De Rham-Witt kompleks

The purpose of this study is to give a comprehensive treatment of big Witt vectors and the de Rham-Witt complex. We begin with an introduction to Witt vectors and cover both the classical p-typical Witt vectors and the big Witt vectors. The latter associates to every ring A and every set S of positive integers stable under division a ring , and the former corresponds to the case where all elements of S are powers of a single prime p. We also give a historical motivation for p-typical Witt vectors.We continue with the definition of the big de Rham-Witt complex mainly focusing on p-typical de Rham-Witt complex. De Rham-Witt complex is a projective system of differential graded algebras. In degree zero, it is the Witt vectors and the first complex in the inverse limit is de Rham complex. It provides a complex which is explicit and computable. We also give an explicit calculation of p-typical de Rham-Witt complex over a polynomial ring.