Der Endlichkeitssatz f�r eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie

Kiehl, Reinhardt

doi:10.1007/bf01425513

Cited by 95 publications

(35 citation statements)

References 4 publications

(2 reference statements)

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“…The rigid-analytic theorem on cohomology and base change (whose proof goes as in the algebraic case, with the help of [31]) implies that the natural map O S → f * O X is an isomorphism and that this property persists after any base change on S and (for quasi-separated or pseudo-separated S) any extension on the base field; we say O S = f * O X universally. Assume that there is given a section e ∈ X(S).…”

Section: Picard Groupsmentioning

confidence: 99%

Relative ampleness in rigid geometry

Conrad¹

2006

Ann. inst. Fourier

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Section: Picard Groupsmentioning

confidence: 99%

Relative ampleness in rigid geometry

Conrad¹

2006

Ann. inst. Fourier

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“…Starting from this and a suitable cover of X by affinoid perfectoid subsets in X proét , one can deduce that H i (Xé t , O + X /p) is almost finitely generated over O K . At this point, one uses that X is proper, and in fact the proof of this finiteness result is inspired by the proof of finiteness of coherent cohomology of proper rigid-analytic varieties, as given by Kiehl [12]. Then one deduces finiteness results for the F p -cohomology by using a variant of the Artin-Schreier sequence…”

Section: Introductionmentioning

confidence: 99%

-Adic Hodge Theory for Rigid-Analytic Varieties

Scholze¹

2013

Forum math. Pi

131

160

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We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-étale site, which makes all constructions completely functorial. Mathematics Subject Classification: 14G22, 14C30 (primary); 14G20, 32J27 (secondary) OverviewThis paper studies p-adic comparison theorems in the general setting of rigid-analytic varieties: that is, the p-adic analogue of complex-analytic varieties. Up to now, such comparison isomorphisms were studied for algebraic varieties over p-adic fields, but we show here that large parts of the theory extend to rigid-analytic varieties over p-adic fields. This was already suggested by Tate in his pioneering work on p-adic Hodge(-Tate) decompositions for p-divisible groups, and is of course in analogy with classical Hodge theory, which works for general (Kähler) complex-analytic spaces.One key problem in the p-adic case, as compared to the complex case, is that rigid-analytic varieties are not locally contractible. However, we show that locally they are K(π, 1)'s (for p-adic coefficients): that is, the higher homotopy groups vanish. However, the π 1 is very big, and contains lots of interesting arithmetic information.Then we proceed to discuss several analogues of known results about complex-analytic spaces. The first is finiteness of p-adicétale cohomology, corresponding to finiteness of singular cohomology in the complex case. For this, we give a proof that is inspired by the Cartan-Serre proof of finiteness of coherent cohomology in the complex case. Next, we establish an analogue of the Riemann-Hilbert correspondence between local systems and modules with an integrable connection. In the p-adic setup, (p-adic) local systems have a much richer arithmetic structure; therefore, in this case, one has to consider modules with an integrable connection together with a Finally, we compare the cohomology of a p-adic local system with the cohomology of the associated filtered module with integrable connection. Moreover, we give a Hodge-Tate decomposition (analogous to the Hodge decomposition over the complex numbers), and we prove degeneration of the Hodge-to-de Rham spectral sequence. Interestingly, the latter two results work in complete generality in the p-adic case, with no analogue of a Kähler condition being necessary. Relative situations are also considered, and can be handled with the same methods.These results generalize many previous results. For algebraic varieties, the results for constant coefficients were known by Faltings (with different proofs given by Tsuji, Niziol and Beilinson). Also, it was known by Brinon how to pass from de Rham local systems to filtered modules with integrable connection, although we improve on his results. However, for example the results comparing cohomology with nontrivial coefficients and the relative results are new even in the algebraic case.Technically, our results rest on our theory of perfectoid spaces, which give...

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“…But we can not restrict ourselves to the case that O(Y ) has a noetherian ring of definition, since the nonarchimedean field k in Theorem 5.1 is not assumed to be discretely valued. Our proof of the GAGA theorem follows ideas of [14], [15], [20]. …”

Section: Gagamentioning

confidence: 99%

A finiteness result for the compactly supported cohomology of rigid analytic varieties, II

Huber¹

2007

Ann. inst. Fourier

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Der Endlichkeitssatz f�r eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie

Cited by 95 publications

References 4 publications

Relative ampleness in rigid geometry

Relative ampleness in rigid geometry

-Adic Hodge Theory for Rigid-Analytic Varieties

A finiteness result for the compactly supported cohomology of rigid analytic varieties, II

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