rigid cohomology
In mathematics, rigid cohomology is a p-adic cohomology theory introduced by {{harvtxt|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 H{{su|p=i|b=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.
The name "rigid cohomology" comes from its relation to rigid analytic spaces.
{{harvtxt|Kedlaya|2006}} used rigid cohomology to give a new proof of the Weil conjectures.
References
- {{Citation | last1=Berthelot | first1=Pierre | author1-link=Pierre Berthelot (mathematician) | title=Géométrie rigide et cohomologie des variétés algébriques de caractéristique p | url=http://www.numdam.org/item?id=MSMF_1986_2_23__R3_0 |mr=865810 | year=1986 | journal=Mémoires de la Société Mathématique de France|series=Nouvelle Série | issn=0037-9484 | issue=23 | pages=7–32}}
- {{Citation | last1=Kedlaya | first1=Kiran S. |authorlink=Kiran Kedlaya | editor1-last=Abramovich | editor1-first=Dan | editor2-last=Bertram | editor2-first=A. | editor3-last=Katzarkov | editor3-first=L. | editor4-last=Pandharipande | editor4-first=Rahul | editor5-last=Thaddeus. | editor5-first=M. | title=Algebraic geometry---Seattle 2005. Part 2 | publisher=Amer. Math. Soc. | location=Providence, R.I. | series=Proc. Sympos. Pure Math. | isbn=978-0-8218-4703-9 | mr=2483951 | year=2009 | volume=80 | chapter=p-adic cohomology | arxiv=math/0601507 | pages=667–684| bibcode=2006math......1507K }}
- {{Citation | last1=Kedlaya | first1=Kiran S. | title=Fourier transforms and p-adic 'Weil II' | doi=10.1112/S0010437X06002338 | mr=2278753 | year=2006 | journal=Compositio Mathematica | issn=0010-437X | volume=142 | issue=6 | pages=1426–1450| arxiv=math/0210149 | s2cid=5233570 }}
- {{Citation | last1=Le Stum | first1=Bernard | title=Rigid cohomology | url=http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521875242 | publisher=Cambridge University Press | series=Cambridge Tracts in Mathematics | isbn=978-0-521-87524-0 |mr=2358812 | year=2007 | volume=172}}
- {{Citation | last1=Tsuzuki | first1=Nobuo | title=Rigid cohomology |mr=2560145 | year=2009 | journal=Mathematical Society of Japan. Sugaku (Mathematics) | issn=0039-470X | volume=61 | issue=1 | pages=64–82}}
External links
- {{citation|url=http://math.mit.edu/~kedlaya/papers/talks.shtml|title=Rigid cohomology and its coefficients|first=Kiran S.|last= Kedlaya}}
- {{citation|url=http://www-irma.u-strasbg.fr/IMG/pdf/CoursLeStum.pdf|title=An introduction to rigid cohomology
|series=Special week – Strasbourg |year=2012|first=Bernard |last=Le Stum}}
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