excision in algebraic de Rham cohomology

Question

Hi friends,

I'm reading the notes by Kedlaya "p-adic cohomology from theory to practice", available here

http://www.math.univ-paris13.fr/~barsky/pieces-jointes/kedlaya-aws2007-notes-p-adic-cohom-11mar-290507.pdf

In Remark 1.3.7 on page 6 he says that if $X$ is a smooth algebraic variety over a field $k \subset \mathbb{C}$ and $Z$ is a smooth subvariety of pure codimension $d$, then there exists a excision sequence in algebraic de Rham cohomology

$$ \cdots \to H^{i−2d}_{dR}(Z) \to H^i_{dR}(X) \to H^i_{dR}(X-Z)\to H^{i−2d+1}_{dR}(Z) \to \cdots $$

Question 1: I was completely unable to find a reference for this. Can anyone provide a reference?

Question 2: I guess one also has an excision sequence of the same form in the singular cohomology of $X(\mathbb{C})$ seen as a complex variety. Are both sequence compatible by the isomorphism period?

Question 3: Is it possible to get rid of the shift by codimension by considering the compact support cohomology of $X-Z$?

Any help will be appreciated!

Answer 1

In addition to the comment above, a brief proof can be found in Proposition 2.2.8 of Abbott-Kedlaya-Roe, a published paper which is one of the main sources for the notes.