Much work has gone into the construction of cohomology theories which are defined on algebraic varieties (étale, crystalline, etc.) and comparison isomorphisms between them.

Say $X$ is an algebraic variety over $\mathbb Z$. I am interested in computing $H^*_{\rm sing}(X_{\mathbb C}, \mathbb{Z}_p)$ using a variant of the de-Rahm complex. However, the obvious integral version of de-Rahm cohomology, gives the completely wrong answer even for $X = \mathbb A^1$.

This is surprising to me, because (if I am reading the literature correctly, as a non-expert) the de-Rahm complex of $X_{\mathbb Z_p}$ computes the crystalline cohomology of $X_{\mathbb F_p}$. And crystalline cohomology is often claimed to be the "correct" $p$-adic replacement for the etale cohomology groups $H^*_{et}(X_{\overline{\mathbb F_p}}, \mathbb Z_l)$. However, it seems that crystalline cohmology/de-Rahm cohomology groups of affine space are "wrong", in the moral sense of not lining up with what I would naively expect.

Is there a correction/explanation for this discrepancy? Is there a variant of de-Rahm/crystalline cohomology which computes the "right" cohomology for $\mathbb A^1$? Morally, is there a reason why we would want integral crystalline cohomology to carry so much extraneous torsion? Is there something I'm missing?

Much work has gone into the construction of cohomology theories which are defined on algebraic varieties (étale, crystalline, etc.) and comparison isomorphisms between them.

Say $X$ is an algebraic variety over $\mathbb Z$. I am interested in computing $H^*_{\rm sing}(X_{\mathbb C}, \mathbb{Z}_p)$ using a variant of the de Rham complex. However, the obvious integral version of de Rham cohomology, gives the completely wrong answer even for $X = \mathbb A^1$.

This is surprising to me, because (if I am reading the literature correctly, as a non-expert) the de Rham complex of $X_{\mathbb Z_p}$ computes the crystalline cohomology of $X_{\mathbb F_p}$. And crystalline cohomology is often claimed to be the "correct" $p$-adic replacement for the etale cohomology groups $H^*_{et}(X_{\overline{\mathbb F_p}}, \mathbb Z_l)$. However, it seems that crystalline cohomology/de Rham cohomology groups of affine space are "wrong", in the moral sense of not lining up with what I would naively expect.

Is there a correction/explanation for this discrepancy? Is there a variant of de Rham/crystalline cohomology which computes the "right" cohomology for $\mathbb A^1$? Morally, is there a reason why we would want integral crystalline cohomology to carry so much extraneous torsion? Is there something I'm missing?

Much work has gone into the construction of cohomology theories which are defined on algebraic varieties (étale, crystalline, etc.) and comparison isomorphisms between them.

Say $X$ is an algebraic variety over $\mathbb Z$. I am interested in computing $H^*_{\rm sing}(X_{\mathbb C}, \mathbb{Z}_p)$ using a variant of the de-Rahm complex. However, the obvious integral version of de-Rahm cohomology, gives the completely wrong answer even for $X = \mathbb A^1$.

This is surprising to me, because (if I am reading the literature correctly, as a non-expert) the de-Rahm complex of $X_{\mathbb Z_p}$ computes the crystalline cohomology of $X_{\mathbb F_p}$. And crystalline cohomology is often claimed to be the "correct" $p$-adic replacement for the etale cohomology groups $H^*_{et}(X_{\overline{\mathbb F_p}}, \mathbb Z_l)$. However, it seems that crystalline cohmology/de-Rahm cohomology groups of affine are "wrong", in the moral sense of not lining up with what I would naively expect.

Is there a correction/explanation for this discrepancy? Is there a variant of de-Rahm/crystalline cohomology which computes the "right" cohomology for $\mathbb A^1$? Morally, is there a reason why we would want integral crystalline cohomology to carry so much extraneous torsion? Is there something I'm missing?

Much work has gone into the construction of cohomology theories which are defined on algebraic varieties (étale, crystalline, etc.) and comparison isomorphisms between them.

Say $X$ is an algebraic variety over $\mathbb Z$. I am interested in computing $H^*_{\rm sing}(X_{\mathbb C}, \mathbb{Z}_p)$ using a variant of the de-Rahm complex. However, the obvious integral version of de-Rahm cohomology, gives the completely wrong answer even for $X = \mathbb A^1$.

This is surprising to me, because (if I am reading the literature correctly, as a non-expert) the de-Rahm complex of $X_{\mathbb Z_p}$ computes the crystalline cohomology of $X_{\mathbb F_p}$. And crystalline cohomology is often claimed to be the "correct" $p$-adic replacement for the etale cohomology groups $H^*_{et}(X_{\overline{\mathbb F_p}}, \mathbb Z_l)$. However, it seems that crystalline cohmology/de-Rahm cohomology groups of affine space are "wrong", in the moral sense of not lining up with what I would naively expect.

Is there a correction/explanation for this discrepancy? Is there a variant of de-Rahm/crystalline cohomology which computes the "right" cohomology for $\mathbb A^1$? Morally, is there a reason why we would want integral crystalline cohomology to carry so much extraneous torsion? Is there something I'm missing?