137
Since fx (mY,y ) mX,x , we can dene a homomorphism OY,y /mY,y OX,x /mX,x
and passing to the elds of quotients we obtain an extension of elds k (x)/k (y ).
Also, fx,y induces a linear map mY,y /m2 mX,x /m2 , where the target space is
Y,y
X,x
considered
27
Thus, our rational map is given by
T1
2T
T2 1
, T2 2
.
T2 + 1
T +1
Next note that the obtained map is birational. The inverse map is given by
T
T2
.
T1 1
In particular, we see that
2
2
R(V (T1 + T2 1) k (T1 ).
=
The next theorem, although sounding as
128
LECTURE 13. TANGENT SPACE
Proof. Obviously, it suces to nd an open subset U of X where dimK T (X )x =
dim X for all x U . Replacing X by an open ane set, we may assume that X is
isomorphic to an open subset of a hypersurface V (F ) An (K ) for some ir
60
LECTURE 7. MORPHISMS OF PROJECTIVE ALGEBRAIC VARIETIES
Let us now state and prove the lemma. Recall rst that for any ring A a local line
M Pn (A) denes a collection cfw_Mai iI of lines in Ani+1 for some covering family
a
cfw_ai iI of elements in A. Let
65
Example 7.6. We already know that P1 is isomorphic to a subvariety of P2 given by
k
k
an equation of degree 2. This result can be generalized as follows. Let N = n+m 1.
m
Let us denote the projective coordinates in PN by
k
Ti = Ti0 .in , i0 + . . . + i
81
number of connected components of the corresponding K -set equals the number of
distinct roots of F (Z ) in K .
Corollary 9.6. Assume k is a perfect eld. Let V be a projective algebraic k -set,
n = #0 (V ). Then there is an isomorphism of k -algebras O
26LECTURE 4. IRREDUCIBLE ALGEBRAIC SETS AND RATIONAL FUNCTIONS
We have P (T1 , . . . , Tn ) 0 on V \ Z . Since V \ Z is dense in the Zariski
topology, P 0 on V , i.e. P I (V ). This shows that under the map R(W )
R(V ), F goes to 0. Since the homomorphis
16
LECTURE 3. MORPHISMS OF AFFINE ALGEBRAIC VARIETIES
Note that it suces to check the previous condition only for generators of the
ideal I (Y ), for example for the polynomials dening the system of equations Y .
In terms of the polynomials (P1 (T ), . .
87
Example 10.4. 1. Let X = cfw_(x, y ) K 2 : y = x2 A2 (K ) and Y = A1 (K ).
Consider the projection map f : X Y, (x, y ) y . Then f is nite. Indeed, O(X )
=
2
k [Z1 , Z2 ]/(Z2 Z1 ), O(Y ) k [Z2 ] and f is the composition of the natural inclusion
=
2
k