TWO VECTORS AND A RECTANGULAR MATRIX 5
equal to the projective dimension of M. The ideal / of R is called perfect if R/I is
a perfect R—module. An excellent reference on perfect modules is [6, Sect. 16C].
For any R—module F, we write F* = Hom#(F, R). If / : F » G is a map of
R—modules, then we define /
r
(/ ) to be the image of the map
f\r

(/\r
G)* R,
which is induced by the map /\T f: /\r F /\ r G. (In particular, if F and G are
free modules, then Ir(f) is the ideal in R which is generated by the r x r minors
of any matrix representation of /.) Let F be a free R—module of finite rank. We
make much use of the exterior algebra /\* F, the symmetric algebra S0F, and the
divided power algebra DmF. In particular,
/\#
F and A* F* are modules over one
another, and S.F and D.F* are modules over one another. Indeed, if a*
f\l
F*,
bj e /\J F, A* ^(F*), and B7- G £,-(F), then
ai(bj) e tf-* F, bjiadetf-fF*, Ai(Bj) e D^F), andBjiAJeSi-jiF*).
(We view /\
l
F, S^F, and JD^F to be meaningful for every integer z; in particular,
these modules are zero whenever i is negative.) The exterior, symmetric, and
divided power algebras A all come equipped with co-multiplication A: A A.
The following facts are well known; see [7, section 1], [8, Appendix], and [16, section
!]•
Proposition 1.1. Let F be a free module ofrank f over a commutative noetherian
ring R and let br e
f\r
F, b'p
f\p
F, and aq 6
f\q
F*.
(a) Ifr = l, then (bT(aq)) %) = br A (aq(b'p)) + (-l)1+^aq(br A b'p).
(b) Ifq = f, then (br(aq)) (b'p) = (_l)(/-r)(/-rt (b'p(aq)) (br).
(c) Ifp = / , then [Ma,)](Vp) = *r A aq(b'p).
(d) If X: F -+ G is a homomorphism of free R—modules and 5s+r A G*,
tten
(As
X*)
[((Ar
X)(br)) (6S+T)} = K
[(tf+r
X') (St+r)].
Note. The exponent which is given in (b) is correct. An incorrect value has ap-
peared elsewhere in the literature.
The following data is in effect throughout most of the paper.
Data 1.2. Let F and G be free modules of rank / and g, respectively, over the
commutative noetherian ring R. Let u £ G*, v G F, and X: F + G be an
R—module homomorphism.
Note 1.3. We will always take Ap SPF*, Bp e DPF, aq e
/\q
F*, br E
f\r
F,
cs A^C?
a n
d $q £ h? G*. In particular, a lower case subscript will give the
position of a homogeneous element, whenever possible.
Convention 1.4- Orient F and G by fixing basis elements ujp £ A F- ^F* f\ F*,
^G AP^5 a n d ^G* £ f\s G* with u;p(ct;F*) = 1 and
UJG{^G*)
= 1- All of our
maps are coordinate free; however, sometimes the easiest way to describe a map is
to tell what it does to a basis. Consequently, we fix bases /W,..., / ^ for F and
g^\ ..., gW for G. Let ( ^ , . . . , cpW and 7W,..., 7 ^ be the corresponding dual
bases for F* and G*, respectively.
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