Lecture 7

Last time we defined noetherian rings and proved Hilbert's Basis Than

Th If R is noetherian,then REX , , . . . , xn) is noeth .

Def " : An R-algebra S is

D of finitelype-vwkcabbrv.f.tt if a a ring mapREX

, .. . .

.xn) → s

D ekcnkallyoffinitetype.ae#Cabbrv.eft) if S is the

localization of a finite type R- algebra .

Example : k Cx, y)- (y, )⇐,g,

is eft 1k ,

where k is a field

and E.g ) is the maximal ideal of KEI generated byX. xy)

the I :-. x t (x', xy ) ; I :-. y t ( x

'

, xy ) .

Upshot of Hilbert 's Basis Theorem o

.

Coro : ① Ayy algenbra which is ef.to?.erafIield is noetherian ;-

③

Recall that R is noeth . iff any non- empty collection of ideals of

R has a maximal element writ . I .

Prpoo. Let R be a meth ring and I FR be an ideal .

I 7 a finite set Lp , , . .. . Pn } E Speck it .

I EP; and P , ooo Pn E I .

Pf : Let E =L I ¥12 : the conclusion of Prop 2 doesn't hold

for I} .

Assume for contradiction that I f- of .

Since R is meth .

,E has

a max element wrt E, say I .

Claim : I is a prime ideal (which would be a contradiction )

Ff of claim : Let a,b ER - I Srt

. ab EI . Then

I ¥ LI , a ) , CI , b ) ⇒ ⇐ al, CI , b) ¢ E .

LI , a) FR .Otherwise I IEI

,x ER s.tl = it xa

⇒ b = ibtxab E I,a contradiction

. Similarly , CI , b) FR .

ooo By def" of E ,F pi , . . . ,Pn , 9 , , . . .

, am E Speck e. t .

① H i = I,. . .

, n , LI , a) I pi and Pio - opn ELI, a)② H j = l, . . .

,m, CI , b) E9j and 9 , .

. . gm E (I, b) .

Then I EP ; ,IE 9J Hi ,j and

Pi . - - Png ,. . - q m E CI , a ) CI , b)

⇐

I.

This contradicts IE 'S .

I

Def " : For an ideal I of R ,a minimal prime of I- is a prime

ideal p it . I Ep and it. If QE Speck it .

I Eg EP .

Ciro. Let R be a meth . ring and I FR be an ideal .- Then the set of minimal primes of I is

finite . In particular , R has finitely many minimal primes .

Pf : By Prop .2

,let Pi , . . . , Pn E Speer s - t .

I EPIand P ,

o . . pn E I . Let g be a minimal prime

of R . Then

P,o . . Pn EI Eg, ⇒ 7- i sit . P ; E g, .

Minimality ⇒ P;= q .

I. { minimal prime of I} E LP , , . . . . Pn} .

R has finitely many min . primes by applying Corollary to I = .

I

Associated:

tefxeru.se. Let 12 be a ring , M be an R-mod and I be-

an ideal of R .TFAE :

① F me M sit . I = Anna (m )⑦ F an injective R- linear map RII → M

.

③ F a submodule of M isomorphic to RII .

Def " : For an R - mod M, PE Speck is an tedpimefM

if it satisfies the equivalent conditions of previous lemma.

Notation : Assam denotes the set of associated primes of M .

Example : Let R = Klay] and M= kcx.gg(x3xy,

i

Let I := x t (x? xy) .

Then x. I =O and y - I = 0

in M ⇒ Ann ,z(I) Z ¢ , y ) .But 4.y) is maximal

and I t O in M ⇒ Anna # I = ( x , y) .

% ④ y ) is an associated prime of M .

Theorem 5 : Let R be a meth . ring and M to be an R- mod.=

Then any maximal element of[ := { Ann ,z(m) : me M

,m to}

is an associated prime of M . ooo Assam to .

Remark : Need R to be meth .to conclude E has a max't element

under E .

Pf : Let I C- I be a max 'd element .

Let me M- Lol at .

I = Anne (m) .

Suppose a,b E R s - t . ab E I , and a ¢ I . Then

am FO

and I I Anna (am) . By maximally ,I = Anne (am) .

But be Ann,z(am) since ab E I = Annie (m) . I. be I .

I

Corollary 68 Let M¥0 be a fig . module over a meth . ring R .

=

Then I an increasing sequence of labmodules

0 = Mo E M,E

. . - E Mn = M e. t . H IE i En,

Milka; . ,I Rlpi ,

with pie Speck .

tf : R meth .t M f - g . ⇒ M is noelh

.

Since M t O, by Thin 5

,choose p

,E Asse M .

Let M,be a lubmod of M t - t . M

, I RIP,

.

If MIM,to

,let Pz C- Asse MIM , and choose a submodule

Mz of M containing M,

s- t .

pyo"21M

,I Rlpz .

11

We have O E M,E Mz . If 141µL to repeat the process

using P,E Assr Mlpaz ,

and so on .

Get an increasing chain of submodule of MO = Mo E M ,

E Mz E . . .

which must stabilize by M is noeth .

Exercise : Check that the chain must stabilize at M by construction .

I

Recall from HH l that Suppa M = L PE Speck : Mp to} .

Lemma 7 : Assr M E Supp ,z M -

I

Pf : PE Assam ⇒ 7 Rip ↳ M

Localizing⇒ (Rip ) p → Mp ⇒ Mp to .

at p511

Frae (Rip) (afield)

I

L_8 : Let R be a meth . ring , SER a multiplicative set-

and Me be an R -mod .Then I a one - to - one correspondence

Ass S- ' M I { pe Assam : p n s =p }S- 'R

p s- 'R 4-1 p expansion

of 1-7 g n R contraction

Pf : It suffices to show via the bijection between specs - 'R

and { PE Speck : pns =p} that the above operationsmap into the desired sets .

PE Assam and pns = of ⇒ I Rip ↳ M

localization⇒ I s

- '

(Rip) → 5 'M ⇒ p s- 'R E Asse, S

-'M .

is exact11

s-' R 1ps- 'R

If g C- Asss . .pe s-' M

,choose my E S

- ' M it. Arms. .ir (F) = g .

Let p = gnr and suppose p = ( r, , ooo, rn ) .

Then ri 7=0⇒ Hi

,2- si Es et .

r; Sim = O.

Let s := s,. . .sn .

Exer : Show p= Annie Csm) ,and so

, p EASSRM .

I