Algebraic de Rham Cohomology and Gauss–Manin Connections

In this note, we discuss algebraic de Rham cohomology and related concepts. The purpose of this note is not to give a very detailed account of the theorems and concepts, but provide a surface level overview. The reason I wanted to write about this is because I want to learn about de Rham–Witt cohomology, for which this is a prerequisite. I have not found a reference that I am completely satisfied with, but a lot of my sources come from Dustin Clausen’s notes.

Algebraic de Rham Cohomology

The de Rham cohomology originated from smooth manifolds, where $\mathrm H^k_{\mathrm{dR}}(M)$ is the cohomology of the de Rham complex $\Omega^{\bullet}(M)$.

(de Rham) There is a canonical isomorphism $\mathrm H^k_{\mathrm{dR}}(M)\cong\mathrm H^k(M,\underline{\mathbb R})$.

Proof. (Sketch)

This is shown with two facts: 1. $\underline{\mathbb R}\rightarrow \Omega^\bullet $ is an quasi-isomorphism, i.e. $\mathcal H^k_{\mathrm{dR}}$ identifies with $\underline{\mathbb R}$ at $0$ and vanishes otherwise. 2. for $p\ge 0$, $\Omega^p$ is acyclic. Together, $\underline{\mathbb R}\rightarrow \Omega^\bullet$ is an acyclic resolution, hence de Rham’s theorem. Fact 1 is shown by Poincare lemma (every closed $k$-form is locally exact), and fact 2 is due to the fact that $M$ has partitions of unity: the sheaves $\Omega^p$ are fine, and fine sheaves are acyclic, see ⟦cite:BT82⟧ and ⟦cite:Bre97⟧ .

There is a different version where the isomorphism is between de Rham cohomology and singular cohomology $\mathrm H^k_{\mathrm{dR}}(M)\cong\mathrm H^k(M,\underline{\mathbb R})$, for which the map is induced by the chain morphism

$$\omega\mapsto \left(\sigma\mapsto \int_{\sigma}\omega\right)$$

its well-definedness is exactly the statement of Stokes theorem.

We want a version of this for algebraic varieties.

For any scheme $X/S$, define the hypercohomology

$$\mathrm{H}_{\mathrm{dR}}^n(X)=\mathbb H^n(X,\Omega^\bullet_X)$$

as the algebraic de Rham cohomology of $X$.

For algebraic varieties (or even complex manifolds), we do not have an adequate partitions of unity, so we can have $\Omega^p$ not acyclic. Hypercohomology captures the higher sheaf cohomology $\mathrm{H}^q(X,\Omega^p)$ when $\Omega^p$ is not necessarily acyclic. This is captured in

$$E_1^{p,q}=\mathrm{H}^q(X,\Omega^p)\Rightarrow \mathrm{H}^{p+q}_{\mathrm{dR}}(X)$$

which is called the Hodge-to-de Rham spectral sequence. For smooth affine $X$, we know that $\Omega_X^p$ is acyclic, so one could compute de Rham cohomology easily with $H^n(\Omega^\bullet_X)$. Grothendieck proved an analogue of de Rham’s theorem for complex algebraic varieties.

(Grothendieck) Let $X/\mathbb C$ be a smooth variety, then there is natural isomorphism $\mathrm{H}^n_{\mathrm{dR}}(X)\cong\mathrm H^n(X^{\mathrm{an}},\mathbb C)$.

Proof. (Sketch)

For simplicity, assume $X$ is proper. The proof combines a comparison map $\Omega^\bullet_X\rightarrow \Omega^\bullet_{X^{\mathrm{an}}}$ via analytification. By the assumption of properness, GAGA applies and identifies sheaf cohomologies $H^q(X,\Omega^p_X)\cong H^q(X,\Omega^p_{X^\mathrm{an}})$, and hence identifies hyper-cohomologies $\mathbb H^{n}(X,\Omega^\bullet)\cong \mathbb H^{n}(X,\Omega^\bullet_{X^{\mathrm{an}}})$ via Hodge-to-de Rham spectral sequence. Finish by Poincare lemma giving an acyclic resolution $\underline{\mathbb C}\rightarrow \Omega^\bullet_{X^\mathrm{an}}$, and done.

If $X$ is non proper, this situation is more difficult. One needs to use log differential forms, which I will not elaborate.

