Logarithmic connections on arithmetic surfaces and cohomology computation

Abstract

De Rham cohomology is important across a broad range of mathematical fields. The good properties of de Rham cohomology on smooth and complex manifolds are also shared by those schemes which most closely resemble complex manifolds, namely schemes that are (1) smooth, (2) proper, and (3) defined over the complex numbers or other another field of characteristic zero. In the absence of one or more of those three properties, one observes more pathological behavior. In particular, for affine morphisms $X/S$, the groups $\Hop^i(X/S)$ may be infinitely generated. In this case, when $S = \Sp(k), \op{char}(k) > 0$, the \textit{Cartier isomorphism} allows one to view the groups as finite dimensional over a different base: $\OO_{X^{(p)}}$. However when $S$ is a Dedekind ring of mixed characteristic, there is no good substitute for the Cartier isomorphism.

In this work we explore a method of calculating the de Rham cohomology of some affine schemes which occur as the complement of certain divisors on arithmetic surfaces over a Dedekind scheme of mixed characteristic. The main tool will be (Koszul) connections on vector bundles, whose primary role is to generalize the exterior derivative $\OO_X \xrightarrow{\D{}} \Omega_{X/S}^1$ to a map $\mathcal{F} \xrightarrow{\nabla} \Omega_{X/S}^1\otimes\mathcal{F}$ defined on more general quasi-coherent modules $\mathcal{F}$.

Given an suitable arithmetic surface $X$ and divisor $D$ with complement $U=X\setminus D$, the de Rham cohomology $\Hop^1(U/S)$ is infinitely generated. We use a natural filtration $\op{Fil}^\bullet\OO_U$ to construct a filtration $\op{Fil}^\bullet\Hop^1(U/S)$. We show that associated graded of this filtration is the direct sum of finitely generated modules, and we give a formula to calculate them in terms of the structure sheaf $\OO_D$ of the divisor as well as the different ideal $\mathcal{D}_D \subset \OO_D$ of the finite, flat extension $D/S$.