# Zeta functions. No.9

Yoshifumi Tsuchimoto

In this lecture we make use of a scheme of finite type over , It is a patchwork of affine schemes of finite type over ,

An affine scheme of finite type over , in turn, is related to a set of polynomial equations of coefficients in , and is written as for a ring of finite type over , We consider quasi coherent sheaves over these objects. When is affine ( ), the category of quasi coherent sheaves over is equivalent to the category of -modules.

For any scheme of finite type over , we put

It is equal to the zeta function of the category of quasi coherent sheaves on .

Recall we have defined the congruent zeta function as

PROPOSITION 9.1   Let be a scheme of finite type over . Then we have

PROOF..

where we put

Now let us put and proceed further.

PROPOSITION 9.2   Let be a scheme of finite type over . then we have

Where we define as a fiber product .

LEMMA 9.3