# on the genus of a function field

Let $K$ be an algebraic function field of one variable. Then we can define its genus. On the other hand, it can also be seen as a scheme, so we can define the arithmetic and geometric genus. Could anyone please tell me the relation between these definitions?

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Closely related: mathoverflow.net/questions/152/… – Qiaochu Yuan Jun 17 '10 at 21:53

@Charles: This is in Artin's book "Algebraic numbers and algebraic functions"; his index points to the def'n and the discussion of behavior under extension of the constant field (at end of Ch. 15 he gives the Tate paper reference). @Yujia: The issue is that if $X$ is a regular proper geom. integral curve over a field $k$ (this is the geometric object intrinsic to a trdeg 1 function field over $k$) and if $k'/k$ is an extension then the base change $X'=X_ {k'}$ is a proper integral curve over $k$ which may not be regular. So its normalization may have different $h^1(\mathcal{O})$! – Boyarsky Jun 18 '10 at 1:45