Crystalline Cohomology

In the entire development of human’s civilization,math has been researched for longest time.And the properties of equations and numbers has attract too many mathe-maticians on it in last two thousands years.Varietie are the zero set of polyomials equa-tions.Mathematicians has gotten many powerful results by researching them.In the last century,the great French mathematician Alexander.Grothendieck proposed the idea of scheme.It is the more generalize and abstract version of variety,was been founded on the language of sheaf.After this,the research of Algebraic geometry has been changed dra-matically.Under this new language,people have solved many difficult problems.Like the Fermat’s least theorem,proved by Wiles.And the Mordell’s Conjecture.proved by G.Faltings.This all showed the power of Algebraic Geometry.It’s known that the homology group is an very important geometry invariant.And it is the base of the Algebraic Topology.In Algebraic geometry,it is also very importan-t.There several ways to define the cohomology.For example.the right derived functor of the global section functor And the etale cohomology.We will introduce a new co-homology theory,named Crystalline Cohomology.lt is very useful in the research of a scheme over a field k with characteristic p.In the first chapter,we will sketch the basic knowledge,including the definition and properties of varieties.sheaf,scheme and the co-homology of sheaf.In the second chapter.we will state the definition of divided power and crystalline cohomology.And how to use the language of category to describe the crystalline cohomology.