Let $X$ be a nonsingular algebraic variety over a field $k$ of characteristic zero. (We may assume $k$ algebraically closed if need be, but I want to avoid specifically demanding $k = \mathbb{C}$.) Let $\mathcal{D}\_X$ be the sheaf of $k$-linear differential operators on $X$, and let $\mathcal{M}$ be a left $\mathcal{D}\_X$-module. Since giving a left $\mathcal{D}\_X$-module structure on $\mathcal{M}$ is equivalent to giving an integrable connection $\nabla$ on $\mathcal{M}$, we can use $\nabla$ to construct the de Rham complex $\mathcal{M} \otimes \Omega^{\bullet}\_{X/k}$ of $X$ with coefficients in $\mathcal{M}$. The corresponding de Rham cohomology is the hypercohomology $\mathbb{H}^{\bullet}(X, \mathcal{M} \otimes \Omega^{\bullet}\_{X/k})$ of this complex. This is the definition given in Hartshorne's 1975 paper "On the de Rham cohomology of algebraic varieties", section III.4.


My main question is the following:



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> If $\mathcal{M}$ is assumed to be a **holonomic** $\mathcal{D}\_X$-module, are all the de Rham cohomology spaces (as defined above) finite-dimensional $k$-spaces? 
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Certainly such finiteness statements are well-known in many cases. For instance, taking $k = \mathbb{C}$ and $X$ to be *affine*, the finiteness of de Rham cohomology is proved in Bjork's 1979 book "Rings of Differential Operators". My interest is in whether I can get away without assuming $k = \mathbb{C}$ and with taking $X$ to be nonsingular but nothing more (not necessarily affine or projective).


In the analytic case, where $X$ is a complex manifold, this finiteness statement is a corollary of a much stronger claim, Kashiwara's constructibility theorem. In the book "$\mathcal{D}$-Modules, Perverse Sheaves, and Representation Theory" by Hotta, Takeuchi, and Tanisaki, the proof of a special case of the constructibility theorem in the algebraic setting (Theorem 4.7.7) uses analytifications everywhere, and so seems to depend in a crucial way on the ground field being $\mathbb{C}$. They remark (Remark 4.7.3) that for smooth algebraic varieties it "is not a good idea" to consider the de Rham complex as defined above, but rather one should analytify everything first. For my own reasons (coming from the study of local cohomology) I am specifically interested in the de Rham cohomology whose purely algebraic definition is given above, rather than passing to analytifications.



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> Is it necessary to pass to analytifications in the *definition* of de Rham cohomology (and in particular, to insist on the ground field being $\mathbb{C}$) in order to obtain the finiteness results for holonomic $\mathcal{D}$-modules? 
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Most general "finiteness of de Rham cohomology" statement for holonomic $D$-modules in the algebraic case?
Your question is a special case of the theorem that states that $D$-module pushforward preserves holonomicity. Note that $\pi\_\ast \mathcal M$ is computed by the de Rham cohomology, (where $\pi:X \to pt$) - this fact is purely algebraic.


Preservation of holonomicity is proved for example in Bernstein's notes (he doesn't assume $k=\mathbb C$). 


