Summary: 1.1. MORPHISMS 1
Lectures 11 and 12
October 6th and October 8th 2008
Scribe: J. Kyle Armstrong
Recall: If Y is a variety we a sheaf of regular functions O(U). If p Y
we had a local ring Op (U p, f) if Y is affine we have that O(Y )
A(Y ) = k[x1, x2, . . . , xn]/I(Y ) and A(Y )mp
Suppose Y is a variety. Consider functions F which are regular on some
open subset (depending on f). These are pairs (U, f) such that U Y is
open and nonempty and f is regular on U. We say that (U, f) (V, f) if
f = g on U V .
Definition 1.1.1. Denote the set of such equivalence classes by K(Y ). Since
the intersection of any two nonempty open subsets in Y is an open nonempty
subset we can make K(Y ) into a ring.
Example 1.1.2. (U, f) + (V, g) = (U V, f + g) and (U, f) · (V, g) = (U
V, f · g) But now we can also invert. If f = 0 is regular on U then 1