Frobenius Endomorphism - Frobenius For Schemes

Frobenius For Schemes

Using the setup above, it is easy to extend the Frobenius map to the context of schemes. Let X be a scheme over a field k of characteristic p. Choose an open affine subset U=Spec R. Since X is a k-scheme, we get an inclusion of k in R. This forces R to be a characteristic p ring, so we can define the Frobenius endomorphism F for R as we did above. It is clear that F commutes with localization, so F glues to give an endomorphism of X, called the absolute Frobenius map.

However, F is not necessarily an endomorphism of k-schemes. If k is not, then F will not fix k, and consequently F will not be a k-algebra map. A partial resolution of this problem is to endow X with a different structure of k-scheme: consider the ring map given by, denote by X' the corresponding k-scheme. Then F induces a morphism of k-schemes .

This solution is not very satisfactory. For example, if X is of finite type over k, then X' is not (if k is not of finite type over its prime field). Even over a function field k over a finite field, the smoothness is lost. To preserve the properties of the k-scheme X, one then introduces the relative Frobenius which is obtained by base change of X by .