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After my another post: algebraic de Rham cohomology - open subvariety and normal crossing, I found a useful reference, this paper: Bounding Picard numbers of surfaces using p-adic cohomology.

On page 12 of that article, the authors compare the algebraic de Rham cohomology of a smooth pair $(X, Z)/K$ with that of $U := X \ \backslash \ Z$. Here is the statement:

alt text

The proof uses 2 facts:

  • The formation of cohomology commutes with direct limits.

  • And the following theorem:

alt text

But I don't understand that theorem well. Here is what I have in mind:

From the map between two "complexes of sheaves"

$$ \Omega^{\cdot}_{(X,Z)/S} \rightarrow \Omega^{\cdot}_{(X,Z)/S}(mZ), $$ there are indcued maps between the "hypercohomology groups of these two complexes of sheaves":

$$ \mathbb{H}^{j}(X, \Omega^{\cdot}_{(X,Z)/S}) \rightarrow \mathbb{H}^{j}(X, \Omega^{\cdot}_{(X,Z)/S}(mZ)), \ \ \ \ \ \ \color{red}{(1)}$$ which are, just by the definition of "algebraic de Rham cohomology of a smooth pair", maps between

$$ H^{j}_{dR} ((X,Z)/S) \rightarrow \mathbb{H}^{j}(X, \Omega^{\cdot}_{(X,Z)/S}(mZ)). $$

Notice that "the collection of complexes of sheaves" $ (\Omega^{\cdot}_{(X,Z)/S}(mZ))_{m \in \mathbb{Z}_{\geq 0}} $ form a direct system, and its direct limit equals to the complex of sheaves $ \Omega^{\cdot}_{U/S} $. From the commutativity of "hypercohomology formation" and "direct limit of complexes of sheaves", one gets a map

$$ H^{j}_{dR} ((X,Z)/S) \rightarrow \varinjlim \mathbb{H}^{j}(X, \Omega^{\cdot}_{(X,Z)/S}(mZ)) = \mathbb{H}^{j}(X, \Omega^{\cdot}_{U/S}) \overset{\mathrm{by \ def.}}{=} H^{j}_{dR}(U/S). $$

My first question is:

In the theorem, the authors said "...the cokernels of the maps on homology sheaves induced by...".

  • Do they mean the cokernels of the maps on the hypercohomology groups in $\color{red}{(1)}$ above?

  • Or, they mean another different thing: The cokernel of the maps on sheaves, which are obtained by taking the cohomology of complexes of shaeves. (We can take the cohomology of any complex in an abelian category). (The reason that this comes into my mind is that: the definition of hypercohomology may uses a complex which is quasi-isomorphic to the original complex and consists of injective objects.)

My second question is: How the theorem is used to prove the isomorphism in that corollary? I believe it is an easy exercise of commutative algebra. But since I am confused with the exact meaning of the theorem, I don't know what algebra details get involved here.

I feel that asking questions on papers on MO site is not so good. But asking such questions on won't be good neither. Finally I am not sure if it is good to bother the authors with my above questions...

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