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Koszul complex

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Algebra

Homological algebra

homological algebra

(also nonabelian homological algebra)

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diagram chasing

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Idea

The Koszul complex of a sequence of elements (x 1,,x d)(x_1, \cdots, x_d) in a commutative ring RR (or more generally of central elements in a non-commutative ring) is a cochain complex whose entry in degree n-n is the exterior power nR d\wedge^n R^d of the free module R d=R dR^d = R^{\oplus_d} over RR of rank dd, and whose differential is given in each degree on the kkth summand by multiplication with x kx_k.

The key property of the Koszul complex is that in good cases (namely if the sequence (x 1,,x d)(x_1, \cdots, x_d) is a regular sequence in RR), it constitutes is a free homological resolution of the quotient R/(x 1,,x d)R/(x_1, \cdots, x_d) of RR 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,,x d)(x_1, \cdots, x_d), regarded as functions on the spectrum Spec(R)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.

Definition

Let RR be a unital ring.

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

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

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

concentrated in degrees 00 and 11, where the map (the differential) 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) \coloneqq K(x_1)\otimes_R K(x_2)\otimes_R\cdots \otimes_R K(x_r)

of complexes of left RR-modules. The 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 basis 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 cochain complex above is obtained by the usual alternating sum rule.

Now let AA be a finitely generated left RR-module. Then the abelian chain homology 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-functor (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).

Properties

Proposition

(Koszul complex of regular sequence is free resolution of quotient ring)

Let RR be a commutative ring and (x 1,,x d)(x_1, \cdots, x_d) a regular sequence of elements in RR. Then the Koszul complex K(x 1,,x d)K(x_1,\cdots, x_d) is a free resolution of the quotient ring R/(x 1,,x d)R/(x_1, \cdots, x_d).

Proposition

(Koszul resolution detected in degree (-1))

Let RR be a commutative ring and (x 1,,x d)(x_1, \cdots, x_d) a sequence of elements in RR, such that

  1. RR is Noetherian;

  2. each x ix_i is contained in the Jacobson radical of RR

then the following are equivalent:

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

  2. the Koszul complex K(x 1,,x d)K(x_1, \cdots, x_d) is a free resolution of the quotient ring R/(x 1,,x d)R/(x_1, \cdots, x_d), hence its cochain cohomology vanishes in all degrees 1\leq -1;

  3. the sequence (x 1,,x d)(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 kk be a field, let R=k[[X 1,,X n]]R = k[ [ X_1,\cdots, X_n ] ] be a formal power series algebra over kk in nn variables, and let (f 1,,f r)(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

(H 1(K(f 1,,f n))=0)(K(f 1,,f n) qik[[X 1,X n]]/(f 1,,f r)). \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. 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).

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

See also

Revised on October 12, 2017 08:02:20 by Urs Schreiber (77.56.177.247)