Koszul and de Rham Complexes

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 November 2009

Example: Singular homology

Let $e_0,\dots ,e_{N–1}$ be the standard basis of $ℝ$. The standard $n$-simplex is

$${\Delta }_n=\{x_0{e}_0+\cdots +x_n{e}_n\mid x_0+\cdots +x_n\le 1\}\text{,}\text{with faces defined by}{\iota }_j:{\Delta }_{n–1}⟶{\Delta }_n$$

where ${\iota }_j(x_0{e}_0+\cdots +x_{n–1}{e}_{n–1})=x_0{e}_0+\cdots +x_{j–1}{e}_{j–1}+x_j{e}_j+x_j{e}_{j+1}+\cdots +x_{n–1}{e}_n$.

Let $X$ be a topological space and let $𝔸$ be a ring. The singular homology of $X$ is the homology $H_i(X;𝔸)$ of the complex with

$$C_n(X;𝔸)=𝔸\text{-span}\{e_f\mid f:{\Delta }_n⟶X\text{ is continuous}\}\text{and}d:C_n(X;𝔸)⟶C_{n–1}(X;𝔸)$$

given by

$$d\left(e_f\right)=\sum _{j=1}^n(–1{)}^j{e}_{f\circ i_j}$$

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: Cell complex homology

Let $B_n$ be the $n$-dimensional open ball in ${ℝ}^n$.

Let $X$ be a Hausdorff topological space. A cellular decomposition of $X$ is a sequence

$$\varnothing \subseteq X_0\subseteq X_1\subseteq \cdots \subseteq X_N=X$$

of closed subspaces of $X$ such that, for each $n$, $X_n–X_{n–1}$ has a finite number of connected components and, for each connected component $C$ of $X_n–X_{n–1}$ there is a

$$\text{homeomorphism}{B}_n\stackrel{\sim }{⟶}C\text{which extends to a continuous map}\overline{B_n}⟶X\text{.}$$

The cellular homology is the homology of the complex

$$\cdots ⟶{\Gamma }_n\stackrel{d_n}{⟶}{\Gamma }_{n–1}⟶\cdots \text{given by}{\Gamma }_n=H_n(X_{n–1},X_n)$$

with the connecting homomorphism $d_n:H_n(X_n,X_{n–1})\mathrm{xrarr}{H}_{n–1}(X_{n–1},X_{n–2})$ coming from the exact sequence

$$0⟶C(X_{n–1},X_{n–2})⟶C(X_n,X_{n–2})⟶C(X_n,X_{n–1})⟶0$$.

Then the singular homology of $X$ is isomorphic to the cellular homology,

$$H_n\left(X\right)\simeq H_n\left(\Gamma \right)\text{.}$$

The Koszul complex and the de Rham complex

Let $𝔸$ be a commutative ring,

$$L\text{an}𝔸\text{-module}\text{and}u:L⟶𝔸$$

an $𝔸$-linear map. The Koszul complex is

$$\cdots ⟶{\Lambda }^{i+1}\left(L\right)\stackrel{d_{i+1}}{⟶}{\Lambda }^i\left(L\right)\stackrel{d_i}{⟶}{\Lambda }^{i-1}\left(L\right)⟶\cdots$$

where $d$ is the unique antiderivation ($d\left(x\wedge y\right)=d\left(x\right)\wedge y–x\wedge d\left(y\right)$) of $\Lambda \left(L\right)$ that extends $u:L\to 𝔸$. Explicitly,

$$d({\ell }_1\wedge \cdots \wedge {\ell }_n)=\sum _{i=1}^n(–1{)}^{i+1}u({\ell }_i)\phantom{,}{\ell }_1\wedge \cdots \wedge {\ell }_{i-1}\wedge {\ell }_{i+1}\wedge \cdots \wedge {\ell }_n$$.

Example

Let $𝕂$ be a commutative ring, $L$ a $𝕂$-module and let $𝔸=S\left(L\right)$. The Koszul complex for the $S\left(L\right)$ module $S\left(L\right){\otimes }_{𝕂}L$ with linear form

$$\begin{array}{rrcl}u:& S\left(L\right){\otimes }_{𝕂}L& ⟶& S\left(L\right)\\ & f\otimes x& ⟼& fx\end{array}\text{is}\Lambda \left(S\right(L\left){\otimes }_{𝕂}L\right)=S\left(L\right){\otimes }_{𝕂}\Lambda \left(L\right)$$

and is the direct sum of complexes

$$0⟶S^{0}L{\otimes }_{𝕂}{\Lambda }^{n}L⟶S^{1}L{\otimes }_{𝕂}{\Lambda }^{n-1}⟶\cdots ⟶S^{n}L{\otimes }_{𝕂}{\Lambda }^{0}L⟶0$$

over $n\in {\mathbb{Z}}_{\ge 0}$ with

$$d\left(\right(x_1\cdots x_p\left)\otimes \right(y_1\wedge \cdots \wedge y_q\left)\right)=\sum _{i=1}^q(–1{)}^{i+1}{y}_i{x}_1\cdots x_p\otimes (y_1\wedge \cdots \wedge y_{i–1}\wedge y_{i+1}\wedge \cdots \wedge y_q)$$.

