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By Manin Yu.I.

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33) p i c k s o u t t h e c o e f f i c i e n t of The l \~ . The e a s i l y v e r i f i e d c o m p a t i b i l i t y p r o p e r t i e s of t h e s t r u c t u r e s i n t r o d u c e d a r e c o l l e c t e d i n t h e f o l l o w i n g lemma. 7. LEMMA, a) The o p e r a t o r s res, d, 8 a r e p a i r w i s e commutative ( i n t h e sense of the commutativity of t h e c o r r e s p o n d i n g d i a g r a m s ) . -1)) . The second assertion follows from the fact that 8 also commutes with d% .

Rings of functions on affine algebraic curves) in rings of differential operators. The fruitfulness of such a reduction will be demonstrated in Chap. III. 2. i dn . Let S3 be a not necessarily commutative Q -algebra which is free of finite rank as a module over its center 33a- Let du .. ,dn:33-+33 be differentiations taking 33o into itself and commuting pairwise. We write i=>(h, • • •. *«). \i 1 = 2 *y d'=dil • • • dn, 33 [d] = { 2 M ' i *<653} with the usual rules of multiplication. , £r6$[d] be a finite family of operators.

1 at/! ~~ a! pi (6—/)! /I J* Here and below a l l i n d i c e s a r e i n t e g e r s c o n s i d e r e d e q u a l to zero for y > 8 b) I n t h e > 0 ; terms w i t h (respectively, - (respectively, -^=-K only t h e c o e f f i c i e n t of a monomial ) of t h e form A tf>KP ). ^^BX(^J) O-th element of t h e column (8—J)[ (26) (cc + p + v + 6 + ti + l)!

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