Gauss-Manin Connection 1

We’ll shift gears for a second and define the relative de Rham cohomology of a scheme. The stuff from the last post will eventually play a role in this. Let {\pi: X\rightarrow S} be a smooth map of (smooth) {k}-schemes. We define the relative de Rham cohomology sheaf to be the quasi-coherent sheaf {\mathcal{H}^q_{dR}(X/S):=\mathbf{R}^q\pi_*(\Omega_{X/S}^\cdot)}. Note that the R is bolded to mean we take hypercohomology, so that the {\mathbf{R}^q\pi_*} is the hyperderived functor of {\pi_*}. Note that {\mathcal{H}^q_{dR}(X/S)} are sheaves of graded anticommutative {\mathcal{O}_S}-algebras.

We now will describe a canonical integrable ({0} curvature) connection on these sheaves called the Gauss-Manin Connection denoted {\nabla :=\nabla(X/S, q)}. First notice that whenever we have a smooth map the fundamental sequence {0\rightarrow \pi^*(\Omega_{S/k})\rightarrow \Omega_{X/k}\rightarrow \Omega_{X/S}\rightarrow 0} is exact. We now consider the filtration on the complex {\Omega_{X/k}^\cdot}.

{\displaystyle\Omega_{X/k}^\cdot=F^0(\Omega_{X/k}^\cdot)\supset F^1(\Omega_{X/k}^\cdot)\supset F^2(\Omega_{X/k}^\cdot)\supset \cdots} where {\displaystyle F^i(\Omega_{X/k}^\cdot)=\mathrm{im}\left(\Omega_{X/k}^\cdot [-i]\otimes_{\mathcal{O}_X}\pi^*\Omega^i_{S/k}\rightarrow \Omega_{X/k}^\cdot\right)}

The smoothness tells us that {\Omega_{X/k}^i} and {\Omega_{S/k}^i} are locally free, so the exactness of the fundamental sequence allows us to compute the associated graded with respect to this filtration as {\mathrm{gr}^i:=\mathrm{gr}^i(\Omega_{X/k}^\cdot)=F^i/F^{i+1}=\pi^*(\Omega_{S/k}^i)\otimes_{\mathcal{O}_X} \Omega^\cdot_{X/S}[-i]}.

Now the functor {\mathbf{R}^0\pi_*} takes complexes of (abelian) sheaves on {X} to complexes of sheaves on {S}, and the derived functors are just {\mathbf{R}^q\pi_*}. Now we can take the spectral sequence associated to a filtration. It takes the form {E_1^{p,q}=\mathbf{R}^{p+q}\pi_*(\mathrm{gr}^p)\Rightarrow \mathbf{R}^q\pi_*(\Omega_{X/k}^\cdot)}.

We can just work out that {E_1} term more explictly using our previous calculation.

{\mathbf{R}^{p+q}\pi_*(\mathrm{gr}^p)=\mathbf{R}^{p+q}\pi_*(\pi^*(\Omega_{S/k}^p)\otimes_{\mathcal{O}_X} \Omega^\cdot_{X/S}[-p])}, then just by shifting for the first equality, and then projection formula for next iso that whole thing is {=\mathbf{R}^{q}\pi_*(\pi^*(\Omega_{S/k}^p)\otimes_{\mathcal{O}_X} \Omega^\cdot_{X/S})\simeq \Omega^p_{S/k}\otimes_{\mathcal{O}_S}\mathbf{R}^q\pi_*(\Omega_{X/S}^\cdot)}

Which is just {\Omega^p_{S/k}\otimes_{\mathcal{O}_S}\mathcal{H}^q_{dR}(X/S)}

Let’s consider the {E_1} terms. The degree of {d_1} is {(1,0)} so we get maps (for every {q}) of the following form {0\rightarrow \mathcal{H}^q_{dR}(X/S)\rightarrow \Omega^1_{S/k}\otimes \mathcal{H}^q_{dR}(X/S)\rightarrow\Omega^2\otimes \mathcal{H}^q_{dR}(X/S)\rightarrow \cdots}. The maps shown here are {d_1^{0,q}} and {d_1^{1,q}}.

It turns out that {\nabla=d_1^{0,q}} is actually a flat connection. We’ll check this next time, but taking this to be true, we get the Gauss-Manin connection on {\mathcal{H}^q_{dR}(X/S)}. I just thought this was a really neat way to construct a canonical flat connection. There is another way to construct this by using Čech computations, and this paper actually goes through and checks that it is actually the same map, but we probably won’t do this.

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