In mathematics, rigid cohomology is a p-adic cohomology theory introduced by Berthelot (1986). It extends crystalline cohomology to schemes that need not be proper or smooth, and extends Monsky–Washnitzer cohomology to non-affine varieties. For a scheme X of finite type over a perfect field k, there are rigid cohomology groups Hi rig(X/K) which are finite dimensional vector spaces over the field K of fractions of the ring of Witt vectors of k. More generally one can define rigid cohomology with compact supports, or with support on a closed subscheme, or with coefficients in an overconvergent isocrystal. If X is smooth and proper over k the rigid cohomology groups are the same as the crystalline cohomology groups.
Berthelot, Pierre (1986), "Géométrie rigide et cohomologie des variétés algébriques de caractéristique p", Mémoires de la Société Mathématique de France, Nouvelle Série (23): 7–32, ISSN 0037-9484, MR 0865810
Kedlaya, Kiran S. (2009), "p-adic cohomology", in Abramovich, Dan; Bertram, A.; Katzarkov, L.; Pandharipande, Rahul; Thaddeus., M. (eds.), Algebraic geometry---Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, Providence, R.I.: Amer. Math. Soc., pp. 667–684, arXiv:math/0601507, Bibcode:2006math......1507K, ISBN978-0-8218-4703-9, MR 2483951
rigid, cohomology, mathematics, rigid, cohomology, adic, cohomology, theory, introduced, berthelot, 1986, extends, crystalline, cohomology, schemes, that, need, proper, smooth, extends, monsky, washnitzer, cohomology, affine, varieties, scheme, finite, type, o. In mathematics rigid cohomology is a p adic cohomology theory introduced by Berthelot 1986 It extends crystalline cohomology to schemes that need not be proper or smooth and extends Monsky Washnitzer cohomology to non affine varieties For a scheme X of finite type over a perfect field k there are rigid cohomology groups Hirig X K which are finite dimensional vector spaces over the field K of fractions of the ring of Witt vectors of k More generally one can define rigid cohomology with compact supports or with support on a closed subscheme or with coefficients in an overconvergent isocrystal If X is smooth and proper over k the rigid cohomology groups are the same as the crystalline cohomology groups The name rigid cohomology comes from its relation to rigid analytic spaces Kedlaya 2006 used rigid cohomology to give a new proof of the Weil conjectures References EditBerthelot Pierre 1986 Geometrie rigide et cohomologie des varietes algebriques de caracteristique p Memoires de la Societe Mathematique de France Nouvelle Serie 23 7 32 ISSN 0037 9484 MR 0865810 Kedlaya Kiran S 2009 p adic cohomology in Abramovich Dan Bertram A Katzarkov L Pandharipande Rahul Thaddeus M eds Algebraic geometry Seattle 2005 Part 2 Proc Sympos Pure Math vol 80 Providence R I Amer Math Soc pp 667 684 arXiv math 0601507 Bibcode 2006math 1507K ISBN 978 0 8218 4703 9 MR 2483951 Kedlaya Kiran S 2006 Fourier transforms and p adic Weil II Compositio Mathematica 142 6 1426 1450 arXiv math 0210149 doi 10 1112 S0010437X06002338 ISSN 0010 437X MR 2278753 S2CID 5233570 Le Stum Bernard 2007 Rigid cohomology Cambridge Tracts in Mathematics vol 172 Cambridge University Press ISBN 978 0 521 87524 0 MR 2358812 Tsuzuki Nobuo 2009 Rigid cohomology Mathematical Society of Japan Sugaku Mathematics 61 1 64 82 ISSN 0039 470X MR 2560145External links EditKedlaya Kiran S Rigid cohomology and its coefficients Le Stum Bernard 2012 An introduction to rigid cohomology PDF Special week Strasbourg This abstract algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Rigid cohomology amp oldid 1140559384, wikipedia, wiki, book, books, library,
