Koszul and de Rham complexes

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 22 June 2012

Example 1: Singular homology

Let e0, ,eN1 be the standard basis of . The standard n-simplex is Δn= {x0e0 ++ xnen x0++ xn1}, with faces defined by ιj: Δn-1 Δn where ιj( x0e0 ++ xn-1 en-1) =x0e0 ++ xj-1 ej-1 +xjej+ xj ej+1 ++ xn-1 en .

Let X be a topological space and let 𝔸 be a ring. The singular homology of X is the homology Hi(X;𝔸) of the complex with Cn(X;𝔸) =𝔸-span {ef|f :Δn X is continuous} and d: Cn(X;𝔸) Cn-1 (X;𝔸) given by d(ef) = j=1n (-1)j efij If Y is a subspace of X let C(X;Y;𝔸) =C(X;𝔸)/ C(Y;𝔸) so that 0 C(Y;𝔸) C(X;𝔸) C(X;Y;𝔸) 0 is an exact sequence of complexes.

Example 2: Cell complex homology

Let Bn be the n-dimensional open ball in n.

Let X be a Hausdorff topological space. A cellular decomposition of X is a sequence X0 X1 XN=X of closed subspaces of X such that, for each n, Xn -Xn-1 has a finite number of connected components and, for each connected component C of Xn- Xn-1 there is a homeomorphism Bn C which extends to a continuous map Bn X.

The cellular homology is the homology of the complex Γn dn Γn-1 given by Γn =Hn( Xn1, Xn) with the connecting homomorphism dn: Hn(Xn, Xn-1) Hn-1 (Xn-1, Xn-2) coming from the exact sequence 0 C(Xn-1, Xn-2) C(Xn, Xn-2) C(Xn, Xn-1) 0. Then the singular homology of X is isomorphic to the cellular homology, Hn(X) Hn(Γ).

The Koszul complex and the de Rham complex

Let 𝔸 be a commutative ring, L an 𝔸-module and u:L𝔸 an 𝔸-linear map. The Koszul complex is Λi+1 (L) di+1 Λi(L) di Λi-1 (L) where d is the unique antiderivation (d(xy) =d(x)y- xd(y)) of Λ(L) that extends u:L𝔸. Explicitly, d(1 n) = i=1n (-1)i+1 u(i) , 1 i-1 i+1 n.


Let 𝕂 be a commutative ring, L a 𝕂-module and let 𝔸=S(L). The Koszul complex for the S(L)-module S(L)𝕂 L with linear form u: S(L)𝕂 L S(L) fx fx is Λ(S(L) 𝕂L) =S(L) 𝕂Λ(L) and is the direct sum of complexes 0S0L 𝕂Λn LS1L 𝕂 Λn-1 SnL 𝕂Λ0L 0 over n 0 with d((x1 xp)( y1 yq)) =i=1 q (-1) i+1yi x1xp (y1 yi-1 yi+1 yq). If L is flat or A is a -algebra these complexes are exact and i=1n (-1)i [Si(L)] [Λn-i (L)]=0.

Example 2

Let 𝕂 be a commutative ring, M a 𝕂-module and x1,, x a set of commuting endomorphisms of M. Then M is a module for the ring 𝔸 =𝕂[x1, ,x] and if L=𝔸 =𝔸-span {e1,, e} with linear map u: L 𝔸 ei Xi If Cp(M)= {alternating maps from {1,,} pM} then Cp(M) Hom𝔸(Λp (L),M) Hom𝕂(Λp (𝕂),M) and Cp(M) M𝕂Λp (𝕂) and this gives a double complex Cp-1 (M)p p-1 Cp(M) p+1 p Cp+1 (M) with (pm) (α1,, αp+1) =j=1 p+1(-1) j+1x αjm( α1,, αj-1, αj+1, , αp+1) and (pm) (α1, ,αp+1) =

The homology and cohomology of the complex are denoted Hr(x1 ,, x;M) and Hr(x1 ,, x;M) .

Let 𝔸 be a commutative ring and let M be an 𝔸-module. A sequence x1,, x of elements of 𝔸 is completely secant for M if Hr(x1 ,, x;M)=0 , for i>0 . An M-regular sequence is a sequence x1,, x of elements of 𝔸 such that M (x1 M++ xi1M) M (x1M+ +xi-1 M) y xiy is injective for i=1,2,,n .

If x1,, x is an M-regular sequence then x1,, x is completely secant for M. If x1,, x is an 𝔸-regular sequence and I=x1 ,,x then I/I2 is free of rank r over 𝔸/I (see [Lang, XXI Sec. 4]).

Example. The special case M =𝕂[x1 ,,xn] with commuting endomorphisms x x1 ,, x xn is the de Rham complex of 𝕂[x1 ,,xn] .

de Rham cohomology

Let A be a commutative algebra. The de Rham cohomology of A is the cohomology of the complex Ωi-1(A) di-1 Ωi(A) di Ωi+1 (A) where the p-differential forms of A is Ωp(A) =Λp( Ω1(A)) ,Ω1(A) =I/I2,I =ker(AA A), and d is the unique antiderivation of degree 1 which extends d: A Ω1(A) x x1-1x and satisfies d2=0.

Example. If A =𝔽[x1, ,xn] then ??????

Let M be an A-module. A connection on M is an 𝔽-linear map :M MA Ω1(A) such that (fm) =f(m)+ mdf, for fA, mM.

Notes and References

These notes are originally from http://researchers.ms.unimelb.edu.au/~aram@unimelb/MathGlossary/index.html the file http://researchers.ms.unimelb.edu.au/~aram@unimelb/MathGlossary/Koszul_deRham.xml


[BouA] N. Bourbaki, Algebra I, Chapters 1-3, Elements of Mathematics, Springer-Verlag, Berlin, 1990.

[BouL] N. Bourbaki, Groupes et Algèbres de Lie, Chapitre IV, V, VI, Eléments de Mathématique, Hermann, Paris, 1968.

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