Let $$T = \mathbb{A}^1_k,\,\,\, Y = \mathbb{P}^1_T,\,\,\, X = \mathscr{B}(Y),$$ the blowup of $Y$ at a point.
I am trying to compute the de Rham cohomology $H^1_{dR}(X/T)$, but I could use some help. I would like to be able to work over a field like $k = \mathbb{Q}$, but ultimately I will be wanting to compute with something in positive characteristic, like $\mathbb{F}_p$.
I have already calculated everything to my satisfaction for the projective line $Y \to T$ (even in positive characteristic), with a $2$x$2$ double complex, but things become substantially messier when using this $3$x$3$ complex (especially with the infinite dimensional terms arising from the positive characteristic stuff).
I have computed a Cech cover and each term and map involved in the de Rham complex, and I have computed bits and pieces of kernels and cokernels of the double complex, but I keep ending up stuck in a big mess.
Is there a better perspective or technique here, like spectral sequences, Mayer-Vietoris, excision sequences, etc? I'm really weak with spectral sequences, so as of yet I haven't been able to understand how to apply them.
We can define the situation as follows:
Let $R = k[t], R_0 = k[t,y^{-1}], R_1 = k[t,y], R_2 = k[t,y,x]/(t - xy), R_3 = k[t,y,x^{-1}]/(x^{-1}t - y)$
and denote $U_i := \mathrm{Spec}R_i$. $$T = \mathrm{Spec} R$$ $$Y = \mathrm{Proj}R[y_0,y_1] = \mathrm{Spec}R_0 \cup \mathrm{Spec}R_1$$ $$X = \mathrm{Spec}R_0 \cup \mathrm{Spec}R_2 \cup \mathrm{Spec}R_3$$
The intersections are given by $U_{ij} = \mathrm{Spec}R_{ij}$ and $U_{ijk} = \mathrm{Spec}R_{ijk}$ with
$$ R_{02} = k[t, y^{\pm 1}]$$ $$ R_{23} = k[t, x^{\pm 1}]$$ $$ R_{03} = k[t^{\pm 1}, y^{\pm 1}]$$
The de Rham cohomology is the hypercohomology of the following double complex:
$$\begin{matrix} \mathcal{O}_X(U_0)\oplus\mathcal{O}_X(U_2) \oplus \mathcal{O}_X(U_3) & \to & \Omega^1_{X/T}(U_0)\oplus\Omega^1_{X/T}(U_2)\oplus\Omega^1_{X/T}(U_3) & \to & \Omega^2_{X/T}(U_0)\oplus\Omega^2_{X/T}(U_2)\oplus\Omega^2_{X/T}(U_3)\\ \downarrow & & \downarrow & & \downarrow\\ \mathcal{O}_X(U_{02})\oplus\mathcal{O}_X(U_{23}) \oplus \mathcal{O}_X(U_{03}) & \to & \Omega^1_{X/T}(U_{02})\oplus\Omega^1_{X/T}(U_{23})\oplus\Omega^1_{X/T}(U_{03}) & \to & \Omega^2_{X/T}(U_{02})\oplus\Omega^2_{X/T}(U_{23})\oplus\Omega^2_{X/T}(U_{03})\\ \downarrow & & \downarrow & & \downarrow\\ \mathcal{O}_X(U_{023}) & \to & \Omega^1_{X/T}(U_{023}) & \to & \Omega^2_{X/T}(U_{023})\\ \end{matrix}$$
Which simplifies to
$$\begin{matrix} R_0\oplus R_2 \oplus R_3 & \to & R_0\cdot \langle dy^{-1} \rangle \oplus M_2 \oplus R_3\cdot \langle dx^{-1} \rangle & \to & 0\oplus N_2 \oplus 0\\ \downarrow & & \downarrow & & \downarrow\\ R_{02}\oplus R_{23} \oplus R_{13} & \to & R_{02} \cdot \langle dy^{-1} \rangle \oplus R_{23} \cdot \langle dx^{-1} \rangle \oplus R_{03} \cdot \langle dy^{-1} \rangle & \to & 0 \oplus 0\oplus 0\\ \downarrow & & \downarrow & & \downarrow\\ R_{023} & \to & \Omega^1_{X/T}(U_{023}) & \to & 0 \\ \end{matrix}$$
Where $M_2, N_2$ are straightfoward to compute.
I doubt it would be helpful for me to put in the rest of the mess that makes up my calculation. I'm not even too sure of it, but I feel there has to be a better approach.
Could anyone provide me some guidance on this?
Thank you.