Gauss–Manin Connection

Recall that a connection on a vector bundle $\mathcal E$ on a scheme $X$ is a map $\nabla:\mathcal E\rightarrow \mathcal E\otimes_{\mathcal O_X}\Omega^1_X$ satisfying Leibniz rule

$$\nabla(U)(fe)=e\otimes \mathrm df+f\nabla(U)(e)$$

where $f\in\mathcal O_X$ and $e\in\mathcal E(U)$ and $U\subseteq X$ an open. For each derivation $D\in \mathcal{Der}_k(\mathcal O_X,\mathcal O_X)\cong \mathcal{Hom}_{\mathcal O_X}(\Omega_X^1, \mathcal O_X)$, define $\nabla_D:\mathcal E\rightarrow\mathcal E$

$$\nabla_D:\mathcal E\xrightarrow{\nabla}\mathcal E\otimes_{\mathcal O_X}\Omega^1_X\xrightarrow{\mathrm{id}\otimes D}\mathcal E\otimes_{\mathcal O_X} \mathcal O_X\cong \mathcal E$$

which we call the covariant derivative in the direction $D$. We can extend the connection to a map $\nabla:\mathcal E\otimes\Omega^p_X\rightarrow \mathcal E\otimes\Omega^{p+1}_X$ by

$$\nabla(s\otimes\omega)=\nabla(s)\land\omega+s\otimes \mathrm dw$$

then the curvature is $\nabla^2:\mathcal E\rightarrow\mathcal E\otimes_{\mathcal O_X}\Omega^2_X$. Recall that a connection is called flat or integrable if the curvature $\nabla^2=0$.

Let $f:X\rightarrow S$ be a smooth proper morphism of smooth schemes over field $k$ of characteristic $0$. The Gauss–Manin connection is the canonical flat connection on the relative de Rham cohomology sheaves

$$\nabla_{\mathrm{GM}}:\mathcal H^n_{\mathrm{dR}}(X/S)\longrightarrow \mathcal H^n_{\mathrm{dR}}(X/S)\otimes \Omega^1_{S/k}$$

where $\mathcal H^n_{\mathrm{dR}}(X/S)=\mathrm{R}^nf_{*}\Omega_{X/S}^\bullet$. They are constructed in the following way: consider the exact sequence

$$0\to f^*\Omega^1_{S/k}\to\Omega^1_{X/k}\to \Omega^1_{X/S}\to 0$$

which induces a Gauss–Manin filtration $\mathrm{Fil}^p\,\Omega^\bullet_X$ defined by

$$\mathrm{Fil}^p\,\Omega^n_X=\mathrm{Im}\left(f^*\Omega^p_{S/k}\otimes_{\mathcal O_X}\Omega^{n-p}_{X/k}\xrightarrow{\eta\otimes\alpha\mapsto f^*\eta\wedge \alpha}\Omega^n_{X/k}\right)$$

then consider the exact sequence

$$0\to f^*\Omega^1_{S/k}\otimes_{\mathcal O_X}\Omega^\bullet_{X/S}[-1]\to \Omega^\bullet_{X/k}/\mathrm{Fil}^2\,\Omega^\bullet_{X/k}\to \Omega^\bullet_{X/S}\to 0$$

In the derived category we have a connecting morphism

$$\Omega^\bullet_{X/S}\longrightarrow f^*\Omega^1_{S/k}\otimes_{\mathcal O_X}\Omega^\bullet_{X/S}$$

applying derived pushforward

$$\mathrm{R}f_*\Omega^\bullet_{X/S}\longrightarrow \mathrm{R}f_*\left(f^*\Omega^1_{S/k}\otimes_{\mathcal O_X}\Omega^\bullet_{X/S}\right)$$

By the projection formula

$$\mathrm{R}f_*\left(f^*\Omega^1_{S/k}\otimes_{\mathcal O_X}\Omega^\bullet_{X/S}\right)\cong \mathrm{R}f_*\Omega^\bullet_{X/S}\otimes_{\mathcal O_S}\Omega^1_{S/k}$$

Therefore we have a morphism

$$\mathrm{R}f_*\Omega^\bullet_{X/S}\longrightarrow \mathrm{R}f_*\Omega^\bullet_{X/S}\otimes_{\mathcal O_S}\Omega^1_{S/k}$$

taking cohomology sheaves gives us

$$\nabla_{\mathrm{GM}}:\mathcal H^n_{\mathrm{dR}}(X/S)\longrightarrow \mathcal H^n_{\mathrm{dR}}(X/S)\otimes_{\mathcal O_S}\Omega^1_{S/k}$$

which is the Gauss–Manin connection.

References

[BT82]Raoul Bott and Loring W. Tu. Differential Forms in Algebraic Topology. Graduate Texts in Mathematics. Vol. 82. Springer-Verlag. New York. 1982.
[Bre97]Glen E. Bredon. Sheaf Theory. Graduate Texts in Mathematics. Vol. 170. Springer-Verlag. New York. 1997.