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),
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