Due Saturday morning .
X - Ha . Sap)
Ree Recall top Atx # = LAT is not product top .
AH.ve#of''Sqaralionaxiom "
Lenny If DC Xxx closed ⇐
V-Q.ch : Z - X then {ZI QCZI=Qc⇒} is closed .
prosit Z F- Xx X 7- 1- ( Q, Cz) , Chez)
{zlQfzI=9uzs} = F- ' CS) .
⇒ X 4z=prz
Introduction to schemes -
domain)
X= Spec CR) As set X := { p f p CR prime ideal} Top . taped Vca) { p Iacp }
Closed
Scp) E Rp
elements in R -
Spencer )
Det ( scheme) A ringed space 'Q)
xp
open
( iso . to )
"
Motivations -
apply geometric method from Ig . gay diff. geo .
to number theory .
not a k - alg . !
' ' curve
spec (Ok ) c- curve
9 : X→ Y morphism X.y quasi - prog
'
as " schemes " instead of suhvarieties .
Ex . X={ ( x . y.tl/x2-ty-- o} Ti : X- Atl (x. y ,t ) ,- t
- y }- ¥0 parabola
XI ty x =D
double line
Tux) ) ms Coor- ring kCx%, = key] But it is really spec ( k"'%g) ms dyke • Morale : "
limits "
story to be regarded as schemes
( defined by ar instead of rank) -
→ Special case of abs .
R = Ri positively graded k - alg ,
i >o
Pick homog .
S={ f , i - - - , fr } finite set of K-alg . gen .
( case of interest is when all fi ER ,) e -g - R = homog .
Coor. ring of a prog '
.
f- become invertible)
g Xf
X C DN proj . var.
One verifies that if R= KCX] homos . Coor. rig
Then Proj ( R ) = X .
-