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

F®

(/\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® 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.