Let $K$ be a finite type field extension of $\mathbf{Q}$ and let 
$Y$ be a smooth quasi-projective variety over $K$. Let $\Omega\_{Y/K}^{\bullet}$ denote the complex
of sheaves of (algebraic) regular differential forms on $Y$. Recall that the $k$-th
De Rham cohomology group of $Y$ is defined as 
$$
H^k(Y):=\mathbb{H}^k(Y/K,\Omega\_{Y/K}^{\bullet}),
$$
where $\mathbb{H}$ stands for the hypercohomology of a complex. 


Q: How do you prove in a **purely algebraic way** that $H^k(Y)$ is a finite dimensional 
$K$-vector space?



On the finitess of algebraic De Rham cohomology of smooth quasi-projective variety
$\def\HH{\mathbb{H}}$The following sketch of an argument is taken from Grothendieck's [On the de Rham cohomology of algebraic varieties](http://www.ams.org/mathscinet-getitem?mr=199194). You can also find a good discussion in Voisin's Hodge Theory book, volume 1, chapter 8. I talked about it a little on the [last day](http://www.math.lsa.umich.edu/~speyer/632/apr-19.pdf) of my Hodge theory class.


Note: Every step here is meant to be nontrivial.


**Step 1:** Using resolution of singularities, compactify $Y$ as $X \setminus \bigcup D\_i$, where $\bigcup D\_i$ is a simple normal crossings divisor and $X$ is smooth and projective.


**Step 2:** Let $M \Omega^{\bullet}$ be the complex of sheaves on $X$ obtained by pushing forward $\Omega^{\bullet}$ from $Y$. Argue that $\HH(X, M\Omega^{\bullet}) = \HH(Y, \Omega^{\bullet})$.


**Step 3:** Let $\Omega^j(\log D)$ be the following subsheaf of $M \Omega^j$: Near a point $x \in X$, with local coordinates $(x\_1, \ldots, x\_n)$, and where $\bigcup D\_i$ is locally cut out by $x\_1 x\_2 \cdots x\_k=0$, a differential form is in $\Omega^j(\log D)$ if it is locally of the form:
$$\sum f\_{i\_1 \ldots i\_r j\_1 \ldots j\_s} \bigwedge\_{1 \leq i\_1 < \cdots < i\_r \leq k} \frac{d x\_{i\_a}}{x\_{i\_a}} \wedge \bigwedge\_{k+1 \leq j\_1 < \cdots < j\_s \leq n} d x\_{j\_b}.$$
where the $f\_{i\_1 \ldots i\_r j\_1 \ldots j\_s}$'s are in the local ring at $x$.


So $\Omega^{\bullet}(\log D)$ is a subcomplex of $M \Omega^{\bullet}$. Define $Z^j$ to be the quotient sheaf $M \Omega^j/\Omega^j(\log D)$. 


Show that $Z^{\bullet}$ is acyclic. Deduce that $\HH(M \Omega^{\bullet}) = \HH(\Omega^{\bullet}(\log D))$.


**Step 4:** Write out the spectral sequence for hypercohomology for $\Omega^{\bullet}(\log D)$. On page $2$ (or is it $1$?), all of the terms look like $H^q(X, \Omega^p(\log D))$. Observe that these terms are all finite dimensional, because they are the cohomology of a coherent sheaf on a projective scheme. Deduce that $\HH(\Omega^{\bullet}(\log D))$ is finite dimensional.