If $L$ is flat or $A$ is a $\mathbb{Q}$-algebra these complexes are exact and

$$\sum _{i=1}^n(–1{)}^i[S^i\left(L\right)\left]\right[{\Lambda }^{n–i}\left(L\right)]=0$$.

Example

Let $𝕂$ be a commutative ring, $M$ a $𝕂$-module and $x_1,\dots ,x_{\ell }$ a set of commuting endomorphisms of $M$. Then $M$ is a module for the ring

$$𝔸=𝕂[x_1,\dots ,x_{\ell }]\text{and if}L={𝔸}^{\oplus \ell }=𝔸\text{-span}\{e_1,\dots ,e_{\ell }\}$$

with linear map

$$\begin{array}{cccc}u:& L& ⟶& 𝔸\\ & e_i& ⟼& X_i\end{array}$$

If $C^p\left(M\right)=\left\{\text{alternating maps from}\right\{1,\dots ,\ell {\}}^p⟶M\}$ then

$$C^p\left(M\right)\simeq {\mathrm{Hom}}_{𝔸}\left({\Lambda }^p\right(L),M)\simeq {\mathrm{Hom}}_{𝕂}\left({\Lambda }^p\right({𝕂}^{\ell }),M)\text{and}{C}^p\left(M\right)\simeq M{\otimes }_{𝕂}{\Lambda }^p\left({𝕂}^{\ell }\right)$$

and this gives a double complex

$$\cdots \leftrightharpoons C^{p–1}\left(M\right)\underset{{\partial }_p}{\overset{{\partial }^{p–1}}{\leftrightharpoons }}{C}^p\left(M\right)\underset{{\partial }_{p+1}}{\overset{{\partial }^p}{\leftrightharpoons }}{C}^{p+1}\left(M\right)\leftrightharpoons \cdots$$with$$\left({\partial }^{p}m\right)({\alpha }_1,\dots ,{\alpha }_{p+1})=\sum _{j=1}^{p+1}(–1{)}^{j+1}{x}_{{\alpha }_j}m({\alpha }_1,\dots ,{\alpha }_{j–1},{\alpha }_{j+1},\dots ,{\alpha }_{p+1})$$and

$\left({\partial }^{p}m\right)({\alpha }_1,\dots ,{\alpha }_{p+1})=$

The homology and cohomology of the complex are denoted $H_r(x_1,\dots ,x_{\ell };M)$ and $H^r(x_1,\dots ,x_{\ell };M)$.

Let $𝔸$ be a commutative ring and let $M$ be an $𝔸$-module. A sequence $x_1,\dots ,x_{\ell }$ of elements of $𝔸$ is completely secant for $M$ if $H_r(x_1,\dots ,x_{\ell };M)=0$, for $i\in {\mathbb{Z}}_{>0}$. An $M$-regular sequence is a sequence $x_1,\dots ,x_{\ell }$ of elements of $𝔸$ such that

$$\begin{array}{r c}\frac{M}{(x_{1}M+\cdots +x_{i–1}M)}& ⟶& \frac{M}{(x_{1}M+\cdots +x_{i–1}M)}\\ y& ⟼& x_{i}y\end{array}$$ is injective for $i=\mathrm{1,2},\dots ,n$.

If $x_1,\dots ,x_{\ell }$is an $M$-regular sequence then $x_1,\dots ,x_{\ell }$ is completely secant for $M$. If $x_1,\dots ,x_{\ell }$is an $𝔸$-regular sequence and $I=(x_1,\dots ,x_{\ell })$ then $I/I^2$ is free of rank $r$ over $𝔸/I$ (see [Lang, XXI Sec. 4]).

Example. The special case $M=𝕂[x_1,\dots ,x_n]$ with commuting endomorphisms $\frac{\partial \phantom{x}}{\partial x_1},\dots ,\frac{\partial \phantom{x}}{\partial x_n}$ is the de Rham complex of $𝕂[x_1,\dots ,x_n]$.

de Rham cohomology

Let $A$ be a commutative algebra. The de Rham cohomology of $A$ is the cohomology of the complex

$$\cdots ⟶{\Omega }^{i–1}\left(A\right)\stackrel{d_{i–1}}{⟶}{\Omega }^i\left(A\right)\stackrel{d_i}{⟶}{\Omega }^{i+1}\left(A\right)⟶\cdots$$

where the $p$-differential forms of $A$ is

$${\Omega }^p\left(A\right)={\Lambda }^p\left({\Omega }^1\right(A\left)\right),{\Omega }^1\left(A\right)=I/I^2,I=\ker (A\otimes A\to A),$$

and $d$ is the unique antiderivation of degree 1 which extends

$$\begin{array}{rrcl}d:& A& ⟶& {\Omega }^1\left(A\right)\\ & x& ⟼& x\otimes 1–1\otimes x\end{array}$$ and satisfies $d^2=0$.

Example. If $A=𝔽[x_1,\dots ,x_n]\mathrm{then}$

Let $M$ be an $A$-module. A connection on $M$ is an $𝔽$-linear map

$$\nabla :M⟶M{\otimes }_A{\Omega }^1\left(A\right)\text{such that}\nabla \left(fm\right)=f\nabla \left(m\right)+m\otimes df\text{,}$$

for $f\in A$, $m\in M$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)