6

ANDREW R. KUSTIN

Convention 1.5. (a) Sometimes we think as the data of 1.2 as matrices:

U — [U\

X =

i n

Xgl

^1/

X

9f

, and v —

vi

Vf

(b) If {u{} U {xjk} U {^} is a list of indeterminates over a commutative noetherian

ring .Ro, and R is the polynomial ring Ro[{ui} U {XJ^} U {t^}], then we say that the

data of 1.2 is generic.

Convention 1.6. The bases and orientation elements of Convention 1.4 are related

by the following equations:

uF =

f[1]

A . . . A /

[ / 1

, uF-

=/[/1

A...A(^

[ 1 ]

,

u,G =

pl1l

A . . .

A#l9},

and u;G* = 7 ^ A...A7

1 1 1

.

If / represents the ordered i—tuple of integers a\ 02 • • • a*, (we write |/| = i),

then let

/ / - /

[ a i ]

A .. . A / M and ^j =

/[ai]

A .. . A

/[ai].

Notice that the element

of /\

l

F* ®

/\l

F is canonical in the sense that it does not depend on the choice

of dual bases

f^l\

. . . , fW and pW,... ,p^. (Indeed, this element corresponds

to the identity map under the canonical identification of Hom(/\

l

F,

f\l

F ) with

/\

l

F* 0

/\z

F.) The above sum is taken over all ordered i—tuples of { 1 , . . . , / } .

(The ambient set in which / lies, in this case ( 1 , . . . , / } , will always be clear from

context.)

Convention 1.7. If bq E

f\q

F , then we use (bq 0 1) * to represent the homomor-

phism

f\q

F* 0 M —• M, which sends aq 0 m to ^ ( a ^ ) • m, for any .R—module

M.

Exampl e 1.8. Adopt Data 1.2. The easiest way to prove the identity

J2 Pi ® X(/z) - X] X*(

7

K) 0 ?K € F* 0 G,

|/|

= 1

|1C|

= 1

is observe that both sides become X(bi), upon application of (61 0 1) * , for an

arbitrary element 61 of F . (Notice that / C { 1 , . . . , / } and K C. { 1 , . . . , p}, and, as

promised, this is clear from the context.)

L e m m a 1.9. Adopt Data 1.2. If k is a fixed integer, ap 6

/\p

F*, bq £

f\q

F, and

br € A

r

F

then

(a) A(ap) = £ Z VI®M*P),