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Ck4 85 Irreduciblevarieties in prime ideals

EI m R2y x axis vs NCxy

tDed Anaffair variety Vale is irreducible if whenever V iswritten in theform

W Uwz where WeWz are varieties then either V W or V wz

If V isnot irreducible wesay V isreducible

union of two half lines

butthesearen'tR varietiesil

Det ft IC k x xn be an ideal Then I isprize if whenever we have

fig c R and fgc I then e ther f c I or g e I

Ez I 5 9 xx ya E R ka y z

X I2axis

J y ai se nd axes

xaxis

KCI is reducible and I is not primesince xyc I but XII yet I

MainPoint I ai Iaaf primePat let c k be an affinevanity Then but CI is irred

is irreducible if ICV is prime

proof

Assume V is irreducible let fg c ICV Wis either facial or g cICU

let W Vn Mf

Wz v n HCg

Claude W and We are varieties ok

Claude V W v wz

w ow CvnNCH uhh s

µjc cumvallucumigl

V n Wcf UNGH then either Pe v nm ftor Pe vnWg

v n Mfg I so p.su andeither PIMor PakCgl

But fge vso Pavn Mf UN g

Sfg a ICY2 similar

Mfg ITV1

So t n cfgI V

so W vwz doneClaim2

Since V is irreducible either we V or Wa V

ce either Un WH V or V n Ng V

ee either f c ICV or gc ICU

So Itv1 is prime

Assume ICV is prime

let V W v wz when We Wz are varieties4

Assume WLOG VFW

wtsvwz.LT v

claim ICH Ich

Since wz EV we get ICV EI wz

So for alain1 wts ICH E ICH

We've assumed V f wi

Subclayin ICH Icw

Slept since we V ICH E ICW

step Icv 7 ICY since if theywereequal

N ICU W cw I W contra

Sosubdacin istruelet f c ICW s t f Cvl ok by subclaim

let g e I Cwr arbitrary

for Claud WTS g c ICV

Recall V W Wz falICwal g c ICW

fg c ICV and fat ICU

Since ICV is prime we musthave gaICVI Done Claim1

Claim Wz

V N ICHI wel Wz Doneproposition

lemme Aprimeideal is radial

proofIf I is prime and fm c I then i

4

f fm I

e ther f c I or f c I since I isprime

So donebyinduction

let k bealg closed The fractions I and 14 induce a 1 I corresp

i.e Syeeton between

fired varieties u k and primeideals in R

radical ideals R affinevarieties

u u

primeideals tired affine varieties