guise the Koszul complex appears prominently in [[Lagrangian field theory]], under the name \emph{[[BV-complex]]}, as a potential [[homological resolution]] of the \emph{[[shell]]} (the solution locus of the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]]). In this case the obstruction to the Koszul complex providing a resolution of the shell is its cochain cohomology in degree -1 (via prop. \ref{KoszulResolutionForNoetherianRngAndElementsInJacobson} below) which has the interpretation as the [[infinitesimal gauge symmetries]] of the [[Lagrangian density]] that have not been made explicit. Making them explicit by promoting them to elements in the [[Chevalley-Eilenberg algebra]] of the corresponding [[action Lie algebroid]] yields what is called the \emph{[[BRST complex]]} of the theory, and \emph{its} Koszul complex then yields the respective [[Koszul-Tate resolution]], now called the \emph{[[BV-BRST complex]]} of the theory.

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

Let $R$ be a [[unital ring]].

Consider also a finite [[sequence]] $(x_1,\ldots,x_r)$ of [[elements]] $x_i \in R$.

Given any [[center|central element]] $x\in Z(R)$, one can define a two term [[cochain complex]]

\begin{displaymath}
K(x) \coloneqq (0\to R\stackrel{x}\to R\to 0)
\end{displaymath}
concentrated in degrees $0$ and $1$, where the map (the [[differential]]) is the left multiplication by $x$. Given a sequence $(x_1,\ldots,x_r)$ of central elements in $R$ one can define the tensor product

\begin{displaymath}
K(x_1,\ldots,x_r) 
   \coloneqq 
  K(x_1)\otimes_R K(x_2)\otimes_R\cdots \otimes_R K(x_r)
\end{displaymath}
of complexes of left $R$-[[modules]]. The degree $p$ part of $K(x_1,\ldots,x_r)$ equals the [[exterior power]] $\Lambda^{p+1}R^r$. Consider the usual [[basis]] elements $e_{i_0}\wedge \cdots \wedge e_{i_p}$ of $\Lambda^{p+1}R^r$, where $1\leq i_0\lt i_1\lt\cdots\lt i_p\leq r$. Then the differential is given by

\begin{displaymath}
d(e_{i_0}\wedge \cdots \wedge e_{i_p}) = \sum_{k = 0}^{p}(-1)^{k+1} x_{i_k} e_{i_0}\wedge \cdots\wedge \hat{e}_{i_k} \wedge \cdots\wedge e_{i_r}
\end{displaymath}
The differential can be obtained from the faces of the obvious Koszul [[semi-simplicial object|semi-simplicial]] $R$-[[module]] and the cochain complex above is obtained by the usual alternating sum rule.

Now let $A$ be a finitely generated left $R$-module. Then the abelian [[chain homology]] [[abelian group|groups]]

\begin{displaymath}
H_q(x_1,\ldots,x_r; A) = H_q(K(x_1,\ldots,x_r)\otimes_R A),
\end{displaymath}
\begin{displaymath}
H^q(x_1,\ldots,x_r;A) = H^q(Hom_R(K(x_1,\ldots,x_r),A)),
\end{displaymath}
together with [[connecting homomorphisms]], form a homological and cohomological [[delta-functor]] (in the sense of [[nlab:Tohoku]]) respectively, deriving the zero parts

\begin{displaymath}
H_0 = A/(x_1,\ldots,x_r)A
\end{displaymath}
\begin{displaymath}
H^0 = Hom_R(R/(x_1,\ldots,x_r)R,A)
\end{displaymath}
where $(x_1,\ldots,x_r)A$ is the left $R$-submodule generated by $x_1,\ldots,x_r$. A Poincare-like duality holds: $H_p(x_1,\ldots,x_r;A) = H^{r-p}(x_1,\ldots,x_r;A)$.

The sequence $\mathbf{x} = (x_1,\ldots,x_r)$ is called \textbf{$A$-regular} (or regular on $A$) if for all $i$ the image of $x_i$ in $A/(x_1,\ldots,x_{i-1})A$ annihilates only zero. This terminology is in accord with calling a non-zero divisor in a ring a ``regular element'' (and is in accord with the terminology regular local rings).

If $\mathbf{x}$ is a regular sequence on/in $R$ then $K(\mathbf{x},R)$ is a free resolution of the module $R/(x_1,\ldots,x_r)R$ and the cohomology $H^q(x_1,\ldots,x_r;A) = Ext^q_R(R/(x_1,\ldots,x_r)R,A)$ while [[Koszul homology]] is $H_q(x_1,\ldots,x_r;A) = Tor_q^R(R/(x_1,\ldots,x_r)R,A)$.

