# nLab Koszul complex

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

Let $R$ be a unital ring.

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

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

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

concentrated in degrees $1$ and $2$, where the map 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

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

of complexes of left $R$-modules. Degree $p$ part of $K(x_1,\ldots,x_r)$ equals the exterior power $\Lambda^{p+1}R^r$. Consider the usual bases 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 chain complex above is obtained by the usual alternating sum rule.

Now let $A$ be a finitely generated left $R$-module. Then the abelian 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 functors (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 $(x_1,\ldots,x_r)$.

## 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

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

Revised on April 24, 2014 23:21:29 by Urs Schreiber (89.204.130.7)