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Koszul complex

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homological algebra

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Definition

Let RR be a unital ring.

Consider also a finite sequence (x 1,,x r)(x_1,\ldots,x_r) of elements in RR.

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

K(x)=(0RxR0) K(x) = (0\to R\stackrel{x}\to R\to 0)

concentrated in degrees 11 and 22, where the map is the left multiplication by xx. Given a sequence (x 1,,x r)(x_1,\ldots,x_r) of central elements in RR one can define the tensor product

K(x 1,,x r)=K(x 1) RK(x 2) R RK(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 RR-modules. Degree pp part of K(x 1,,x r)K(x_1,\ldots,x_r) equals the exterior power Λ p+1R r\Lambda^{p+1}R^r. Consider the usual bases elements e i 0e i pe_{i_0}\wedge \cdots \wedge e_{i_p} of Λ p+1R r\Lambda^{p+1}R^r, where 1i 0<i 1<<i pr1\leq i_0\lt i_1\lt\cdots\lt i_p\leq r. Then the differential is given by

d(e i 0e i p)= k=0 p(1) k+1x i ke i 0e^ i ke 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 RR-module and the chain complex above is obtained by the usual alternating sum rule.

Now let AA be a finitely generated left RR-module. Then the abelian groups

H q(x 1,,x r;A)=H q(K(x 1,,x r) RA), H_q(x_1,\ldots,x_r; A) = H_q(K(x_1,\ldots,x_r)\otimes_R A),
H q(x 1,,x r;A)=H q(Hom R(K(x 1,,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,,x r)A H_0 = A/(x_1,\ldots,x_r)A
H 0=Hom R(R/(x 1,,x r)R,A) H^0 = Hom_R(R/(x_1,\ldots,x_r)R,A)

where (x 1,,x r)A(x_1,\ldots,x_r)A is the left RR-submodule generated by x 1,,x rx_1,\ldots,x_r. A Poincare-like duality holds: H p(x 1,,x r;A)=H rp(x 1,,x r;A)H_p(x_1,\ldots,x_r;A) = H^{r-p}(x_1,\ldots,x_r;A).

The sequence x=(x 1,,x r)\mathbf{x} = (x_1,\ldots,x_r) is called AA-regular (or regular on AA) if for all ii the image of x ix_i in A/(x 1,,x i1)AA/(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 x\mathbf{x} is a regular sequence on/in RR then K(x,R)K(\mathbf{x},R) is a free resolution of the module R/(x 1,,x r)RR/(x_1,\ldots,x_r)R and the cohomology H q(x 1,,x r;A)=Ext R q(R/(x 1,,x r)R,A)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,,x r;A)=Tor q R(R/(x 1,,x r)R,A)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,,x r)RR/(x_1,\ldots,x_r)R can be written

0Λ r(R r)Λ 2(R r)R rRR/(x 1,,x r)R0 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 RR-linear map R rRR^r\to R is given by the row vector (x 1,,x r)(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)