Crystalline cohomology and de rham cohomology pdf

In this approach you prove the independence of the choice of the particular algebra directly. Our first goal is the following result, from which all the finiteness and base changing properties to come will be deduced. The exact poincare lemma in crystalline cohomology of. Crystalline cohomology and applications nicol as ojeda b ar.

Crystalline cohomology of superschemes sciencedirect. Crystalline cohomology stacks project columbia university. A gentle introduction to homology, cohomology, and sheaf cohomology jean gallier and jocelyn quaintance department of computer and information science. Discussion of this in terms of cech cohomology is in. In general one uses local embeddings to build a doublecomplex, as in fa2, iv, e. Integral padic hodge theory, and qde rham cohomology. The idea of crystalline cohomology what happens in characteristic p0. Crystalline cohomology 5 thus torb 1 b0,bib 0 implies that a. A gentle introduction to homology, cohomology, and sheaf. Crystalline cohomology of towers of curves international. It will also be the case that q crystalline cohomology of sidenti es with prismatic. Kato, semistable reduction and crystalline cohomology with logarithmic poles, in periodes padiques buressuryvette, 1988, asterisque 223 1994, 221268. It is a cohomology theory based on the existence of differential forms with.

Degree, linking numbers and index of vector fields 12. Kedlaya department of mathematics, massachusetts institute of technology. Milne, duality in the flat cohomology of a surface. Xs is compared to the crystalline cohomology if x is smooth over s and p is nilpotent on s. Request pdf on the crystalline cohomology of delignelusztig varieties let xy0 be an abelian primetop galois covering of smooth schemes over a perfect field k of characteristic p0. Y is a closed immersion of sschemes, with ys smooth. Bb 0 o this diagram is exact even with 0s added at the top and the right. Following lieberman l, berthelot suggests in b3 to use the socalled jet complex. In crystalline cohomology, the main ingredient in order to show that. Then we start by looking at algebraic and geometric basics around the crystalline theory. In particular, you dont have to consider the crystalline site at all. The rough idea of crystalline cohomology then is, for a variety xover a perfect eld kof positive. Crystalline cohomology is a padic cohomology theory for varieties in characteristic p created by berthelot ber74.

These results transfer calculations from crystalline cohomology to coherent cohomology, where it is much easier to localise calculations at the singular. Nygaard, closedness of regular 1forms on algebraic surfaces. Algebraic ktheory and crystalline cohomology springerlink. Most notably, we reprove berthelots comparison result without using pdstratifications, linearisations, and pddifferential operators. Crystalline cohomology is the abelian sheaf cohomology with respect to the crystalline site of a scheme. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Torsion in the crystalline cohomology of singular varieties. I hope the reader will share my belief that there is something to be gained from each of the approaches, and.

Most notably, we reprove berthelots comparison result without using pdstrati. Unfortunately, this will only give numerical information modulo p. In fact, the fundamental structural result of berthelot ber74, theorem v. To prove the claim we note that it holds for x ax i. Crystalline cohomology of algebraic stacks and hyodokato cohomology martin c.

Pdf a minicourse on crystalline cohomology june 15. Because of this, we would like a theory with values in characteristic zero. At last, we apply our theory to several questions about rational points in arithmetic geometry. We show that a suitable generalization of crystalline cohomology to superschemes, denoted by h scris.

On the crystalline cohomology of delignelusztig varieties. Spencer bloch, kazuya kato, padic etale cohomology. Before doing so, however, let us consider the case of smooth a ne varieties. Our work is partially motivated by the connection between supergeometry and crystalline cohomology developed in. Hence we see that the claim holds for a set of generators of ias an abelian group.