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As far as I understand Deligne's far reaching generalisation of Čech cohomology is called cohomological descent and is used to endow any variety with a (mixed) Hodge structure. Again, AFAIU, the idea is to resolve the variety, then take intersections of the exceptional pieces, resolve those and so on.

As you can see I don't understand it well, so can someone please help? Also, what is the most ridiculously easy example one can have? (I guess the thing I'm interested in most is very simple examples illustrating the possible behaviours)

The only places I know that discuss this are an SGA, Brian Conrad's notes and Peters-Steenbrink.

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Luc Illusie "La descente galoisienne", Moscow Math. Journal 9-1 (2009), 47-55. – Niels Mar 10 '13 at 8:44
thanks for that Niels, it seems a bit terse but I'll have a closer look. – Jacob Bell Mar 10 '13 at 20:40
up vote 11 down vote accepted

As you said, it is a generalization of Cech theory. The standard example to understand first is a divisor $D=\bigcup D_i$ with simple normal crossings. A resolution of singularities is obtained by simply taking a disjoint union of components $X_0= \coprod D_i$. Let $\pi_0:X_0\to D$ be the obvious map. The cohomology of $X_0$ with your favourite coefficients will not (usually) be the same as $D$. So you want to correct this by adding in "higher simplicies". Let $X_1$ be the disjoint union $\coprod D_i\cap D_j$. This has a pair of "face" maps to $X_1\to X_0$. We can continue this process to get a (strict) simplicial object $$ \ldots X_1\rightrightarrows X_0\to D$$ with an augmentation to $D$. Given a sheaf $F$ on $D$, we can pull it back to a get a collection of sheaves $F_i$ on $X_i$ with various structure maps. We can define the cohomolgy $H^i(X_\bullet, F_\bullet)$ by taking (for example) by compatible injective resolutions, applying $\Gamma$, and taking cohomology of the total complex of the resulting double complex. Now the point is that the machinery of descent tells you that $$H^i(D, F)\cong H^i(X_\bullet, F_\bullet)$$ or if you prefer, there is a spectral sequence relating $H^*(X_p, F_p)$ and $H^*(D,F)$. This reduces down to a Mayer-Vietoris type sequence $$\ldots H^i(D,F)\to H^i(D_1, F)\oplus H^i(D_2,F)\to H^i(D_1\cap D_2, F)\ldots$$ when $D$ has two components. Why is this good? Because for certain things, e.g.constructing mixed Hodge structures, it's better to replace the singular space by a bunch of smooth spaces.

As per request, I'm expanding my comment, although this may be a bit too concise. If $X_\bullet\to X$ and $Y_\bullet\to X$ are two simplicial resolutions, take the fibre product to get simplicial scheme $Z_n =\coprod_{i+j=n}X_i\times_X Y_j$ dominating both. However, it will be singular. You can build simplicial resolution $\tilde Z_\bullet$ of this inductively. First resolve singularites of $Z_0$ to get $\tilde Z_0$. For the next step, choose a resolution of $Z_1$ which maps to $\tilde Z_0\times_{X}\tilde Z_0$ etc.

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thanks for the answer! may I ask what happens if I change the simplicial resolution? (I assume that the end-result of the MHS does not depend on a choice, no?) I always heard that the proper base change theorem plays a key role, but I don't explicitly see why. – Jacob Bell Mar 9 '13 at 22:30
Right, the MHS is independent of the choice. The idea is that any two simplicial resolutions are dominated by a third, so the resulting MHS's are comparable... – Donu Arapura Mar 9 '13 at 22:38
I see, thanks. If you have time to edit your answer and give an example of two resolutions being dominated by a third (and how base change fits in) that would be great. Anyway, before accepting I'll wait in case someone else wants to chip in. – Jacob Bell Mar 9 '13 at 22:54

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