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Finiteness of rigid cohomology with coefficients. (English) Zbl 1133.14019

In this very important article, the author proves the finiteness of rigid cohomology with coefficients in full generality. More precisely, if \(k\) is a field of characteristic \(p>0\) and \(K\) is a \(p\)-adic field with residue field \(k\) and \(X\) is a separated scheme of finite type over \(k\) and if \(\mathcal{E}\) is an overconvergent \(F\)-isocrystal over \(X\) then the author proves that both the rigid cohomology \(H^i_{\text{rig}}(X/K,\mathcal{E})\) and the rigid cohomology with compact supports \(H^i_{c,\text{rig}}(X/K,\mathcal{E})\) are finite dimensional vector spaces over \(K\) for all \(i\). He also proves Poincaré duality and a Künneth formula with coefficients.
The article begins with a very clear and informative introduction, giving in particular an overview of rigid cohomology and of the previously known cases of the article’s main results as well as comments on possible future work, all of which we highly recommend for more details.

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
11G25 Varieties over finite and local fields
12H25 \(p\)-adic differential equations
14G22 Rigid analytic geometry
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