Talk:Function field (scheme theory)

The link at the bottom of the page is dead. —Preceding unsigned comment added by 79.182.16.179 (talk) 10:09, 16 October 2008 (UTC)[reply]

Hmmm - now for anyone who needs to read dimension of an algebraic variety, plenty of sheaf theory will be needed. A more elementary intro is surely called for. Charles Matthews 11:51, 28 September 2005 (UTC)[reply]

Terminology

This article defines the sheaf of rational functions on a scheme X. Except for integral schemes, is there a place where this sheaf is called the function field of X ? As explained in the article, $K_{X}$ is not a field nor constant in general. It does not make sense to call it a function field imho. Liu (talk) 20:24, 9 January 2011 (UTC)[reply]

Scheme theory is a generalization of classical algebraic geometry. In classical algebraic geometry, the function field is defined as follows. Let V be an (affine) algebraic variety defined as the set of zeros of a prime ideal I in a polynomial ring

R=K[x_{1},\ldots ,x_{n}]

. Then the function field of V is the field of fractions of R/I, which may be considered as the field of rational functions of V. Such definition make sense, as it does not depend on the way of defining V as the zero set of an ideal.

It is a pity that

It is not mentioned that the scheme theory term is the natural generalization of a classical term
The classical meaning of function field is nowhere defined in Wikipedia. D.Lazard (talk) 09:37, 14 December 2011 (UTC)[reply]