nLab Koszul complex

Idea

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. 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. 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.

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 central element $x\in Z(R)$, one can define a two term cochain complex

$$K(x) \coloneqq (0\to R\stackrel{x}\to R\to 0)$$

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

$$K(x_1,\ldots,x_r) \coloneqq K(x_1)\otimes_R K(x_2)\otimes_R\cdots \otimes_R K(x_r)$$

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

$$d(e_{i_0}\wedge \cdots \wedge e_{i_p}) = \sum_{k = 0}^{p}(-1)^{k+1} x_{i_k} e_{i_0}\wedge \cdots\wedge \hat{e}_{i_k} \wedge \cdots\wedge e_{i_r}$$

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

$$H_q(x_1,\ldots,x_r; A) = H_q(K(x_1,\ldots,x_r)\otimes_R A),$$

$$H^q(x_1,\ldots,x_r;A) = H^q(Hom_R(K(x_1,\ldots,x_r),A)),$$

together with connecting homomorphisms, form a homological and cohomological delta-functor (in the sense of Tohoku) respectively, deriving the zero parts

$$H_0 = A/(x_1,\ldots,x_r)A$$

$$H^0 = Hom_R(R/(x_1,\ldots,x_r)R,A)$$

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

$$0 \to \Lambda^r(R^r)\to \cdots \to \Lambda^2(R^r)\to R^r \to R \to R/(x_1,\ldots,x_r)R\to 0$$

and the $R$-linear map $R^r\to R$ is given by the row vector $(x_1,\ldots,x_r)$.

Properties

Proposition

(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 $R/(x_1, \cdots, x_d)$.

Proposition

(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

1. $R$ is Noetherian;

2. each $x_i$ is contained in the Jacobson radical of $R$

then the following are equivalent:

1. the cochain cohomology of the Koszul complex $K(x_1, \cdots, x_d)$ vanishes in degree $-1$;

2. 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 $\leq -1$;

3. the sequence $(x_1, \cdots, x_d)$ is a regular sequence.

A proof is spelled out on Wikipedia - Properties of Koszul homology

Examples

Example

(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

$$\left( H^{-1}(K(f_1, \cdots, f_n)) = 0 \right) \;\Leftrightarrow\; \left( K(f_1, \cdots, f_n) \overset{\simeq_{qi}}{\longrightarrow} k[ [X_1, \cdots X_n] ]/(f_1, \cdots, f_r) \right) \,.$$

This is because the assumptions of prop. 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).

Related concepts

• Koszul-Tate resolution

• syzygy

• BV-formalism

References

The original reference is

• Jean-Louis Koszul, Homologie et cohomologie des algèbres de Lie , Bulletin de la Société Mathématique de France, 78, 1950, pp 65-127.

A standard textbook reference is

• Charles Weibel, section 4.5 of Homological algebra (pdf)

A generalization of Koszul complexes to (appropriate resolutions of) algebras over operads is in

• Joan Millès, The Koszul complex is the cotangent complex, MPIM2010-32, pdf

See also

• Wikipedia, Koszul complex