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Let $k$ be an algebraically closed field and let $X$ be an $n$-dimensional smooth projective variety over $k$.

If $k= \mathbb C$, there is a natural injective morphism of vector spaces

$$H^0(X,\Omega^n_X) \to H^n(X_{an},\mathbb C) $$ given by Hodge theory.

My question is whether this natural map exists in a more broader context. To be precise:

Does there exist, for all prime numbers $\ell$ invertible in $k$, a natural morphism of vector spaces

$$ H^0(X,\Omega^n_X) \to H^n(X_{et},\overline {\mathbb Q_\ell})?$$

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    $\begingroup$ No. The global differentials are a vector space over $k$, while the etale cohomology is a vector space over the l-adic numbers. If $k$ has positive characteristic this already implies any such map is zero. $\endgroup$ – Felipe Voloch Sep 22 '15 at 15:03
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    $\begingroup$ As written, Felipe is right, of course. But I'd love to see some answers about how $H^0(X, \Omega^j)$ is related to the various $p$-adic cohomology theories. $\endgroup$ – David E Speyer Sep 22 '15 at 15:49
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Although Felipe answered your question, let me come back to David's wish, and give an answer using and advocating crystalline cohomology:

First, if $X$ is, say, smooth and proper over a field $k$, then we always have a Hodge-to-deRham spectral sequence $$ E_1^{i,j}=H^j(X,\Omega_{X/k}^i)\Rightarrow H^{i+j}_{{\rm dR}}(X/k), $$ which may or may not degenerate at $E_1$. For example, this is always the case if $k$ is of characteristic zero (Hodge theory), or if $k$ is of characteristic $p$, $X$ is of dimension $\leq p$, and $X$ lifts to $W_2(k)$ (Deligne-Illusie). So let's assume $E_1$-degeneration. Then, we obtain a filtration on deRham cohomology, whose subquotients are isomorphic to the $H^j(\Omega_X^i)$. (Aside: the well-known Hodge decomposition in direct sums over the complex numbers comes from the Laplacian, it does not behave well in families or in positive characteristic - so we really should look at the filtration.)

Now, deRham cohomology is still a $k$-vector space. In many situations, we want to have a cohomology theory with characteristic zero coefficients: for example, when computing fixed points using a Lefschetz fixed point formula and with a cohomology theory that takes values in a field of positive characteristic $p$, the best we can hope for is the number of fixed points as a congruence modulo $p$. If $k$ is perfect, then there is a unique local, complete, minimal and functorial in $k$ DVR of characteristic zero having $k$ as residue field: the Witt ring $W(k)$. Now, if (and this in general not the case) $X$ lifts from $k$ over $W(k)$, one could consider the deRham cohomology of a lift, which then is a $W(k)$-module. It is true (but highly non-trivial) that the deRham-cohomology of a lift (if a lift exists) does not depend on the lift, and that is so canonical that it makes sense to talk about deRham cohomology of a lift even if the lift does not exist. This cohomology theory is crystalline cohomology. Let me also note that $W({\mathbb F}_p)={\mathbb Z}_p$, the ring of $p$-adics, so that if $k$ is of characteristic $p$, $W(k)$ contains ${\mathbb Z}_p$ as subring.

It is related to deRham-cohomology via a universal coefficient formula $$0\to H^n_{{\rm cris}}(X/W(k))\otimes_{W(k)}k \to H^n_{{\rm dR}}(X/k) \to {\rm Tor}_1^{W(k)}(H^{n+1}_{{\rm cris}}(X/W(k)),k)\to 0,$$ which is actually easy to see if you have a lift and believe that crystalline cohomology is deRham cohomology of this lift. In particular, if the crystalline cohomology groups of $X$ are torsion-free, then deRham cohomology is just crystalline cohomology modulo $p$.

Finally, there are comparison theorems between $\ell$-adic cohomology and crystalline cohomology. However, in view of the above, I would like to argue that crystalline cohomology is the perfect answer - because here, the connection to $H^n_{{\rm dR}}(X/k)$ and the $H^i(\Omega_X^j)$ is very natural and visible.

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