The resolution of $R/(x_1,\ldots,x_r)R$ can be written

\begin{displaymath}
0 \to \Lambda^r(R^r)\to \cdots \to \Lambda^2(R^r)\to R^r \to R \to R/(x_1,\ldots,x_r)R\to 0
\end{displaymath}
and the $R$-linear map $R^r\to R$ is given by the row vector $(x_1,\ldots,x_r)$.

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\begin{prop}
\label{KoszulComplexOfRegularSequenceIsFreeResolutionOfQuotientRing}\hypertarget{KoszulComplexOfRegularSequenceIsFreeResolutionOfQuotientRing}{}
\textbf{(Koszul complex of [[regular sequence]] is [[free resolution]] of [[quotient ring]])}

Let $R$ be a [[commutative ring]] and $(x_1, \cdots, x_d)$ a [[regular sequence]] of elements in $R$. Then the Koszul complex $K(x_1,\cdots, x_d)$ is a [[free resolution]] of the [[quotient ring]] $R/(x_1, \cdots, x_d)$.

\end{prop}
\begin{prop}
\label{KoszulResolutionForNoetherianRngAndElementsInJacobson}\hypertarget{KoszulResolutionForNoetherianRngAndElementsInJacobson}{}
\textbf{(Koszul resolution detected in degree (-1))}

Let $R$ be a [[commutative ring]] and $(x_1, \cdots, x_d)$ a [[sequence]] of elements in $R$, such that

\begin{enumerate}%
\item $R$ is [[Noetherian ring|Noetherian]];


\item each $x_i$ is contained in the [[Jacobson radical]] of $R$



\end{enumerate}
then the following are equivalent:

\begin{enumerate}%
\item the [[cochain cohomology]] of the Koszul complex $K(x_1, \cdots, x_d)$ vanishes in degree $-1$;


\item the Koszul complex $K(x_1, \cdots, x_d)$ is a [[free resolution]] of the [[quotient ring]] $R/(x_1, \cdots, x_d)$, hence its [[cochain cohomology]] vanishes in all degrees $\leq -1$;


\item the sequence $(x_1, \cdots, x_d)$ is a [[regular sequence]].



\end{enumerate}
\end{prop}
A \textbf{proof} is spelled out on \href{https://en.wikipedia.org/wiki/Koszul_complex#Properties_of_a_Koszul_homology}{Wikipedia - Properties of Koszul homology}

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

\begin{example}
\label{KoszulComplexForFormalPowerSeriesAlgebras}\hypertarget{KoszulComplexForFormalPowerSeriesAlgebras}{}
\textbf{(Koszul complex for formal power series algebras)}

Let $k$ be a [[field]], let $R = k[ [ X_1,\cdots, X_n ] ]$ be a [[formal power series algebra]] over $k$ in $n$ [[variables]], and let $(f_1, \cdots, f_r)$ be formal power series whose constant term vanishes. Then the Koszul complex is a homological resolution precisely already if its cohomology in degree -1 vanishes

\begin{displaymath}
\left(
    H^{-1}(K(f_1, \cdots, f_n))
    = 0
  \right)
   \;\Leftrightarrow\;
  \left(
    K(f_1, \cdots, f_n)
      \overset{\simeq_{qi}}{\longrightarrow}
    k[ [X_1, \cdots X_n] ]/(f_1, \cdots, f_r)
  \right)
  \,.
\end{displaymath}
This is because the assumptions of prop. \ref{KoszulResolutionForNoetherianRngAndElementsInJacobson} are met: A formal power series ring over a field is Noetherian (\href{noetherian+ring#PolynomialAlgebraOverNoetherianRingIsNoetherian}{this example}) and an element of a formal power series algebra is in the Jacobson radical precisely if its constant term vanishes (\href{Jacobson+radical#JacobsonRadicalOfFormalPowerSeriesAlgebra}{this example}).

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

\begin{itemize}%
\item [[Koszul-Tate resolution]]


\item [[syzygy]]


\item [[BV-formalism]]



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

The original reference is

\begin{itemize}%
\item [[Jean-Louis Koszul]], \emph{Homologie et cohomologie des alg\`e{}bres de Lie} , Bulletin de la Soci\'e{}t\'e{} Math\'e{}matique de France, 78, 1950, pp 65-127.

\end{itemize}
A standard textbook reference is

\begin{itemize}%
\item [[Charles Weibel]], section 4.5 of \emph{Homological algebra} (\href{http://www.math.unam.mx/javier/weibel.pdf}{pdf})

\end{itemize}
A generalization of Koszul complexes to (appropriate resolutions of) [[algebras over operads]] is in

\begin{itemize}%
\item Joan Mill\`e{}s, \emph{The Koszul complex is the cotangent complex}, MPIM2010-32, \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=4143}{pdf}

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Koszul_complex}{Koszul complex}}

\end{itemize}
[[!redirects Koszul complexes]]



\end{document}