# algebraic de Rham cohomology - open subvariety and normal crossing

I need to do some explicit computation with algebraic de Rham cohomology on some projective varieties, also some open subsets of them.

I don't know the theory well, but I just search some notes to have an idea how to do the computation.

On p.15 (p.18 on PDF file) of Periods and algebraic deRham cohomology by Benjamin Friedrich, there is a long sequence

$\cdots \rightarrow H_{dR}^{p-1} (D/k) \rightarrow H_{dR}^{p}(X, D/k) \rightarrow H_{dR}^{p} (X/k) \rightarrow H_{dR}^{p} (D/k)\rightarrow H_{dR}^{p+1}(X, D/k) \rightarrow \cdots$

, here $H_{dR}^{p+1}(X, D/k)$ is the relative algebraic de Rham cohomology.

But on pp.10-11 of Note On Algebraic de Rham Cohomology, there is

$\ \ \ \ \ \ \ \ \ H_{dR}^{i-1} (U) \rightarrow H_{dR}^{i-2}(Y) \rightarrow H_{dR}^{i} (X/k) \rightarrow H_{dR}^{i} (U)$

which is obtained from logarithmic de Rham complex.

It seems to me that these 2 exact sequences are very different. Is the relative algebraic de Rham cohomology $H_{dR}^{p}(X, D/k)$ the same or related to $H_{dR}^{p}(U, D/k)$ if $U = X \ \backslash \ D$ ? And when one wants to do the computation for the open subvariety, which exact sequence is used?

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The second sequence is correct only if Y is smooth in which case the Gysin isomorphism will identify its cohomology with the relative cohomology of (X,U). – ulrich Jun 4 '13 at 10:04
@ulrich: Could you give a reference for Gysin isomorphism and (if necessary) how it identifies these things. Also, I think what I need is an identification of the cohomology of $U$ to something, rather than an identification of the cohomology of a closed subscheme to something. – user565739 Jun 4 '13 at 16:02