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Koszul complex

Definition

Let $R$ be a unital ring.

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

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

$$K(x)=(0\to R\stackrel{x}{\to }R\to 0)$$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 $(x_1,\dots ,x_r)$ of central elements in $R$ one can define the tensor product

$$K(x_1,\dots ,x_r)=K(x_1){\otimes }_{R}K(x_2){\otimes }_R\cdots {\otimes }_{R}K(x_r)$$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 $K(x_1,\dots ,x_r)$ 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 $1\le i_0<i_1<\cdots <i_p\le r$. Then the differential is given by

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

$$H_q(x_1,\dots ,x_r;A)=H_q(K(x_1,\dots ,x_r){\otimes }_{R}A),$$H_q(x_1,\ldots,x_r; A) = H_q(K(x_1,\ldots,x_r)\otimes_R A),

$$H^q(x_1,\dots ,x_r;A)=H^q({\mathrm{Hom}}_R(K(x_1,\dots ,x_r),A)),$$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

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

$$H^0={\mathrm{Hom}}_R(R/(x_1,\dots ,x_r)R,A)$$H^0 = Hom_R(R/(x_1,\ldots,x_r)R,A)

where $(x_1,\dots ,x_r)A$ is the left $R$-submodule generated by $x_1,\dots ,x_r$. A Poincare-like duality holds: $H_p(x_1,\dots ,x_r;A)=H^{r-p}(x_1,\dots ,x_r;A)$.

The sequence $x=(x_1,\dots ,x_r)$ is called $A$-regular (or regular on $A$) if for all $i$ the image of $x_i$ in $A/(x_1,\dots ,x_{i-1})A$ 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 $x$ is a regular sequence on/in $R$ then $K(x,R)$ is a free resolution of the module $R/(x_1,\dots ,x_r)R$ and the cohomology $H^q(x_1,\dots ,x_r;A)={\mathrm{Ext}}_R^q(R/(x_1,\dots ,x_r)R,A)$ while homology $H_q(x_1,\dots ,x_r;A)={\mathrm{Tor}}_q^R(R/(x_1,\dots ,x_r)R,A)$.

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

$$0\to {\Lambda }^r(R^r)\to \cdots \to {\Lambda }^2(R^r)\to R^r\to R\to R/(x_1,\dots ,x_r)R\to 0$$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 $(x_1,\dots ,x_r)$.

Related concepts

• Koszul-Tate resolution

• syzygy

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, Homological algebra

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