Kiran kedlaya, padic cohomology, from theory to practice pdf. We shall also see that this theorem is true on smooth manifolds with corners. Finiteness of crystalline cohomology of higher level. Rigid cohomology does seem to be a universal padic cohomology with. Some questions from the audience have been included. Global stringy orbifold cohomology 167 conventions we will use at least coef. General setup we will work in the same setup as in the global part of 12. For most of our calculations, we work with monskywashnitzer. Then clearly d dat dx so that ax, da, dx is a double complex and. Cohomology of projective varieties, excision and comparison with topological cohomology 5 3. Finiteness conditions in group cohomology by martin hamilton a thesis submitted to the faculty of information and mathematical sciences at the university of glasgow. Dmodules and lyubezniks finiteness theorems for local cohomology by melvin hochster 0. We do not assume kalgebraically closed since the most interesting case of.
A pole of order mis a singularity that could be \controlled with a polynomial of order m. This formalism also provides a convenient tool to understand the comparison of such cohomology theories. One of the many strong connections between local cohomology of modules and cohomology. It also ventures into deeper waters, such as the role of posets and brations. Im told that mebkhout used some of the tools from my paper in giving a local proof of such finiteness theorems. Grothendieck, local cohomology lc, springer lecture notes, 41 1966.
Thus knowing seemingly unrelated properties about existence of. The zetafunction of an algebraic variety over a finite field can be expressed. I think kedlaya is the person who can tell you about this. The result as stated in 1931 is very di erent from the. Thus knowing seemingly unrelated properties about existence of closed but not exact forms gives us. Monsky and 1vashni tzer s method ltas to first lift affine schemes. Introduction we have been introduced to the idea of homology, which derives from a chain complex of singular or simplicial chain groups together with some map. Mike shulmans extensive appendix x5 clari es many puzzles raised in the talks. We prove the finiteness of crystalline cohomology of higher level. Let z be a projective hypersurface over a finite field. Finiteness for the homology of extended koszul complex let a be a vector space over k and di.
Monsky 39 has given an entirely different proof of the finiteness ofde rham. This is closely related to other constructions in algebraic topology such as simplicial homology and cohomology, singular homology and cohomology, and cech cohomology 15. When is a closed kform on an open subset of rn or, more generally on a submanifold of rn exact. Kamp, eine topologische eigenschaft steinscher raume, nachr. Topics include nonabelian cohomology, postnikov towers, the theory of nstu, and ncategories for n 1 and 2.
Monskywashnitzer cohomology and overconvergent derhamwitt. Id like it very much because it seems to explain in concrete t. We will see later that stokes theorem explains this duality. By a variety over k, we will mean a kscheme of finite type which is reduced and. The slope spectral sequence and finiteness results 202 4. Berthelots rigid cohomology also in the nonsmooth case. By variety over k, we mean a separated and integral scheme of finite type. There are versions of the axioms for a homology theory as well as for a cohomology theory. Crystalline cohomology is the theory which lls the gap among the adic cohomology theories for p, proposed. A more abstract perspective on all of this is the notion of a weil cohomology theory with coe.
We establish some functorial properties and a finiteness result, and discuss the relation. The monskywashnitzer and the overconvergent realizations. We will focus, in particular, on various aspects of when the hodgetode rham spectral sequence on the first page, the most interesting case of which happens in positive characteristic. Because of the fact d2 0, we have a very special algebraic structure. Here we are considering schemes over a perfect eld kof positive characteristic p. To investigate this question more systematically than weve done heretofore, let xbe an ndimensional. An introduction to the cohomology of groups peter j. We strongly urge the reader to read this online at instead of reading the old material.
This implies a finiteness theorem and a poincare duality theorem for such a cohomology with respect to smooth and projective sschemes which can be extended to smooth sschemes when s is the spectrum of a perfect field. Asymptotics for the wave equation on differential forms on kerrde sitter space hintz, peter and vasy, andras, journal of differential geometry, 2018. We establish some functorial properties and a finiteness result, and discuss the relation to the rigid cohomology as defined by p. Therefore, the higher the order of the pole, the harder to extend the function. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. Since we are limited to rational expressions of the form pz qz dz with qz only divisible by the z. It is a cohomology theory based on the existence of differential forms with. Kedlayas estimates for ppowers in the reduction process on hyperelliptic curves 8 3. Crystalline cohomology of algebraic stacks and hyodokato cohomology martin c. This note is a detailed analysis of monsky s paper. Also let d, and dx denote the exterior differential in the trespectively xvariables. A third approach, introduced by crew, attempts to mediate the two extremes. Through this memoire we will only assume a basic knowledge of scheme theory and of category theory. Introduction and background we shall restrict attention to the situation where i is an ideal of a noetherian ring r.
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