symmetric monoidal (∞,1)-category of spectra
(also nonabelian homological algebra)
The Koszul complex of a sequence of elements $(x_1, \cdots, x_d)$ in a commutative ring $R$ (or more generally of 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 homological resolution of the quotient $R/(x_1, \cdots, x_d)$ of $R$ by the ideal generated by these elements (see prop. 1 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 Koszul-Tate resolution.
From the perspective of derived algebraic geometry the Koszul complex may be interpreted as the formal dual of the derived critical locus of the elements $(x_1, \cdots, x_d)$, regarded as functions on the spectrum $Spec(R)$.
In this guise the Koszul complex appears prominently in Lagrangian field theory, under the name BV-complex, as a potential homological resolution of the shell (the solution locus of the 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. 2 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 BRST complex of the theory, and its Koszul complex then yields the respective Koszul-Tate resolution, now called the BV-BRST complex of the theory.
Let $R$ be a unital ring.
Consider also a finite sequence $(x_1,\ldots,x_r)$ of elements $x_i R$.
Given any central element $x\in Z(R)$, one can define a two term cochain complex
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
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
The differential can be obtained from the faces of the obvious Koszul 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 groups
together with connecting homomorphisms, form a homological and cohomological delta-functor (in the sense of Tohoku) respectively, deriving the zero parts
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 $A$-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
and the $R$-linear map $R^r\to R$ is given by the row vector $(x_1,\ldots,x_r)$.
(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 $R/(x_1, \cdots, x_d)$.
(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
$R$ is Noetherian;
each $x_i$ is contained in the Jacobson radical of $R$
then the following are equivalent:
the cochain cohomology of the Koszul complex $K(x_1, \cdots, x_d)$ vanishes in degree $-1$;
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 $\leq -1$;
the sequence $(x_1, \cdots, x_d)$ is a regular sequence.
A proof is spelled out on Wikipedia - Properties of Koszul homology
(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
This is because the assumptions of prop. 2 are met: A formal power series ring over a field is Noetherian (this example) and an element of a formal power series algebra is in the Jacobson radical precisely if its constant term vanishes (this example).
The original reference is
A standard textbook reference is
A generalization of Koszul complexes to (appropriate resolutions of) algebras over operads is in
See also