LINEAR IDEAL S 3

operators, namely the elements of degree one in the variables Y^'K A typical one looks

like

This L i s monk i n a n obviou s sense . Th e order of L i s £.

When w e thin k o f L a s a n operator , it s actio n o n a n elemen t y o f R wil l b e

denoted L(y).

Consistent wit h the above notation, i f R C S i s an extension o f differential ring s

and J i s a subse t o f S, the n R{X} denote s th e differentia l sub-i^-algebr a o f S

generated b y X; i f R an d S ar e fields, the n R(X) denote s th e differentia l subfiel d

generated b y R an d X.

Linear ideal s

In thi s section , w e consider th e universal solutio n o f a linea r homogeneou s dif -

ferential equatio n ove r F; tha t is , a ring extension o f F i n which th e equation ha s a

solution suc h tha t ther e i s a unique differentia l homomorphis m fro m thi s extensio n

to an y other extensio n i n which a solution exists . A s we will see, this extension i s an

integral domai n finitely generate d a s a n (ordinary , no t differential ) algebr a ove r F ,

and moreove r these algebras admit a nice characterization (the y are the quotients o f

differential polynomia l ring s b y what wil l b e terme d linea r differentia l ideals) .

PROPOSITION 1.1Let . L £ F{Y} be a monic homogeneous linear differential

operator, and let I be the ideal of F{Y} generated by {D

1

L\i — 0,1,... }. Then

I is a differential ideal and the quotient algebra F{Y}/I has the property that:

(1) L(Y + I) = 0;

(2) If S is a differential F-algebra and y G F satisfies L(y) = 0 , then there is a

unique differential homomorphism F{Y} — » S sending Y + I to y.

We omit th e obviou s proo f o f Propositio n 1.1.

Ideals lik e the idea l / i n Propositio n 1.1hav e a n intrinsi c characterization . W e

first defin e th e relevan t terminolog y an d the n establis h th e characterization .

DEFINITION 1.2. Le t F{Y}\ denot e th e homogeneou s element s o f degree on e in

F{ Y} = F[F( 0 ) . F ( l ) ... . ] . A differential idea l I CF{Y} i s linear if I i s generated by

/ DF{Y}\. Notic e tha t thi s latter i s a Z)-stable subspac e of F{Y}. Th e dimensio n

of a linea r differentia l idea l / i s the codimensio n of I P\ F{Y}\ i n F{Y}\.

Next, w e prove tha t th e idea l / o f Propositio n 1.1i s linear o f finite dimension .

PROPOSITION 1.3. Let L E F{Y} be a monic homogeneous linear differential

operator of order £, and let I be the ideal of F{ Y} generated by {D

1

L\i — 0. 1 } .

Then I is a linear differential ideal of dimension £.

PROOF.

/ i s clearly a differential ideal . W e claim that F{ Y}\/(I n F { Y}\) ha s

basis Y ,... . Y , where th e overba r denote s th e imag e modul o / .

As remarked above , D preserve s both F{Y}\ an d I DF{Y}\ an d thu s also act s

on th e quotient . Sinc e L = Y^ + Af , wher e M i s an F -linear combinatio n o f th e

F(/)

o f orde r les s tha n €, i t follow s tha t D

nL

= Y

{i+n)

4 - JV, wher e N i s a n F-

linear combinatio n o f 7^' s o f orde r les s tha n £ + n. Sinc e D

nL

belong s t o / fo r

n — 0,, 1...., i t follow s fro m thi s tha t ou r putativ e basi s i s a spannin g set .